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- Parallels
Given one rail of a railroad track, is there always a second rail whose (perpendicular) distance from the first rail is exactly the width across the tires of a train, so that the two rails never intersect? Of course we believe this is so. Let’s state it as a theorem.
Theorem 4.1. Given a line and a point P not on.
Then there exists a line through P parallel to `.
Okay, how do we go about proving Theorem 4.1? One idea is to use penpendiculars. Construct a line t through P and perpendicular to . Now construct a line m through P and perpendicular to t. Surely, lines m and must be parallel. Suppose you are driving on 8th Avenue in New York City. You turn right on 34th Street, go two blocks, and then turn right on 6th Avenue. Sixth Avenue must be parallel to 8th Avenue—and it is. But can we prove it?
Here goes. Suppose, to play Devil’s Advocate, that m and are not parallel. Then they must meet at a point which we will call X (the “mysterious” point). Let Q be the point of intesection of line t and. Note that ∠QP X and ∠P QX are both right angles. Now let Y be a point on the opposite ray of − QX−→ so that QY = QX.
P
Q
Y X
m
`
s (^) t s
s
s
Consider the triangles 4 P QY and 4 P QX. Since the angles ∠P QY and ∠P QX are sup- plementary and m∠P QX = 90, it follows that m∠P QY = 90. By construction QY = QX and finally the two triangles share the side P Q. By Side–Angle–Side,
4 P QY ∼= 4 P QX.
Consequently,
m∠QP Y = m∠QP X = 90. 103
104
By angle addition,
m∠Y P X = m∠Y P Q + m∠QP X = 90 + 90 = 180.
In other words, ∠Y P X is a straight angle, implying that Y , P , and X are collinear. We now have two different lines m and passing through the points X and Y. That ain’t right! One of our basic axioms states that only one line passes through two distinct points. We were led to this contradiction by assuming that m and intersect; so, in fact, m and ` must be parallel.
Now here comes the question that mystefied mathematicians for over two thousand years:
Is m the only parallel line through P?
It certainly seems plausible that any other line, different than the line constructed by per- pendiculars in the previous theorem, must meet `.
Theorem 4.2. [The Parallel Postulate] Given a line and a point P not on.
Then there exists a unique line through P parallel to `.
A line t which crosses two given lines ` and m is called a transversal. In general, eight angles are formed:
`
m
t
XXX
XXXX
XXXX
XXXX
XXXX
XXXX
X
Angles formed between the lines ` and m are called interior angles. In the above diagram, angles ∠1, ∠2, ∠5, and ∠6 are all interior angles. Note that interior angles come in pairs: ∠1 and ∠5 both lie on the left side of t, while ∠2 and ∠6 both lie on the right side of t.
106
4.11 Exercises
- Given an isosceles triangle 4 ABC, with AB = AC. Suppose
is a line parallel to ← AB→ and that meets segment AC at point P and ` meets segment BC at point Q. Prove that 4 P CQ is isosceles. - Given: ← AB→ is parallel to ← DE→ AB = DE BE = CF Prove: ← AC→ is parallel to ← DF .→
QQ
QQ
QQ
QQ
QQ
QQ
A
B C
D
E F
- One way of constructing a line through a point P parallel to a given line ` is to use the Alternate Interior Angles Theorem. Here is another way.
Given a line and a point P not on. Let A be any point on . Locate the midpoint M of AP. Draw any line through M which meets at a point C other than A. Find the point D on − CM−→ such that CD = 2 CM. Show that ← BP→ is parallel to line `.
- Angle Sums of Triangles
We are now ready to prove the most famous of all theorems in geometry.
Theorem 4.4. In 4 ABC
m∠A + m∠B + m∠C = 180.
The idea of the proof is to use the line m through A parallel to ← BC→.
- ANGLE SUMS OF TRIANGLES 107
A B
C
m ∠ 1 ∠ 2
q q
q
J
J
J
J
J
J
J
J
J
J
J
By the Alternate Interior Angle Theorem,
m∠A = m∠ 1
and
m∠B = m∠ 2.
Thus
180 = m∠1 + m∠C + m∠ 2 = m∠A + m∠C + m∠B.
We’re done.
Definition 4.1. Two angles ∠A and ∠B are said to be complementary if m∠A + m∠B = 90.
Theorem 4.5. In triangle 4 ABC, if ∠C is a right angle, then ∠A and ∠B are com- plementarry.
4.12 Exercises
- If m∠A = 40 and m∠B = 70, what is m∠C of 4 ABC?
- If m∠A = 2m∠B and m∠C of 4 ABC is 120, what is m∠A and m∠B?
- If one angle of an isosceles triangle has measure 45, what are the measures of the other two angles?
- What is the measure of an angle of an equilateral triangle?
- ANGLE SUMS OF TRIANGLES 109
- Given: B − C − D ∠ABC is a right angle ∠CDE is a right angle AB = CD BC = DE Prove: ∠ACE is a right angle.
q B
q C
A q
q
q
bb bb bb bb bb
b
a (^) b D
a
E
In the Euclidean model of the x−y plane, suppose the points A − E are given by
A = (0, b), B = (0, 0), C = (a, 0), D = (a + b, 0), E = (a + b, a)
(a) What is the slope of line ← AC→? (b) What is the slope of line ← CE→? (c) What conclusion can you make about the slopes of perpendicular lines?
