FARMING An X-brace on a rectangular barn
door is both decorative and functional. It helps
to prevent the door from warping over time.
If feet, PS = 7 feet, and
, find each measure.
1.QR
SOLUTION:
The opposite sides of a rectangle are parallel and
congruent. Therefore, QR = PS = 7 ft.
ANSWER:
7 ft
2.SQ
SOLUTION:
The diagonals of a rectangle bisect each other. So,
ANSWER:
3.
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, isanisoscelestriangle.
Then,
By the Exterior Angle Theorem,
Therefore,
ANSWER:
33.5
4.
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, isanisoscelestriangle.
Then,
By the Vertical Angle Theorem,
Therefore,
ANSWER:
56.5
ALGEBRA Quadrilateral DEFG is a rectangle.
5.If FD = 3x – 7 and EG = x + 5, find EG.
SOLUTION:
The diagonals of a rectangle are congruent to each
other. So, FD = EG.
Use the value of x to find EG.
EG = 6 + 5 = 11
ANSWER:
11
6.If ,
find .
SOLUTION:
All four angles of a rectangle are right angles. So,
ANSWER:
51
7.PROOF If ABDE is a rectangle and ,
provethat .
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. You are given ABDE is a rectangle and
. You need to prove . Use the
properties that you have learned about rectangles to
walk through the proof.
Given: ABDE is a rectangle;
Prove:
Statements(Reasons)
1. ABDE is a rectangle; .
(Given)
2. ABDE is a parallelogram. (Def. of
rectangle)
3. (Opp.sidesofa .)
4. arerightangles.(Def.ofrectangle)
5. (Allrt .)
6. (SAS)
7. (CPCTC)
ANSWER:
Given: ABDE is a rectangle;
Prove:
Statements(Reasons)
1. ABDE is a rectangle; .
(Given)
2. ABDE is a parallelogram. (Def. of
rectangle)
3. (Opp.sidesofa .)
4. arerightangles.(Def.ofrectangle)
5. (Allrt .)
6. (SAS)
7. (CPCTC)
COORDINATE GEOMETRY Graph each
quadrilateral with the given vertices. Determine
whether the figure is a rectangle. Justify your
answer using the indicated formula.
8.W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula
SOLUTION:
Use the slope formula to find the slope of the sides of
the quadrilateral.
There are no parallel sides, so WXYZ is not a
parallelogram. Therefore, WXYZ is not a rectangle.
ANSWER:
No; slope of , slope of , slope of
, and slope of . Slope of
slope of , and slope of slopeof
, so WXYZ is not a parallelogram. Therefore,
WXYZ is not a rectangle.
9.A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance
Formula
SOLUTION:
First, Use the Distance formula to find the lengths of
the sides of the quadrilateral.
so ABCD is a
parallelogram.
A parallelogram is a rectangle if the diagonals are
congruent. Use the Distance formula to find the
lengths of the diagonals.
So the diagonals are congruent. Thus, ABCD is a
rectangle.
ANSWER:
Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a
parallelogram. , so the diagonals are
congruent. Thus, ABCD is a rectangle.
FENCING X-braces are also used to provide
support in rectangular fencing. If AB = 6 feet,
AD = 2 feet, and , find each
measure.
10.BC
SOLUTION:
The opposite sides of a rectangle are parallel and
congruent. Therefore, BC = AD = 2 ft.
ANSWER:
2 ft
11.DB
SOLUTION:
All four angles of a rectangle are right angles. So,
isarighttriangle.BythePythagorean
Theorem, DB2 = BC2 + DC2.
BC = AD = 2
DC = AB = 6
ANSWER:
6.3 ft
12.
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, isanisosceles
triangle.
Then,
By the Vertical Angle Theorem,
ANSWER:
50
13.
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, isanisosceles
triangle.
Then,
All four angles of a rectangle are right angles.
ANSWER:
25
CCSS REGULARITY Quadrilateral WXYZ is a
rectangle.
14.If ZY = 2x + 3 and WX = x + 4, find WX.
SOLUTION:
The opposite sides of a rectangle are parallel and
congruent. So, ZY = WX.
2x + 3 = x + 4
x = 1
Use the value of x to find WX.
WX = 1 + 4 = 5.
ANSWER:
5
15.If PY = 3x – 5 and WP = 2x + 11, find ZP.
SOLUTION:
The diagonals of a rectangle bisect each other. So,
PY = WP.
The diagonals of a rectangle are congruent and
bisect each other. So, isanisoscelestriangle.
ANSWER:
43
16.If , find
.
SOLUTION:
All four angles of a rectangle are right angles. So,
ANSWER:
39
17.If ZP = 4x – 9 and PY = 2x + 5, find ZX.
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, ZP = PY.
Then ZP = 4(7) – 9 = 19.
Therefore, ZX = 2(ZP) = 38.
ANSWER:
38
18.If , find
.
SOLUTION:
All four angles of a rectangle are right angles. So,
ANSWER:
48
19.If , find
.
SOLUTION:
All four angles of a rectangle are right angles. So,
isarighttriangle.Then,
Therefore, Since is a
transversal of parallel sides and , alternate
interior angles WZX and ZXY are congruent. So,
= = 46.
ANSWER:
46
PROOF Write a two-column proof.
20.Given: ABCD is a rectangle.
Prove:
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a rectangle.
You need to prove . Use the
properties that you have learned about rectangles to
walk through the proof.
Given: ABCD is a rectangle.
Prove:
Proof:
Statements (Reasons)
1. ABCD is a rectangle. (Given)
2. ABCD is a parallelogram. (Def. of rectangle)
3. (Opp.sidesofa .)
4. (Refl.Prop.)
5. (Diagonalsofarectangleare .)
6. (SSS)
ANSWER:
Proof:
Statements (Reasons)
1. ABCD is a rectangle. (Given)
2. ABCD is a parallelogram. (Def. of rectangle)
3. (Opp.sidesofa .)
4. (Refl.Prop.)
5. (Diagonalsofarectangleare .)
6. (SSS)
21.Given: QTVW is a rectangle.
Prove:
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given QTVW is a rectangle and
. You need to prove . Use
the properties that you have learned about rectangles
to walk through the proof.
Given: QTVW is a rectangle.
Prove:
Proof:
Statements (Reasons)
1. QTVW is a rectangle; .(Given)
2. QTVW is a parallelogram. (Def. of rectangle)
3. (Oppsidesofa .)
4. arerightangles.(Def.ofrectangle)
5. (Allrt .)
6. QR = ST (Def. of segs.)
7. (Refl.Prop.)
8. RS = RS (Def. of segs.)
9. QR + RS = RS + ST (Add. prop.)
10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.)
11. QS = RT (Subst.)
12. (Def.of segs.)
13. (SAS)
ANSWER:
Proof:
Statements (Reasons)
1. QTVW is a rectangle; .(Given)
2. QTVW is a parallelogram. (Def. of rectangle)
3. (Oppsidesofa .)
4. arerightangles.(Def.ofrectangle)
5. (Allrt .)
6. QR = ST (Def. of segs.)
7. (Refl.Prop.)
8. RS = RS (Def. of segs.)
9. QR + RS = RS + ST (Add. prop.)
10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.)
11. QS = RT (Subst.)
12. (Def.of segs.)
13. (SAS)
COORDINATE GEOMETRY Graph each
quadrilateral with the given vertices. Determine
whether the figure is a rectangle. Justify your
answer using the indicated formula.
22.W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula
SOLUTION:
Use the slope formula to find the slope of the sides of
the quadrilateral.
The slopes of each pair of opposite sides are equal.
So, the two pairs of opposite sides are parallel.
Therefore, the quadrilateral WXYZ is a parallelogram.
The products of the slopes of the adjacent sides are
–1. So, any two adjacent sides are perpendicular to
each other. That is, all the four angles are right
angles. Therefore, WXYZ is a rectangle.
ANSWER:
Yes; slope of slope of , slope
of slopeof . So WXYZ is a
parallelogram. The product of the slopes of
consecutive sides is –1, so the consecutive sides are
perpendicular and form right angles. Thus, WXYZ is a
rectangle.
23.J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance
Formula
SOLUTION:
First, Use the Distance formula to find the lengths of
the sides of the quadrilateral.
soJKLM is a
parallelogram.
A parallelogram is a rectangle if the diagonals are
congruent. Use the Distance formula to find the
lengths of the diagonals.
So the diagonals are not congruent. Thus, JKLM is
not a rectangle.
ANSWER:
No; soJKLM is a
parallelogram; so
the diagonals are not congruent. Thus, JKLM is not a
rectangle.
24.Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula
SOLUTION:
First, use the Distance Formula to find the lengths of
the sides of the quadrilateral.
soQRST is a
parallelogram.
A parallelogram is a rectangle if the diagonals are
congruent. Use the Distance Formula to find the
lengths of the diagonals.
So the diagonals are congruent. Thus, QRST is a
rectangle.
ANSWER:
Yes; so QRST is a
parallelogram. sothediagonalsare
congruent. QRST is a rectangle.
25.G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula
SOLUTION:
First, use the slope formula to find the lengths of the
sides of the quadrilateral.
The slopes of each pair of opposite sides are equal.
So, the two pairs of opposite sides are parallel.
Therefore, the quadrilateral GHJK is a
parallelogram.
None of the adjacent sides have slopes whose
product is –1. So, the angles are not right angles.
Therefore, GHJK is not a rectangle.
ANSWER:
No; slope of slopeof andslopeof
slopeof . So, GHJK is a
parallelogram. The product of the slopes of
consecutivesides≠–1, so the consecutive sides are
not perpendicular. Thus, GHJK is not a rectangle.
Quadrilateral ABCD is a rectangle. Find each
measure if .
26.
SOLUTION:
All four angles of a rectangle are right angles. So,
ANSWER:
50
27.
SOLUTION:
The measures of angles 2 and 3 are equal as they
are alternate interior angles.
Since the diagonals of a rectangle are congruent and
bisect each other, the triangle with the angles 3, 7
and 8 is an isosceles triangle. So, m∠7 = m∠3.
Therefore, m∠7 = m∠2 = 40.
ANSWER:
40
28.
SOLUTION:
The measures of angles 2 and 3 are equal as they
are alternate interior angles.
Therefore, m∠3 = m∠2 = 40.
ANSWER:
40
29.
SOLUTION:
All four angles of a rectangle are right angles. So,
The measures of angles 1 and 4 are equal as they
are alternate interior angles.
Therefore, m∠4 = m∠1 = 50.
Since the diagonals of a rectangle are congruent and
bisect each other, the triangle with the angles 4, 5
and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50.
The sum of the three angles of a triangle is 180.
Therefore, m∠5 = 180 – (50 + 50) = 80.
ANSWER:
80
30.
SOLUTION:
All four angles of a rectangle are right angles. So,
The measures of angles 1 and 4 are equal as they
are alternate interior angles.
Therefore, m∠4 = m∠1 = 50.
Since the diagonals of a rectangle are congruent and
bisect each other, the triangle with the angles 4, 5
and 6 is an isosceles triangle. Therefore, m∠6 =
m∠4 = 50.
ANSWER:
50
31.
SOLUTION:
The measures of angles 2 and 3 are equal as they
are alternate interior angles.
Since the diagonals of a rectangle are congruent and
bisect each other, the triangle with the angles 3, 7
and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40.
The sum of the three angles of a triangle is 180.
Therefore, m∠8 = 180 – (40 + 40) = 100.
ANSWER:
100
32.CCSS MODELING Jody is building a new
bookshelf using wood and metal supports like the one
shown. To what length should she cut the metal
supports in order for the bookshelf to be square,
which means that the angles formed by the shelves
and the vertical supports are all right angles? Explain
your reasoning.
SOLUTION:
In order for the bookshelf to be square, the lengths of
all of the metal supports should be equal. Since we
know the length of the bookshelves and the distance
between them, use the Pythagorean Theorem to find
the lengths of the metal supports.
Let x be the length of each support. Express 3 feet
as 36 inches.
Therefore, the metal supports should each be 39
inches long or 3 feet 3 inches long.
ANSWER:
3 ft 3 in.; Sample answer: In order for the bookshelf
to be square, the lengths of all of the metal supports
should be equal. Since I knew the length of the
bookshelves and the distance between them, I used
the Pythagorean Theorem to find the lengths of the
metal supports.
PROOF Write a two-column proof.
33.Theorem 6.13
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given WXYZ is a rectangle with
diagonals . You need to prove
. Use the properties that you have learned
about rectangles to walk through the proof.
Given: WXYZ is a rectangle with diagonals
.
Prove:
Proof:
1. WXYZ is a rectangle with diagonals .
(Given)
2. (Opp.sidesofa .)
3. (Refl.Prop.)
4. arerightangles.(Def.of
rectangle)
5. (Allright .)
6. (SAS)
7. (CPCTC)
ANSWER:
Given: WXYZ is a rectangle with diagonals
.
Prove:
Proof:
1. WXYZ is a rectangle with diagonals .
(Given)
2. (Opp.sidesofa .)
3. (Refl.Prop.)
4. arerightangles.(Def.of
rectangle)
5. (Allright .)
6. (SAS)
7. (CPCTC)
34.Theorem 6.14
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given
. You need to
prove that WXYZ is a rectangle.. Use the properties
that you have learned about rectangles to walk
through the proof.
Given:
Prove: WXYZ is a rectangle.
Proof:
1. (Given)
2. (SSS)
3. (CPCTC)
4. (Def.of )
5. WXYZ is a parallelogram. (If both pairs of opp.
sides are , then quad. is .)
6. aresupplementary.(Cons.
aresuppl.)
7. (Defofsuppl.)
8. arerightangles.(If
2 and suppl., each isart. .)
9. arerightangles.(Ifa has1
rt. , it has 4 rt. .)
10. WXYZ is a rectangle. (Def. of rectangle)
ANSWER:
Given:
Prove: WXYZ is a rectangle.
Proof:
1. (Given)
2. (SSS)
3. (CPCTC)
4. (Def.of )
5. WXYZ is a parallelogram. (If both pairs of opp.
sides are , then quad. is .)
6. aresupplementary.(Cons.
aresuppl.)
7. (Defofsuppl.)
8. arerightangles.(If
2 and suppl., each isart. .)
9. arerightangles.(Ifa has1
rt. , it has 4 rt. .)
10. WXYZ is a rectangle. (Def. of rectangle)
PROOF Write a paragraph proof of each
statement.
35.If a parallelogram has one right angle, then it is a
rectangle.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given a parallelogram with one
right angle. You need to prove that it is a rectangle.
Let ABCD be a parallelogram with one right angle.
Then use the properties of parallelograms to walk
through the proof.
ABCD is a parallelogram, and is a right angle.
Since ABCD is a parallelogram and has one right
angle, then it has four right angles. So by the
definition of a rectangle, ABCD is a rectangle.
ANSWER:
ABCD is a parallelogram, and is a right angle.
Since ABCD is a parallelogram and has one right
angle, then it has four right angles. So by the
definition of a rectangle, ABCD is a rectangle.
36.If a quadrilateral has four right angles, then it is a
rectangle.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given a quadrilateral with four
right angles. You need to prove that it is a rectangle.
Let ABCD be a quadrilateral with four right angles.
Then use the properties of parallelograms and
rectangles to walk through the proof.
ABCD is a quadrilateral with four right angles.
ABCD is a parallelogram because both pairs of
opposite angles are congruent. By definition of a
rectangle, ABCD is a rectangle.
ANSWER:
ABCD is a quadrilateral with four right angles.
ABCD is a parallelogram because both pairs of
opposite angles are congruent. By definition of a
rectangle, ABCD is a rectangle.
37.CONSTRUCTION Construct a rectangle using the
construction for congruent segments and the
construction for a line perpendicular to another line
through a point on the line. Justify each step of the
construction.
SOLUTION:
Sampleanswer:
Step 1: Graph a line and place points P and Q on the
line.
Step 2: Place the compass on point P. Using the
same compass setting, draw an arc to the left and
right of P that intersects .
Step 3: Open the compass to a setting greater than
the distance from P to either point of intersection.
Place the compass on each point of intersection and
draw arcs that intersect above .
Step 4: Draw line a through point P and the
intersection of the arcs drawn in step 3.
Step 5: Repeat steps 2 and 3 using point Q.
Step 6: Draw line b through point Q and the
intersection of the arcs drawn in step 5.
Step 7: Using any compass setting, put the compass
at P and draw an arc above it that intersects line a.
Label the point of intersection R.
Step 8: Using the same compass setting, put the
compass at Q and draw an arc above it that
intersects line b. Label the point of intersection S.
Step 9: Draw .
Since and , the measure of
angles P and Q is 90 degrees. Lines that are
perpendicular to the same line are parallel, so
. The same compass setting was used to
locate points R and S, so . If one pair of
opposite sides of a quadrilateral is both parallel and
congruent, then the quadrilateral is a parallelogram.
A parallelogram with right angles is a rectangle.
Thus, PRSQ is a rectangle.
ANSWER:
Sample answer: Since and , the
measure of angles P and Q is 90 degrees. Lines that
are perpendicular to the same line are parallel, so
. The same compass setting was used to
locate points R and S, so . If one pair of
opposite sides of a quadrilateral is both parallel and
congruent, then the quadrilateral is a parallelogram.
A parallelogram with right angles is a rectangle.
Thus, PRSQ is a rectangle.
38.SPORTS The end zone of a football field is 160 feet
wide and 30 feet long. Kyle is responsible for
painting the field. He has finished the end zone.
Explain how Kyle can confirm that the end zone is
the regulation size and be sure that it is also a
rectangle using only a tape measure.
SOLUTION:
Sample answer: If the diagonals of a parallelogram
are congruent, then it is a rectangle. He should
measurethediagonalsoftheendzoneandboth
pairs of opposite sides. If both pairs of opposite sides
are congruent, then the quadrilateral is a
parallelogram. If the diagonals are congruent, then
the end zone is a rectangle.
ANSWER:
Sample answer: He should measure the diagonals of
the end zone and both pairs of opposite sides. If the
diagonals and both pairs of opposite sides are
congruent, then the end zone is a rectangle.
ALGEBRA Quadrilateral WXYZ is a rectangle.
39.If XW = 3, WZ = 4, and XZ = b, find YW.
SOLUTION:
The diagonals of a rectangle are congruent to each
other. So, YW = XZ = b. All four angles of a
rectangle are right angles. So, isaright
triangle. By the Pythagorean Theorem, XZ2 = XW2 +
WZ2.
Therefore, YW = 5.
ANSWER:
5
40.If XZ = 2c and ZY = 6, and XY = 8, find WY.
SOLUTION:
The diagonals of a rectangle are congruent to each
other. So, WY = XZ = 2c. All four angles of a
rectangle are right angles. So, isaright
triangle. By the Pythagorean Theorem, XZ2 = XY2 +
ZY2.
Therefore, WY = 2(5) = 10.
ANSWER:
10
41.SIGNS The sign below is in the foyer of Nyoko’s
school. Based on the dimensions given, can Nyoko
be sure that the sign is a rectangle? Explain your
reasoning.
SOLUTION:
Both pairs of opposite sides are congruent, so the
sign is a parallelogram, but no measure is given that
can be used to prove that it is a rectangle. If the
diagonals could be proven to be congruent, or if the
angles could be proven to be right angles, then we
could prove that the sign was a rectangle.
ANSWER:
No; sample answer: Both pairs of opposite sides are
congruent, so the sign is a parallelogram, but no
measure is given that can be used to prove that it is a
rectangle.
PROOF Write a coordinate proof of each
statement.
42.The diagonals of a rectangle are congruent.
SOLUTION:
Begin by positioning quadrilateral ABCD on a
coordinate plane. Place vertex D at the origin. Let
the length of the bases be a units and the height be b
units. Then the rest of the vertices are A(0, b), B(a,
b), and C(a, 0). You need to walk through the proof
step by step. Look over what you are given and what
you need to prove. Here, you are given ABCD is a
rectangle and you need to prove . Use the
properties that you have learned about rectangles to
walk through the proof.
Given: ABCD is a rectangle.
Prove:
Proof:
Use the Distance Formula to find the lengths of the
diagonals.
The diagonals, havethesamelength,so
they are congruent.
ANSWER:
Given: ABCD is a rectangle.
Prove:
Proof:
Use the Distance Formula to find
and .
havethesamelength, so they are
congruent.
43.If the diagonals of a parallelogram are congruent,
then it is a rectangle.
SOLUTION:
Begin by positioning parallelogram ABCD on a
coordinate plane. Place vertex A at the origin. Let
the length of the bases be a units and the height be c
units. Then the rest of the vertices are B(a, 0), C(b +
a, c), and D(b, c). You need to walk through the
proof step by step. Look over what you are given
and what you need to prove. Here, you are given
ABCD and and you need to prove that
ABCD is a rectangle. Use the properties that you
have learned about rectangles to walk through the
proof.
Given: ABCD and
Prove: ABCD is a rectangle.
Proof:
AC = BD. So,
Because A and B are different points,
a ≠0.Thenb = 0. The slope of isundefined
and the slope of . Thus, . ∠DAB is
a right angle and ABCD is a rectangle.
ANSWER:
Given: ABCD and
Prove: ABCD is a rectangle.
Proof:
But AC = BD. So,
Because A and B are different points,
a ≠0.Thenb = 0. The slope of isundefined
and the slope of . Thus, . ∠DAB is
a right angle and ABCD is a rectangle..
44.MULTIPLE REPRESENTATIONS In the
problem, you will explore properties of other special
parallelograms.
a. GEOMETRIC Draw three parallelograms, each
with all four sides congruent. Label one
parallelogram ABCD, one MNOP, and one WXYZ.
Draw the two diagonals of each parallelogram and
label the intersections R.
b. TABULAR Use a protractor to measure the
appropriate angles and complete the table below.
c. VERBAL Make a conjecture about the diagonals
of a parallelogram with four congruent sides.
SOLUTION:
a. Sample answer:
b.Use a protractor to measure each angle listed in
the table.
c. Sample answer: Each of the angles listed in the
table are right angles. These angles were each
formed by diagonals. The diagonals of a
parallelogram with four congruent sides are
perpendicular.
ANSWER:
a. Sample answer:
b.
c. Sample answer: The diagonals of a parallelogram
with four congruent sides are perpendicular.
45.CHALLENGE In rectangle ABCD,
Findthe
values of x and y.
SOLUTION:
All four angles of a rectangle are right angles. So,
The diagonals of a rectangle are congruent and
bisect each other. So, isanisoscelestriangle
with
So,
The sum of the angles of a triangle is 180.
So,
By the Vertical Angle Theorem,
ANSWER:
x = 6, y = −10
46.CCSS CRITIQUE Parker says that any two
congruent acute triangles can be arranged to make a
rectangle. Tamika says that only two congruent right
triangles can be arranged to make a rectangle. Is
either of them correct? Explain your reasoning.
SOLUTION:
When two congruent triangles are arranged to form a
quadrilateral, two of the angles are formed by a
single vertex of a triangle.
In order for the quadrilateral to be a rectangle, one of
the angles in the congruent triangles has to be a right
angle.
So, Tamika is correct.
ANSWER:
Tamika; Sample answer: When two congruent
triangles are arranged to form a quadrilateral, two of
the angles are formed by a single vertex of a triangle.
In order for the quadrilateral to be a rectangle, one of
the angles in the congruent triangles has to be a right
angle.
47.REASONING In the diagram at the right, lines n, p,
q,and r are parallel and lines l and m are parallel.
How many rectangles are formed by the intersecting
lines?
SOLUTION:
Let the lines n, p, q, and r intersect the line l at the
points A, B, C, and D respectively. Similarly, let the
lines intersect the line m at the points W, X, Y, and Z.
Then the rectangles formed are ABXW, ACYW,
BCYX, BDZX, CDZY, and ADZW. Therefore, 6
rectangles are formed by the intersecting lines.
ANSWER:
6
48.OPEN ENDED Write the equations of four lines
having intersections that form the vertices of a
rectangle. Verify your answer using coordinate
geometry.
SOLUTION:
Sample answer:
A rectangle has 4 right angles and each pair of
opposite sides is parallel and congruent. Lines that
run parallel to the x-axis will be parallel to each other
and perpendicular to lines parallel to the y-axis.
Selecting y = 0 and x = 0 will form 2 sides of the
rectangle. The lines y = 4 and x = 6 form the other 2
sides.
x = 0, x = 6, y = 0,y = 4;
The length of is6−0 or 6 units and the length of
is6– 0 or 6 units. The slope of AB is 0 and the
slope of DC is 0. Since a pair of sides of the
quadrilateral is both parallel and congruent, by
Theorem 6.12, the quadrilateral is a parallelogram.
Since is horizontal and isvertical, the lines
are perpendicular and the measure of the angle they
form is 90. By Theorem 6.6, if a parallelogram has
one right angle, it has four right angles. Therefore, by
definition, the parallelogram is a rectangle.
ANSWER:
Sample answer: x = 0, x = 6, y = 0,
y = 4;
The length of is6−0 or 6 units and the length of
is6– 0 or 6 units. The slope of AB is 0 and the
slope of DC is 0. Since a pair of sides of the
quadrilateral is both parallel and congruent, by
Theorem 6.12, the quadrilateral is a parallelogram.
Since is horizontal and isvertical, the lines
are perpendicular and the measure of the angle they
form is 90. By Theorem 6.6, if a parallelogram has
one right angle, it has four right angles. Therefore, by
definition, the parallelogram is a rectangle.
49.WRITING IN MATH Explain why all rectangles
are parallelograms, but all parallelograms are not
rectangles.
SOLUTION:
Sample answer: A parallelogram is a quadrilateral in
which each pair of opposite sides are parallel. By
definition of rectangle, all rectangles have both pairs
of opposite sides parallel. So rectangles are always
parallelograms.
Parallelograms with right angles are rectangles, so
some parallelograms are rectangles, but others with
non-right angles are not.
ANSWER:
Sample answer: All rectangles are parallelograms
because, by definition, both pairs of opposite sides
are parallel. Parallelograms with right angles are
rectangles, so some parallelograms are rectangles,
but others with non-right angles are not.
50.If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM =
13, what values of x and y make parallelogram
FGHJ a rectangle?
A x = 3, y=4
B x = 4, y = 3
C x = 7, y = 8
D x = 8, y = 7
SOLUTION:
The opposite sides of a parallelogram are congruent.
So, FJ = GH.
–3x + 5y=11FJ = GH
If the diagonals of a parallelogram are congruent,
then it is a rectangle. Since FGHJ is a parallelogram,
the diagonals bisect each other. So, if it is a
rectangle, then willbeanisoscelestriangle
with Thatis,3x + y = 13.
Add the two equations to eliminate the x-term and
solve for y.
–3x + 5y = 11
3
x + y = 13
6y=24Add to eliminate x-term.
y =4Divide each side by 6.
Use the value of y to find the value of x.
3x+4=13Substitute.
3x=9Subtract 4 from each side.
x =3Divide each side by 3.
So, x = 3, y = 4, the correct choice is A.
ANSWER:
A
51.ALGEBRA A rectangular playground is surrounded
by an 80-foot fence. One side of the playground is 10
feet longer than the other. Which of the following
equations could be used to find r, the shorter side of
the playground?
F 10r + r=80
G 4r+10=80
H r(r + 10) = 80
J 2(r + 10) + 2r = 80
SOLUTION:
The formula to find the perimeter of a rectangle is
given by P = 2(l + w). The perimeter is given to be 8
ft. The width is r and the length is r + 10. So, 2r + 2
(r + 10) = 80.
Therefore, the correct choice is J.
ANSWER:
J
52.SHORT RESPONSE What is the measure of
?
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, isanisoscelestriangle
with
So,
By the Exterior Angle Theorem,
ANSWER:
112
53.SAT/ACT If p is odd, which of the following must
also be odd?
A 2p
B 2p + 2
C
D
E
p + 2
SOLUTION:
Analyze each answer choice to determine which
expression is odd when p is odd.
ChoiceA2
p
Thisisamultipleof2soitiseven.
ChoiceB2p + 2
Thiscanberewrittenas2(p + 1) which is also a
multiple of 2 so it is even.
ChoiceC
If pisodd,isnotawholenumber.
ChoiceD2
p – 2
Thiscanberewrittenas2(p - 1) which is also a
multiple of 2 so it is even.
ChoiceE
p + 2
Adding2toanoddnumberresultsinanodd
number. This is odd.
The correct choice is E.
ANSWER:
E
ALGEBRA Find x and y so that the
quadrilateral is a parallelogram.
54.
SOLUTION:
The opposite sides of a parallelogram are congruent.
In a parallelogram, consecutive interior angles are
supplementary.
ANSWER:
x = 2, y = 41
55.
SOLUTION:
The opposite sides of a parallelogram are congruent.
The opposite sides of a parallelogram are parallel to
each other. So, the angles with the measures (3y +
3)°and(4y – 19)°arealternateinterioranglesand
hence they are equal.
ANSWER:
x = 8, y = 22
56.
SOLUTION:
The diagonals of a parallelogram bisect each other.
So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6.
Solve the first equation to find the value of y.
2y = 14
y = 7
Use the value of y in the second equation and solve
for x.
7 + 3 = 2x + 6
4=2x
2=x
ANSWER:
x = 2, y = 7
57.COORDINATE GEOMETRY Find the
coordinates of the intersection of the diagonals of
parallelogram ABCD with vertices A(1, 3), B(6, 2), C
(4, –2), and D(–1, –1).
SOLUTION:
The diagonals of a parallelogram bisect each other,
so the intersection is the midpoint of either diagonal.
Find the midpoint of or .
Therefore, the coordinates of the intersection of the
diagonals are (2.5, 0.5).
ANSWER:
(2.5, 0.5)
Refer to the figure.
58.If , name two congruent angles.
SOLUTION:
Since is an isosceles triangle. So,
ANSWER:
59.If , name two congruent segments.
SOLUTION:
In a triangle with two congruent angles, the sides
opposite to the congruent angles are also congruent.
So, .
ANSWER:
60.If , name two congruent segments.
SOLUTION:
In a triangle with two congruent angles, the sides
opposite to the congruent angles are also congruent.
So, .
ANSWER:
FARMING An X-brace on a rectangular barn
door is both decorative and functional. It helps
to prevent the door from warping over time.
If feet, PS = 7 feet, and
, find each measure.
1.QR
SOLUTION:
The opposite sides of a rectangle are parallel and
congruent. Therefore, QR = PS = 7 ft.
ANSWER:
7 ft
2.SQ
SOLUTION:
The diagonals of a rectangle bisect each other. So,
ANSWER:
3.
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, isanisoscelestriangle.
Then,
By the Exterior Angle Theorem,
Therefore,
ANSWER:
33.5
4.
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, isanisoscelestriangle.
Then,
By the Vertical Angle Theorem,
Therefore,
ANSWER:
56.5
ALGEBRA Quadrilateral DEFG is a rectangle.
5.If FD = 3x – 7 and EG = x + 5, find EG.
SOLUTION:
The diagonals of a rectangle are congruent to each
other. So, FD = EG.
Use the value of x to find EG.
EG = 6 + 5 = 11
ANSWER:
11
6.If ,
find .
SOLUTION:
All four angles of a rectangle are right angles. So,
ANSWER:
51
7.PROOF If ABDE is a rectangle and ,
provethat .
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. You are given ABDE is a rectangle and
. You need to prove . Use the
properties that you have learned about rectangles to
walk through the proof.
Given: ABDE is a rectangle;
Prove:
Statements(Reasons)
1. ABDE is a rectangle; .
(Given)
2. ABDE is a parallelogram. (Def. of
rectangle)
3. (Opp.sidesofa .)
4. arerightangles.(Def.ofrectangle)
5. (Allrt .)
6. (SAS)
7. (CPCTC)
ANSWER:
Given: ABDE is a rectangle;
Prove:
Statements(Reasons)
1. ABDE is a rectangle; .
(Given)
2. ABDE is a parallelogram. (Def. of
rectangle)
3. (Opp.sidesofa .)
4. arerightangles.(Def.ofrectangle)
5. (Allrt .)
6. (SAS)
7. (CPCTC)
COORDINATE GEOMETRY Graph each
quadrilateral with the given vertices. Determine
whether the figure is a rectangle. Justify your
answer using the indicated formula.
8.W(–4, 3), X(1, 5), Y(3, 1), Z(–2, –2); Slope Formula
SOLUTION:
Use the slope formula to find the slope of the sides of
the quadrilateral.
There are no parallel sides, so WXYZ is not a
parallelogram. Therefore, WXYZ is not a rectangle.
ANSWER:
No; slope of , slope of , slope of
, and slope of . Slope of
slope of , and slope of slopeof
, so WXYZ is not a parallelogram. Therefore,
WXYZ is not a rectangle.
9.A(4, 3), B(4, –2), C(–4, –2), D(–4, 3); Distance
Formula
SOLUTION:
First, Use the Distance formula to find the lengths of
the sides of the quadrilateral.
so ABCD is a
parallelogram.
A parallelogram is a rectangle if the diagonals are
congruent. Use the Distance formula to find the
lengths of the diagonals.
So the diagonals are congruent. Thus, ABCD is a
rectangle.
ANSWER:
Yes; AB = 5 = CD and BC = 8 = AD. So, ABCD is a
parallelogram. , so the diagonals are
congruent. Thus, ABCD is a rectangle.
FENCING X-braces are also used to provide
support in rectangular fencing. If AB = 6 feet,
AD = 2 feet, and , find each
measure.
10.BC
SOLUTION:
The opposite sides of a rectangle are parallel and
congruent. Therefore, BC = AD = 2 ft.
ANSWER:
2 ft
11.DB
SOLUTION:
All four angles of a rectangle are right angles. So,
isarighttriangle.BythePythagorean
Theorem, DB2 = BC2 + DC2.
BC = AD = 2
DC = AB = 6
ANSWER:
6.3 ft
12.
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, isanisosceles
triangle.
Then,
By the Vertical Angle Theorem,
ANSWER:
50
13.
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, isanisosceles
triangle.
Then,
All four angles of a rectangle are right angles.
ANSWER:
25
CCSS REGULARITY Quadrilateral WXYZ is a
rectangle.
14.If ZY = 2x + 3 and WX = x + 4, find WX.
SOLUTION:
The opposite sides of a rectangle are parallel and
congruent. So, ZY = WX.
2x + 3 = x + 4
x = 1
Use the value of x to find WX.
WX = 1 + 4 = 5.
ANSWER:
5
15.If PY = 3x – 5 and WP = 2x + 11, find ZP.
SOLUTION:
The diagonals of a rectangle bisect each other. So,
PY = WP.
The diagonals of a rectangle are congruent and
bisect each other. So, isanisoscelestriangle.
ANSWER:
43
16.If , find
.
SOLUTION:
All four angles of a rectangle are right angles. So,
ANSWER:
39
17.If ZP = 4x – 9 and PY = 2x + 5, find ZX.
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, ZP = PY.
Then ZP = 4(7) – 9 = 19.
Therefore, ZX = 2(ZP) = 38.
ANSWER:
38
18.If , find
.
SOLUTION:
All four angles of a rectangle are right angles. So,
ANSWER:
48
19.If , find
.
SOLUTION:
All four angles of a rectangle are right angles. So,
isarighttriangle.Then,
Therefore, Since is a
transversal of parallel sides and , alternate
interior angles WZX and ZXY are congruent. So,
= = 46.
ANSWER:
46
PROOF Write a two-column proof.
20.Given: ABCD is a rectangle.
Prove:
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given ABCD is a rectangle.
You need to prove . Use the
properties that you have learned about rectangles to
walk through the proof.
Given: ABCD is a rectangle.
Prove:
Proof:
Statements (Reasons)
1. ABCD is a rectangle. (Given)
2. ABCD is a parallelogram. (Def. of rectangle)
3. (Opp.sidesofa .)
4. (Refl.Prop.)
5. (Diagonalsofarectangleare .)
6. (SSS)
ANSWER:
Proof:
Statements (Reasons)
1. ABCD is a rectangle. (Given)
2. ABCD is a parallelogram. (Def. of rectangle)
3. (Opp.sidesofa .)
4. (Refl.Prop.)
5. (Diagonalsofarectangleare .)
6. (SSS)
21.Given: QTVW is a rectangle.
Prove:
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given QTVW is a rectangle and
. You need to prove . Use
the properties that you have learned about rectangles
to walk through the proof.
Given: QTVW is a rectangle.
Prove:
Proof:
Statements (Reasons)
1. QTVW is a rectangle; .(Given)
2. QTVW is a parallelogram. (Def. of rectangle)
3. (Oppsidesofa .)
4. arerightangles.(Def.ofrectangle)
5. (Allrt .)
6. QR = ST (Def. of segs.)
7. (Refl.Prop.)
8. RS = RS (Def. of segs.)
9. QR + RS = RS + ST (Add. prop.)
10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.)
11. QS = RT (Subst.)
12. (Def.of segs.)
13. (SAS)
ANSWER:
Proof:
Statements (Reasons)
1. QTVW is a rectangle; .(Given)
2. QTVW is a parallelogram. (Def. of rectangle)
3. (Oppsidesofa .)
4. arerightangles.(Def.ofrectangle)
5. (Allrt .)
6. QR = ST (Def. of segs.)
7. (Refl.Prop.)
8. RS = RS (Def. of segs.)
9. QR + RS = RS + ST (Add. prop.)
10. QS = QR + RS, RT = RS + ST (Seg. Add. Post.)
11. QS = RT (Subst.)
12. (Def.of segs.)
13. (SAS)
COORDINATE GEOMETRY Graph each
quadrilateral with the given vertices. Determine
whether the figure is a rectangle. Justify your
answer using the indicated formula.
22.W(–2, 4), X(5, 5), Y(6, –2), Z(–1, –3); Slope Formula
SOLUTION:
Use the slope formula to find the slope of the sides of
the quadrilateral.
The slopes of each pair of opposite sides are equal.
So, the two pairs of opposite sides are parallel.
Therefore, the quadrilateral WXYZ is a parallelogram.
The products of the slopes of the adjacent sides are
–1. So, any two adjacent sides are perpendicular to
each other. That is, all the four angles are right
angles. Therefore, WXYZ is a rectangle.
ANSWER:
Yes; slope of slope of , slope
of slopeof . So WXYZ is a
parallelogram. The product of the slopes of
consecutive sides is –1, so the consecutive sides are
perpendicular and form right angles. Thus, WXYZ is a
rectangle.
23.J(3, 3), K(–5, 2), L(–4, –4), M(4, –3); Distance
Formula
SOLUTION:
First, Use the Distance formula to find the lengths of
the sides of the quadrilateral.
soJKLM is a
parallelogram.
A parallelogram is a rectangle if the diagonals are
congruent. Use the Distance formula to find the
lengths of the diagonals.
So the diagonals are not congruent. Thus, JKLM is
not a rectangle.
ANSWER:
No; soJKLM is a
parallelogram; so
the diagonals are not congruent. Thus, JKLM is not a
rectangle.
24.Q(–2, 2), R(0, –2), S(6, 1), T(4, 5); Distance Formula
SOLUTION:
First, use the Distance Formula to find the lengths of
the sides of the quadrilateral.
soQRST is a
parallelogram.
A parallelogram is a rectangle if the diagonals are
congruent. Use the Distance Formula to find the
lengths of the diagonals.
So the diagonals are congruent. Thus, QRST is a
rectangle.
ANSWER:
Yes; so QRST is a
parallelogram. sothediagonalsare
congruent. QRST is a rectangle.
25.G(1, 8), H(–7, 7), J(–6, 1), K(2, 2); Slope Formula
SOLUTION:
First, use the slope formula to find the lengths of the
sides of the quadrilateral.
The slopes of each pair of opposite sides are equal.
So, the two pairs of opposite sides are parallel.
Therefore, the quadrilateral GHJK is a
parallelogram.
None of the adjacent sides have slopes whose
product is –1. So, the angles are not right angles.
Therefore, GHJK is not a rectangle.
ANSWER:
No; slope of slopeof andslopeof
slopeof . So, GHJK is a
parallelogram. The product of the slopes of
consecutivesides≠–1, so the consecutive sides are
not perpendicular. Thus, GHJK is not a rectangle.
Quadrilateral ABCD is a rectangle. Find each
measure if .
26.
SOLUTION:
All four angles of a rectangle are right angles. So,
ANSWER:
50
27.
SOLUTION:
The measures of angles 2 and 3 are equal as they
are alternate interior angles.
Since the diagonals of a rectangle are congruent and
bisect each other, the triangle with the angles 3, 7
and 8 is an isosceles triangle. So, m∠7 = m∠3.
Therefore, m∠7 = m∠2 = 40.
ANSWER:
40
28.
SOLUTION:
The measures of angles 2 and 3 are equal as they
are alternate interior angles.
Therefore, m∠3 = m∠2 = 40.
ANSWER:
40
29.
SOLUTION:
All four angles of a rectangle are right angles. So,
The measures of angles 1 and 4 are equal as they
are alternate interior angles.
Therefore, m∠4 = m∠1 = 50.
Since the diagonals of a rectangle are congruent and
bisect each other, the triangle with the angles 4, 5
and 6 is an isosceles triangle. So, m∠6 = m∠4 = 50.
The sum of the three angles of a triangle is 180.
Therefore, m∠5 = 180 – (50 + 50) = 80.
ANSWER:
80
30.
SOLUTION:
All four angles of a rectangle are right angles. So,
The measures of angles 1 and 4 are equal as they
are alternate interior angles.
Therefore, m∠4 = m∠1 = 50.
Since the diagonals of a rectangle are congruent and
bisect each other, the triangle with the angles 4, 5
and 6 is an isosceles triangle. Therefore, m∠6 =
m∠4 = 50.
ANSWER:
50
31.
SOLUTION:
The measures of angles 2 and 3 are equal as they
are alternate interior angles.
Since the diagonals of a rectangle are congruent and
bisect each other, the triangle with the angles 3, 7
and 8 is an isosceles triangle. So, m∠3 = m∠7 = 40.
The sum of the three angles of a triangle is 180.
Therefore, m∠8 = 180 – (40 + 40) = 100.
ANSWER:
100
32.CCSS MODELING Jody is building a new
bookshelf using wood and metal supports like the one
shown. To what length should she cut the metal
supports in order for the bookshelf to be square,
which means that the angles formed by the shelves
and the vertical supports are all right angles? Explain
your reasoning.
SOLUTION:
In order for the bookshelf to be square, the lengths of
all of the metal supports should be equal. Since we
know the length of the bookshelves and the distance
between them, use the Pythagorean Theorem to find
the lengths of the metal supports.
Let x be the length of each support. Express 3 feet
as 36 inches.
Therefore, the metal supports should each be 39
inches long or 3 feet 3 inches long.
ANSWER:
3 ft 3 in.; Sample answer: In order for the bookshelf
to be square, the lengths of all of the metal supports
should be equal. Since I knew the length of the
bookshelves and the distance between them, I used
the Pythagorean Theorem to find the lengths of the
metal supports.
PROOF Write a two-column proof.
33.Theorem 6.13
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given WXYZ is a rectangle with
diagonals . You need to prove
. Use the properties that you have learned
about rectangles to walk through the proof.
Given: WXYZ is a rectangle with diagonals
.
Prove:
Proof:
1. WXYZ is a rectangle with diagonals .
(Given)
2. (Opp.sidesofa .)
3. (Refl.Prop.)
4. arerightangles.(Def.of
rectangle)
5. (Allright .)
6. (SAS)
7. (CPCTC)
ANSWER:
Given: WXYZ is a rectangle with diagonals
.
Prove:
Proof:
1. WXYZ is a rectangle with diagonals .
(Given)
2. (Opp.sidesofa .)
3. (Refl.Prop.)
4. arerightangles.(Def.of
rectangle)
5. (Allright .)
6. (SAS)
7. (CPCTC)
34.Theorem 6.14
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given
. You need to
prove that WXYZ is a rectangle.. Use the properties
that you have learned about rectangles to walk
through the proof.
Given:
Prove: WXYZ is a rectangle.
Proof:
1. (Given)
2. (SSS)
3. (CPCTC)
4. (Def.of )
5. WXYZ is a parallelogram. (If both pairs of opp.
sides are , then quad. is .)
6. aresupplementary.(Cons.
aresuppl.)
7. (Defofsuppl.)
8. arerightangles.(If
2 and suppl., each isart. .)
9. arerightangles.(Ifa has1
rt. , it has 4 rt. .)
10. WXYZ is a rectangle. (Def. of rectangle)
ANSWER:
Given:
Prove: WXYZ is a rectangle.
Proof:
1. (Given)
2. (SSS)
3. (CPCTC)
4. (Def.of )
5. WXYZ is a parallelogram. (If both pairs of opp.
sides are , then quad. is .)
6. aresupplementary.(Cons.
aresuppl.)
7. (Defofsuppl.)
8. arerightangles.(If
2 and suppl., each isart. .)
9. arerightangles.(Ifa has1
rt. , it has 4 rt. .)
10. WXYZ is a rectangle. (Def. of rectangle)
PROOF Write a paragraph proof of each
statement.
35.If a parallelogram has one right angle, then it is a
rectangle.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given a parallelogram with one
right angle. You need to prove that it is a rectangle.
Let ABCD be a parallelogram with one right angle.
Then use the properties of parallelograms to walk
through the proof.
ABCD is a parallelogram, and is a right angle.
Since ABCD is a parallelogram and has one right
angle, then it has four right angles. So by the
definition of a rectangle, ABCD is a rectangle.
ANSWER:
ABCD is a parallelogram, and is a right angle.
Since ABCD is a parallelogram and has one right
angle, then it has four right angles. So by the
definition of a rectangle, ABCD is a rectangle.
36.If a quadrilateral has four right angles, then it is a
rectangle.
SOLUTION:
You need to walk through the proof step by step.
Look over what you are given and what you need to
prove. Here, you are given a quadrilateral with four
right angles. You need to prove that it is a rectangle.
Let ABCD be a quadrilateral with four right angles.
Then use the properties of parallelograms and
rectangles to walk through the proof.
ABCD is a quadrilateral with four right angles.
ABCD is a parallelogram because both pairs of
opposite angles are congruent. By definition of a
rectangle, ABCD is a rectangle.
ANSWER:
ABCD is a quadrilateral with four right angles.
ABCD is a parallelogram because both pairs of
opposite angles are congruent. By definition of a
rectangle, ABCD is a rectangle.
37.CONSTRUCTION Construct a rectangle using the
construction for congruent segments and the
construction for a line perpendicular to another line
through a point on the line. Justify each step of the
construction.
SOLUTION:
Sampleanswer:
Step 1: Graph a line and place points P and Q on the
line.
Step 2: Place the compass on point P. Using the
same compass setting, draw an arc to the left and
right of P that intersects .
Step 3: Open the compass to a setting greater than
the distance from P to either point of intersection.
Place the compass on each point of intersection and
draw arcs that intersect above .
Step 4: Draw line a through point P and the
intersection of the arcs drawn in step 3.
Step 5: Repeat steps 2 and 3 using point Q.
Step 6: Draw line b through point Q and the
intersection of the arcs drawn in step 5.
Step 7: Using any compass setting, put the compass
at P and draw an arc above it that intersects line a.
Label the point of intersection R.
Step 8: Using the same compass setting, put the
compass at Q and draw an arc above it that
intersects line b. Label the point of intersection S.
Step 9: Draw .
Since and , the measure of
angles P and Q is 90 degrees. Lines that are
perpendicular to the same line are parallel, so
. The same compass setting was used to
locate points R and S, so . If one pair of
opposite sides of a quadrilateral is both parallel and
congruent, then the quadrilateral is a parallelogram.
A parallelogram with right angles is a rectangle.
Thus, PRSQ is a rectangle.
ANSWER:
Sample answer: Since and , the
measure of angles P and Q is 90 degrees. Lines that
are perpendicular to the same line are parallel, so
. The same compass setting was used to
locate points R and S, so . If one pair of
opposite sides of a quadrilateral is both parallel and
congruent, then the quadrilateral is a parallelogram.
A parallelogram with right angles is a rectangle.
Thus, PRSQ is a rectangle.
38.SPORTS The end zone of a football field is 160 feet
wide and 30 feet long. Kyle is responsible for
painting the field. He has finished the end zone.
Explain how Kyle can confirm that the end zone is
the regulation size and be sure that it is also a
rectangle using only a tape measure.
SOLUTION:
Sample answer: If the diagonals of a parallelogram
are congruent, then it is a rectangle. He should
measurethediagonalsoftheendzoneandboth
pairs of opposite sides. If both pairs of opposite sides
are congruent, then the quadrilateral is a
parallelogram. If the diagonals are congruent, then
the end zone is a rectangle.
ANSWER:
Sample answer: He should measure the diagonals of
the end zone and both pairs of opposite sides. If the
diagonals and both pairs of opposite sides are
congruent, then the end zone is a rectangle.
ALGEBRA Quadrilateral WXYZ is a rectangle.
39.If XW = 3, WZ = 4, and XZ = b, find YW.
SOLUTION:
The diagonals of a rectangle are congruent to each
other. So, YW = XZ = b. All four angles of a
rectangle are right angles. So, isaright
triangle. By the Pythagorean Theorem, XZ2 = XW2 +
WZ2.
Therefore, YW = 5.
ANSWER:
5
40.If XZ = 2c and ZY = 6, and XY = 8, find WY.
SOLUTION:
The diagonals of a rectangle are congruent to each
other. So, WY = XZ = 2c. All four angles of a
rectangle are right angles. So, isaright
triangle. By the Pythagorean Theorem, XZ2 = XY2 +
ZY2.
Therefore, WY = 2(5) = 10.
ANSWER:
10
41.SIGNS The sign below is in the foyer of Nyoko’s
school. Based on the dimensions given, can Nyoko
be sure that the sign is a rectangle? Explain your
reasoning.
SOLUTION:
Both pairs of opposite sides are congruent, so the
sign is a parallelogram, but no measure is given that
can be used to prove that it is a rectangle. If the
diagonals could be proven to be congruent, or if the
angles could be proven to be right angles, then we
could prove that the sign was a rectangle.
ANSWER:
No; sample answer: Both pairs of opposite sides are
congruent, so the sign is a parallelogram, but no
measure is given that can be used to prove that it is a
rectangle.
PROOF Write a coordinate proof of each
statement.
42.The diagonals of a rectangle are congruent.
SOLUTION:
Begin by positioning quadrilateral ABCD on a
coordinate plane. Place vertex D at the origin. Let
the length of the bases be a units and the height be b
units. Then the rest of the vertices are A(0, b), B(a,
b), and C(a, 0). You need to walk through the proof
step by step. Look over what you are given and what
you need to prove. Here, you are given ABCD is a
rectangle and you need to prove . Use the
properties that you have learned about rectangles to
walk through the proof.
Given: ABCD is a rectangle.
Prove:
Proof:
Use the Distance Formula to find the lengths of the
diagonals.
The diagonals, havethesamelength,so
they are congruent.
ANSWER:
Given: ABCD is a rectangle.
Prove:
Proof:
Use the Distance Formula to find
and .
havethesamelength, so they are
congruent.
43.If the diagonals of a parallelogram are congruent,
then it is a rectangle.
SOLUTION:
Begin by positioning parallelogram ABCD on a
coordinate plane. Place vertex A at the origin. Let
the length of the bases be a units and the height be c
units. Then the rest of the vertices are B(a, 0), C(b +
a, c), and D(b, c). You need to walk through the
proof step by step. Look over what you are given
and what you need to prove. Here, you are given
ABCD and and you need to prove that
ABCD is a rectangle. Use the properties that you
have learned about rectangles to walk through the
proof.
Given: ABCD and
Prove: ABCD is a rectangle.
Proof:
AC = BD. So,
Because A and B are different points,
a ≠0.Thenb = 0. The slope of isundefined
and the slope of . Thus, . ∠DAB is
a right angle and ABCD is a rectangle.
ANSWER:
Given: ABCD and
Prove: ABCD is a rectangle.
Proof:
But AC = BD. So,
Because A and B are different points,
a ≠0.Thenb = 0. The slope of isundefined
and the slope of . Thus, . ∠DAB is
a right angle and ABCD is a rectangle..
44.MULTIPLE REPRESENTATIONS In the
problem, you will explore properties of other special
parallelograms.
a. GEOMETRIC Draw three parallelograms, each
with all four sides congruent. Label one
parallelogram ABCD, one MNOP, and one WXYZ.
Draw the two diagonals of each parallelogram and
label the intersections R.
b. TABULAR Use a protractor to measure the
appropriate angles and complete the table below.
c. VERBAL Make a conjecture about the diagonals
of a parallelogram with four congruent sides.
SOLUTION:
a. Sample answer:
b.Use a protractor to measure each angle listed in
the table.
c. Sample answer: Each of the angles listed in the
table are right angles. These angles were each
formed by diagonals. The diagonals of a
parallelogram with four congruent sides are
perpendicular.
ANSWER:
a. Sample answer:
b.
c. Sample answer: The diagonals of a parallelogram
with four congruent sides are perpendicular.
45.CHALLENGE In rectangle ABCD,
Findthe
values of x and y.
SOLUTION:
All four angles of a rectangle are right angles. So,
The diagonals of a rectangle are congruent and
bisect each other. So, isanisoscelestriangle
with
So,
The sum of the angles of a triangle is 180.
So,
By the Vertical Angle Theorem,
ANSWER:
x = 6, y = −10
46.CCSS CRITIQUE Parker says that any two
congruent acute triangles can be arranged to make a
rectangle. Tamika says that only two congruent right
triangles can be arranged to make a rectangle. Is
either of them correct? Explain your reasoning.
SOLUTION:
When two congruent triangles are arranged to form a
quadrilateral, two of the angles are formed by a
single vertex of a triangle.
In order for the quadrilateral to be a rectangle, one of
the angles in the congruent triangles has to be a right
angle.
So, Tamika is correct.
ANSWER:
Tamika; Sample answer: When two congruent
triangles are arranged to form a quadrilateral, two of
the angles are formed by a single vertex of a triangle.
In order for the quadrilateral to be a rectangle, one of
the angles in the congruent triangles has to be a right
angle.
47.REASONING In the diagram at the right, lines n, p,
q,and r are parallel and lines l and m are parallel.
How many rectangles are formed by the intersecting
lines?
SOLUTION:
Let the lines n, p, q, and r intersect the line l at the
points A, B, C, and D respectively. Similarly, let the
lines intersect the line m at the points W, X, Y, and Z.
Then the rectangles formed are ABXW, ACYW,
BCYX, BDZX, CDZY, and ADZW. Therefore, 6
rectangles are formed by the intersecting lines.
ANSWER:
6
48.OPEN ENDED Write the equations of four lines
having intersections that form the vertices of a
rectangle. Verify your answer using coordinate
geometry.
SOLUTION:
Sample answer:
A rectangle has 4 right angles and each pair of
opposite sides is parallel and congruent. Lines that
run parallel to the x-axis will be parallel to each other
and perpendicular to lines parallel to the y-axis.
Selecting y = 0 and x = 0 will form 2 sides of the
rectangle. The lines y = 4 and x = 6 form the other 2
sides.
x = 0, x = 6, y = 0,y = 4;
The length of is6−0 or 6 units and the length of
is6– 0 or 6 units. The slope of AB is 0 and the
slope of DC is 0. Since a pair of sides of the
quadrilateral is both parallel and congruent, by
Theorem 6.12, the quadrilateral is a parallelogram.
Since is horizontal and isvertical, the lines
are perpendicular and the measure of the angle they
form is 90. By Theorem 6.6, if a parallelogram has
one right angle, it has four right angles. Therefore, by
definition, the parallelogram is a rectangle.
ANSWER:
Sample answer: x = 0, x = 6, y = 0,
y = 4;
The length of is6−0 or 6 units and the length of
is6– 0 or 6 units. The slope of AB is 0 and the
slope of DC is 0. Since a pair of sides of the
quadrilateral is both parallel and congruent, by
Theorem 6.12, the quadrilateral is a parallelogram.
Since is horizontal and isvertical, the lines
are perpendicular and the measure of the angle they
form is 90. By Theorem 6.6, if a parallelogram has
one right angle, it has four right angles. Therefore, by
definition, the parallelogram is a rectangle.
49.WRITING IN MATH Explain why all rectangles
are parallelograms, but all parallelograms are not
rectangles.
SOLUTION:
Sample answer: A parallelogram is a quadrilateral in
which each pair of opposite sides are parallel. By
definition of rectangle, all rectangles have both pairs
of opposite sides parallel. So rectangles are always
parallelograms.
Parallelograms with right angles are rectangles, so
some parallelograms are rectangles, but others with
non-right angles are not.
ANSWER:
Sample answer: All rectangles are parallelograms
because, by definition, both pairs of opposite sides
are parallel. Parallelograms with right angles are
rectangles, so some parallelograms are rectangles,
but others with non-right angles are not.
50.If FJ = –3x + 5y, FM = 3x + y, GH = 11, and GM =
13, what values of x and y make parallelogram
FGHJ a rectangle?
A x = 3, y=4
B x = 4, y = 3
C x = 7, y = 8
D x = 8, y = 7
SOLUTION:
The opposite sides of a parallelogram are congruent.
So, FJ = GH.
–3x + 5y=11FJ = GH
If the diagonals of a parallelogram are congruent,
then it is a rectangle. Since FGHJ is a parallelogram,
the diagonals bisect each other. So, if it is a
rectangle, then willbeanisoscelestriangle
with Thatis,3x + y = 13.
Add the two equations to eliminate the x-term and
solve for y.
–3x + 5y = 11
3
x + y = 13
6y=24Add to eliminate x-term.
y =4Divide each side by 6.
Use the value of y to find the value of x.
3x+4=13Substitute.
3x=9Subtract 4 from each side.
x =3Divide each side by 3.
So, x = 3, y = 4, the correct choice is A.
ANSWER:
A
51.ALGEBRA A rectangular playground is surrounded
by an 80-foot fence. One side of the playground is 10
feet longer than the other. Which of the following
equations could be used to find r, the shorter side of
the playground?
F 10r + r=80
G 4r+10=80
H r(r + 10) = 80
J 2(r + 10) + 2r = 80
SOLUTION:
The formula to find the perimeter of a rectangle is
given by P = 2(l + w). The perimeter is given to be 8
ft. The width is r and the length is r + 10. So, 2r + 2
(r + 10) = 80.
Therefore, the correct choice is J.
ANSWER:
J
52.SHORT RESPONSE What is the measure of
?
SOLUTION:
The diagonals of a rectangle are congruent and
bisect each other. So, isanisoscelestriangle
with
So,
By the Exterior Angle Theorem,
ANSWER:
112
53.SAT/ACT If p is odd, which of the following must
also be odd?
A 2p
B 2p + 2
C
D
E
p + 2
SOLUTION:
Analyze each answer choice to determine which
expression is odd when p is odd.
ChoiceA2
p
Thisisamultipleof2soitiseven.
ChoiceB2p + 2
Thiscanberewrittenas2(p + 1) which is also a
multiple of 2 so it is even.
ChoiceC
If pisodd,isnotawholenumber.
ChoiceD2
p – 2
Thiscanberewrittenas2(p - 1) which is also a
multiple of 2 so it is even.
ChoiceE
p + 2
Adding2toanoddnumberresultsinanodd
number. This is odd.
The correct choice is E.
ANSWER:
E
ALGEBRA Find x and y so that the
quadrilateral is a parallelogram.
54.
SOLUTION:
The opposite sides of a parallelogram are congruent.
In a parallelogram, consecutive interior angles are
supplementary.
ANSWER:
x = 2, y = 41
55.
SOLUTION:
The opposite sides of a parallelogram are congruent.
The opposite sides of a parallelogram are parallel to
each other. So, the angles with the measures (3y +
3)°and(4y – 19)°arealternateinterioranglesand
hence they are equal.
ANSWER:
x = 8, y = 22
56.
SOLUTION:
The diagonals of a parallelogram bisect each other.
So, 4y – 9 = 2y + 5 and y + 3 = 2x + 6.
Solve the first equation to find the value of y.
2y = 14
y = 7
Use the value of y in the second equation and solve
for x.
7 + 3 = 2x + 6
4=2x
2=x
ANSWER:
x = 2, y = 7
57.COORDINATE GEOMETRY Find the
coordinates of the intersection of the diagonals of
parallelogram ABCD with vertices A(1, 3), B(6, 2), C
(4, –2), and D(–1, –1).
SOLUTION:
The diagonals of a parallelogram bisect each other,
so the intersection is the midpoint of either diagonal.
Find the midpoint of or .
Therefore, the coordinates of the intersection of the
diagonals are (2.5, 0.5).
ANSWER:
(2.5, 0.5)
Refer to the figure.
58.If , name two congruent angles.
SOLUTION:
Since is an isosceles triangle. So,
ANSWER:
59.If , name two congruent segments.
SOLUTION:
In a triangle with two congruent angles, the sides
opposite to the congruent angles are also congruent.
So, .
ANSWER:
60.If , name two congruent segments.
SOLUTION:
In a triangle with two congruent angles, the sides
opposite to the congruent angles are also congruent.
So, .
ANSWER:
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6-4 Rectangles
