Download Finding Unknown Angles and more Lecture notes Geometry in PDF only on Docsity!
CHAPTER 3
Finding Unknown Angles
Geometry becomes more interesting when students start using geometric facts to find unknown
lengths and angles. During this stage, roughly grades 5-8, students work on āunknown angle
problemsā. These problems are learning bonanzas. They initiate students in the art of deduc-
tive reasoning, solidify their understanding of geometry and measurement, and help introduce
algebra.
You have already solved some unknown angle problems and seen how they are integrated
into the Primary Math curriculum in grades 5 and 6. This chapter examines how unknown angle
problems are used to develop geometry in grades 6 and 7.
From a teaching perspective, unknown angle problems are not just part of the geometry
curriculum, they are the curriculum in grades 5-8; everything else is secondary. In these grades,
teachers and textbooks introduce facts about angles within triangles and polygons, about par-
allel lines, about congruent and similar figures, and about circles. These are not simply facts
to memorize: understanding emerges as students use them to solve problems. Thus teaching
centers on solving problems.
Unknown angle problems are superbly suited for this purpose. Solutions require several
steps, each applying a known fact to the given figure. As students do these problems the ge-
ometric facts spring to life; these facts become friends that can be called upon to help solve
problems. Unknown angle problems are also enormous fun!
3.1 Unknown Angle Problems
An unknown angle problem is a puzzle consisting of a figure with the measures of some
sides and angles given and with one angle ā the unknown angle ā marked with a letter. The
studentās task is to find the measure of the unknown angle by applying basic geometric facts.
Beginning exercises require only rudimentary facts, such as the fact that angles around a point
add to 360
ā¦
. As new geometric facts are introduced, they are added to the list of facts that
are available as tools to solve unknown angle problems. As more knowledge is integrated, the
problems become more challenging and more interesting.
56 ā¢ CHAPTER 3. FINDING UNKNOWN ANGLES
This section examines the role of unknown angle problems in the Primary Math and New
Elementary Math textbooks for grades 5-7. It includes a list of the geometric facts learned
during this stage and a format for presenting āTeacherās Solutionsā to unknown angle problems.
You will be asked to use this format for many homework problems.
Many elementary textbooks, including the Primary Math books, introduce new concepts
using the following specific process.
Teaching sequence for introducing geometric facts
- Review background knowledge and introduce any new terms needed.
- Introduce the fact by an activity (measuring, folding, or cutting-and-rearranging) that
serves to clarify what the fact says and convince students that it is true.
- Summarize by stating the geometric fact in simple clear language.
- Have students solve dozens of unknown angle problems:
a) simple problems using the fact alone,
b) multi-step problems using the fact alone,
c) multi-step problems combining the fact with previously-learned facts.
Step 3 takes only a few minutes, but it is the teacherās most important input. In geometry,
words have precise meanings; studentsā success depends on knowing definitions and knowing
how to apply them. One can even argue that geometry is included in the K-12 curriculum to
teach students that giving words precise meaning fosters clear thinking. This lesson is applica-
ble to all subjects.
After these preliminaries, the fun begins as students solve increasingly challenging prob-
lems (Step 4). As always in mathematics, the real learning occurs as students solve problems.
Geometry Facts ā First List
As you have seen in homework problems, the basic facts about angles, triangles and quadri-
laterals are presented in Primary Mathematics 5A and 5B. Below is a list of the facts learned at
that stage. Each has a simple abbreviation. You will be expected to be consistent in using these
abbreviations in your homework solutions.
The list of facts is built around three exercises. These questions ask you to observe how
these facts are justified at the grade 5 level (using folding, cutting, and measuring exercises) and
to observe the type of problems students are asked to solve.
EXERCISE 1.1 (Angle Facts). The following three facts are introduced on pages 85ā88 of Pri-
mary Mathematics 5A. How are these facts justified?
58 ā¢ CHAPTER 3. FINDING UNKNOWN ANGLES
Base angles of an isosceles triangle are equal.
(Abbreviation: base ā s of isos. ā.) a Ā°
a = b
b Ā°
Each interior angle of an equilateral triangle is 60
ā¦
.
(Abbreviation: equilat. ā.)
60 Ā° 60 Ā°
60 Ā°
EXERCISE 1.3 (Quadrilateral Facts). The next section of Primary Math 5B (pages 68ā71)
introduces two facts about 4-sided figures. Study the folding and cutting exercises given on
page 70. How would you use these exercises in your class?
Opposite angles in a parallelogram are equal.
(Abbreviation: opp. ā s ā„ -ogram .)
a Ā°
a = b.
b Ā°
Interior angles between two parallel sides in a trape-
zoid (or a parallelogram) are supplementary.
(Abbreviation: int. ā s, BC ā„ AD .)
a Ā°
c Ā°
d Ā°
b Ā°
a + b = 180.
c + d = 180.
A
B C
D
The āTeacherās Solutionā Format for Unknown Angle Problems
Teachers are obliged to present detailed solutions to problems for the benefit of their stu-
dents. The teacherās solutions must meet a different standard than the studentsā solutions. Both
teachers and students are expected to get the reasoning and the answer correct. But teacher-
presented solutions must also communicate the thought process as clearly as possible.
In this book, solutions that meet this high standard are called Teacherās Solutions. You will
frequently be asked to write such Teacherās Solutions in homework. If you are unsure how to do
this, look in the textbooks : almost every solution presented in the Primary Math books, and all
of the āWorked Examplesā in the New Elementary Mathematics book, are Teacherās Solutions.
You are already familiar with one type of Teacherās Solution ā bar diagrams. Bar diagrams
are extraordinarily useful for communicating ideas about arithmetic. Teachers need similar
devices for communicating geometric ideas. As a start, in this chapter you will be writing
Teacherās Solutions for unknown angle problems. Here is a simple example.
SECTION 3.1 UNKNOWN ANGLE PROBLEMS ā¢ 59
EXAMPLE 1.4. The figure shows angles around a point. Find the value of x.
..
.
3x + 15 = 75
3x = 60
x = 20.
Teacherās Solution to an Unknown Angle Problem
Answer clearly stated on the last line.
Diagram shows
all needed
information.
A new line for each step.
Each fact used is stated
on the same line
using our abbreviations.
75Āŗ 3xĀŗ
15Āŗ
vert. s_._
}
Arithmetic and algebra
done one step at a time
(no reasons needed).
This solution is short and clear, yet displays all the reasoning. It always begins with a picture
showing all points, lines, and angles used in the solution, and it always ends with a clear answer
to the question asked.
Notice what happens with the degree signs. The angles in the picture have degree signs, so
75, 15 and 3 x are all numbers. Thus we can drop the degree signs in the equations. This saves
work and makes the solution clearer. Degree signs are handled in the same way in the next two
examples.
EXAMPLE 1.5. The figure shows a parallelogram. Find the value of x.
Teacherās Solution:
x Ā°
a Ā°
74 Ā°
67 Ā°
A D
B
C
a = 74 opp. ā s ofā„-ogram
x = a + 67 ext. ā of ā ABD
ā“ x = 74 + 67
EXAMPLE 1.6. Find the values of a and b in the following figure.
Teacherās Solution:
bĀŗ aĀŗ
44Āŗ
66Āŗ
95Āŗ
a + 44 + 95 = 180 ā sum of ā
a + 139 = 180 ,
ā“ a = 41.
a + b = 66 ext. ā of ā
41 + b = 66 ,
ā“ b = 25.
SECTION 3.2 FINDING ANGLES USING PARALLEL LINES ā¢ 61
3.2 Finding Angles Using Parallel Lines
After completing Primary Mathematics 5 and 6, students enter grade 7 with two yearsā
experience solving unknown angle problems. They understand that geometry is a game in
which one uses a few simple facts to find relations among the lengths and angles in figures. The
grade 7 textbook (New Elementary Mathematics 1) revisits the material learned earlier, putting
it in a condensed, structured form. It then moves on to new ideas.
The revision begins with lines and angles (Chapter 9 of NEM1). Folding and measuring ex-
ercises are no longer primary; the discussion is built on facts about parallel lines and congruent
triangles, and is broadened to include area. The transition is subtle: the spirit of the subject is
unchanged and student work remains focused on solving short geometric puzzles.
This section describes the basic facts about parallel lines. It ends with discussions of two
sources of student confusion: the distinction between a statement and its converse, and the
technique of drawing auxiliary lines.
Transversals and Parallel Lines
In the figure below, T is a transversal to the lines L and M. More precisely: Given a pair of
lines L and M in a plane, a third line T is a transversal if it intersects L at a single point P and
intersects M in a different point Q.
L
M
T
P
Q
T is not a
transversal!
Itās not a line.
T
M
L
A transversal forms eight angles with the two lines. Pairs of angles are named according to
their relative positions. Angles on the same side of the transversal and on the same side of the
lines are called corresponding angles. Four pairs of corresponding angles are shown below.
aĀ°
pĀ°
bĀ°
qĀ°
cĀ°
rĀ°
dĀ°
sĀ°
The key fact about parallel lines is that when a transversal intersects parallel lines, corre-
sponding angles are equal. Here is an activity introducing the idea.
A sheet of lined paper has many parallel lines.
Draw a slanted line.
Compare the corresponding angles.
Are they equal?
62 ā¢ CHAPTER 3. FINDING UNKNOWN ANGLES
This lined paper activity provides experimental evidence, but it is not the kind of logical
argument required in geometry. Instead, the value of the activity is psychological: it serves to
make the stated fact clear and believable to students.
If a transversal intersects two parallel lines, then corre-
sponding angles are equal, i.e.,
if AB ā„ CD , then a = p.
(Abbreviation: corr. ā s, AB ā„ CD .)
aĀ°
pĀ°
A B
C D
In the blue box, the same fact is stated three times: first in words, then as a labeled picture,
and then again as an abbreviation. This presentation gives students three ways to understand
and remember the fact.
Notice that the abbreviation says ācorr. ā sā and then identifies which pair of parallel lines
we are using. Asking students to name the parallel lines is important for clarity. It reminds them
that this fact requires parallel lines. It is also a courtesy to the teacher who is trying to follow
the studentās reasoning.
The statement in the box above has a partner called its converse. For the converse, we
consider a transversal intersecting two lines that are not necessarily parallel. We then measure
corresponding angles. If these are equal, then we can conclude that the lines are parallel.
If a = p , then AB ā„ CD.
(Abbreviation: corr. ā s converse .)
aĀ°
pĀ°
A B
C D
Recall the principle that when two lines cross, each of the angles formed determines the
other three. The above fact gives a similar principle about the eight angles formed by a transver-
sal that intersects two parallel lines: any one angle determines all 8. In fact, among the eight
angles, four have the same measure as the given angle, and the other four have the supplemen-
tary measure.
EXAMPLE 2.1. In the figure below, AB ā„ CD. Find b , c , d , p , q , r, and s.
Teacherās Solution:
30 Ā°
qĀ° pĀ°
rĀ° sĀ°
bĀ°
cĀ° dĀ°
A B
C D
30 + b = 180 , ā s on a line
b = 150.
c = 30
d = 150
vert. ā s.
p = 30
q = 150
r = 30
s = 150
corr. ā s, AB ā„ CD.
64 ā¢ CHAPTER 3. FINDING UNKNOWN ANGLES
If a transversal intersects two parallel lines, then alter-
nate interior angles are equal, i.e.,
if AB ā„ CD , then c = p.
(Abbreviation: alt. ā s, AB ā„ CD .)
pĀ°
cĀ°
A B
C D
Conversely, if c = p , then AB ā„ CD.
(Abbreviation: alt. ā s converse .)
pĀ°
cĀ°
A B
D C
EXERCISE 2.2. Read Class Activity 4 (pages 245-246) of New Elementary Mathematics 1. What
terms are introduced in this activity? On page 246, match the statements in the blue box with
the statements in the blue boxes above.
Converses
Each fact in this section is paired with its converse. All students should be able to distinguish
a statement from its converse. The distinction is a simple point of logic that arises in many
contexts, but is rarely taught outside of geometry. A complete explanation requires only a few
sentences and some examples.
Every āif... thenā statement has a converse. The converse of the statement āIf A is true then
B is trueā is the statement obtained by reversing the roles of A and B , namely āIf B is true then
A is trueā. Here are three simple examples.
Statement: If an animal is a woman then it is human. (TRUE)
Converse: If an animal is human then it is a woman. (FALSE)
Statement: If two angles are vertically opposite, then they have equal measure. (TRUE)
Converse: If two angles have equal measure, then they are vertically opposite. (FALSE)
Statement: If a triangle is isosceles then it has two equal sides. (TRUE)
Converse: If a triangle has two equal sides then it is isosceles. (TRUE)
In the first two cases, the statement is true but its converse is false. In the third case, both the
statement and its converse are true. Thus one should not confuse a statement with its converse.
Knowing that one is true tells us nothing about the other.
The blue boxes in this section give three statements about parallel lines, each paired with its
converse. For all three facts, both the statement and its converse are true.
SECTION 3.2 FINDING ANGLES USING PARALLEL LINES ā¢ 65
Auxiliary Lines
Some geometry problems can be solved easily after modifying the given figure. If you are
stuck, it may help to extend an existing line or to add a new line; both are called auxiliary lines.
You can also give letter names to angles not named in the original figure.
Hint: Look for an auxiliary line that forms a new triangle , or one that is parallel or perpendic-
ular to an existing line.
EXAMPLE 2.3. In the figure, AB ā„ CD.
Find the value of x.
75Ā°
A B
C D
37Ā°
x Ā°
Below are three student solutions. Each uses an auxiliary line (the dotted line).
a)
75Ā°
A B
C D
37Ā°
x Ā°
a Ā° b Ā°
a = x , corr. ā s, AB ā„ CD ,
b = 37 , vert. ā s,
a + b = 75 , ext. ā s of a ā,
x + 37 = 75 ,
ā“ x = 38.
b)
75Ā°
A B
C D
37Ā°
x Ā°
b Ā°
a Ā° a + 37 = 90 , ā sum of rt. ā,
ā“ a = 53.
a + b + 75 = 180 , ā sum of ā,
53 + b + 75 = 180 ,
b + 128 = 180 ,
ā“ b = 52.
x + 52 = 90 , ā sum of rt. ā,
ā“ x = 38.
c)
75Ā°
A B
C D
37Ā°
x Ā°
b Ā°
a Ā°
L
a = 37 , corr. ā s, AB ā„ L.
37 + b = 75 , ā s add,
ā“ b = 38.
ā“ x = 38 , corr. ā s, L ā„ DC.
Caution: In c), the line L was draw parallel to AB. It does not bisect the center angle. (One
could draw the actual angle bisector, but then it would not be parallel to AB or CD ).
SECTION 3.3 ANGLES OF A POLYGON ā¢ 67
curricula. It is a simple example of ābuilding new facts from known onesā, and the facts learned
can be used to solve interesting unknown angle problems. Here, for the first time, students learn
to make statements about n -sided polygons without specifying the number n. It is an ideal place
in the curriculum for discussing the distinction between inductive and deductive reasoning.
In the early grades, polygons are usually regarded as regions in the plane whose boundary
is a union of straight segments. Students are given the school definition below, and lots of
examples (as described in Section 2.2). In the school definition, polygons have interiors, so the
meaning of āinterior anglesā is clear.
School Definition: A region enclosed by 3 straight segments is a triangle.
A region enclosed by n straight segments is an n -sided polygon.
Children also learn to use the correct mathematical definition, in which a polygon is a col-
lection of line segments. The definition is based on the same idea as connect-the-dots puzzles:
draw a segment from the first point to the second, from the second to the third, etc., and end by
connecting the last to the first point, thereby āclosing upā the figure. The definition is clearest
when given in two parts.
DEFINITION 3.1. Given n ā„ 3 points P 1
, P
2
,... , P
n
, all di ff erent, the corresponding closed
polygonal path is the collection of segments P 1
P
2
, P
2
P
3
,... , P
n
P
1
. The points are called the
vertices and the segments are called the sides of the polygonal path.
a polygon!
a closed polygonal path
P
3
P
4
P
5
P
1
P 2
DEFINITION 3.2. An n -sided polygon or n -gon is a closed polygonal path in a plane with n ā„ 3
vertices such that
(i) the sides intersect only at their endpoints and
(ii) no adjacent sides are collinear.
Conditions (i) and (ii) may seem awkward, but they are needed to
make this definition compatible with the school definition above.
Polygons separate the plane into two regions, the interior and the
exterior. A polygon together with its interior is called a polygonal
region ; the school definition is actually the definition of a polygonal
region. Condition (ii) ensures that the count of sides and vertices is
the same as the count obtained from the school definition.
polygonal region
polygon
68 ā¢ CHAPTER 3. FINDING UNKNOWN ANGLES
EXERCISE 3.3. Which of the figures below is a polygon? Which violates the requirement that
the vertices be distinct? Is figure D a triangle or a quadrilateral?
A B C D
Definition 3.2 is different from the school definition, yet the two coexist through elementary
and middle school. Sometimes the word āpolygonā refers to a union of segments, and some-
times it means a region. In most textbooks, including the always-careful Primary Mathematics
and New Elementary Mathematics books, the meaning of words like ātriangleā, ārectangleā and
āpolygonā shifts according to the topic being covered.
- When discussing area, āpolygonsā are regions.
- When finding unknown angles, polygons are unions of segments.
- For some topics, the distinction is not important. One example is the topic on the next
page: the sum of interior angles.
Teachers should be alert to possible confusion. When clarity is needed, both teachers and
students should speak of ātriangular regionsā and āpolygonal regionsā. This long-winded ter-
minology becomes tiresome in studying topics, such as area, where one is always considering
regions. In such situations, it is fine to say ātriangleā and āquadrilateralā instead of ātriangular
regionā and āquadrilateral regionā provided that all students are aware that the words refer to
regions.
The parts of a polygon are named using terms students learned when studying triangles and
quadrilaterals: vertex, side, diagonal, interior angle, and exterior angle. Look at page 271 in
New Elementary Mathematics 1 to see how the terms ( vertex , side , diagonal , interior angle ,
and exterior angle ) are reviewed for 7th grade students simply by drawing a single picture.
Two other terms frequently enter discussions of polygons. The first gives a name to the
most commonly-seen examples of polygons.
DEFINITION 3.4. A polygon is regular if (i) all sides have equal length, and (ii) all angles have
equal measure.
Most children are familiar with the regular polygons below. In fact, teachers should be sure
that students realize that words like āpentagonā do not automatically refer to a regular pentagon.
equilateral triangle square regular pentagon regular hexagon
70 ā¢ CHAPTER 3. FINDING UNKNOWN ANGLES
Interior Point Method. Choose a point in the interior. Draw lines from that interior point to
every vertex, thereby decomposing the n -gon into n triangles.
There are 180Āŗ in each triangle.
The remaining angles form the interior angles of the polygon.
Sum of interior angles = 180Ć n - 360
= 180 Ć (n-2).
Center angles total 360Āŗ.
The case of non-convex polygons is more complicated. But by drawing pictures like the one
below, you should be able to convince yourself that every non-convex polygonal region with n
sides can be partitioned into ( n ā 2) triangular regions ā in fact there are often many ways of
doing this.
A 7-gon decomposed into 5 triangles (in two different ways).
The sum of the interior angles of an n -gon is 180( n ā 2) degrees.
(Abbreviation: ā sum of n -gon. )
EXERCISE 3.7. Do Problem 1 in Class Activity 3 of NEM1 on page 272.
a) What do you conclude about the sum of interior angles of an n-sided polygon?
b) Is this activity an example of building a fact inductively or deductively?
Sum of the Exterior Angles
At each vertex of a convex polygon there is an interior angle and also two exterior angles.
There is a formula for the sum of the exterior angles that is analogous to the one for interior
angles, but simpler. Here are three classroom explanations.
Racetrack Method. Imagine the polygon as a racetrack.
A car starts on one side and moves around the track counter-
clockwise. At the first vertex it turns left through an angle
equal to the exterior angle at that vertex. When it gets to
the second vertex it turns again, by an amount equal to the
second exterior angle. When the car returns to its starting
point it has completed one full turn ā 360 degrees. Thus
the sum of the exterior angles of the polygon is 360
ā¦
.
turn
STA
RT
SECTION 3.3 ANGLES OF A POLYGON ā¢ 71
Zoom-out Method. Moving around the polygon in one direction, extend each side to a ray.
Then āzoom-outā, looking at the polygon from farther and farther away. From 1000 miles away,
the figure looks like a point and the exterior angles clearly add to 360
ā¦
.
360Āŗ
up close far away very far away
Base Point Method. Fix a point P and draw
segments at P parallel to the sides of the poly-
gon as shown. This creates angles congruent
to the exterior angles of the polygon whose
sum is 360
ā¦
.
P
a
a
b
b
c
c
d
d
e
e
The sum of the exterior angles, one at each vertex, of a convex polygon is 360
ā¦
.
(Abbreviation: ext. ā s of polygon. )
EXERCISE 3.8. In the above āBase Point Methodā picture, why are the two angles labeled a
congruent to each other?
Recall that there are two exterior angles at each vertex. In the above explanations, the
exterior angles were chosen in a consistent manner (the ones that arise by āturning leftā at each
vertex). But there is no need to be consistent: because the two exterior angles at each vertex are
congruent, we can arbitrarily choose one external angle at each vertex and still have a sum of
ā¦
.
EXAMPLE 3.9. What is the measure of each interior angle of a regular 9-gon?
Teacherās Solution:
140Āŗ
40Āŗ
Sum of exterior angles: 360 ,
Each exterior angle: 360Ć· 9 = 40 ,
Each interior angle: 180-40=.
Each interior angle of a regular 9 -gon is 140 Āŗ.
In your homework, you will consider whether the sum of the exterior angles is 360
ā¦
for
polygons that are not convex.
