1. 3.63.6 Prove Theorems about Perpendicular Lines
Bell Thinger
1. What is the distance between the points (2, 3)
and (5, 7)?
ANSWER 5
2. If m DBC = 90°, what is m ABD?
ANSWER 90°
3. 3.6Example 1
SOLUTION
AB and BC are perpendicular, so by Theorem 3.9, they
form four right angles. You can conclude that 1 and
2 are right angles, so 1 2.
In the diagram, AB BC. What
can you conclude about 1 and 2?
5. 3.6Example 2
Prove that if two sides of two adjacent
acute angles are perpendicular, then the
angles are complementary.
GIVEN: ED EF
PROVE: 7 and 8 are complementary.
6. 3.6Guided Practice
Given that ABC ABD, what can
you conclude about 3 and 4?
Explain how you know.
1.
They are complementary.
Sample Answer: ABD is a right angle since 2 lines
intersect to form a linear pair of congruent angles
(Theorem 3.8), so BA CD. Then 3 and 4 are
complementary by Theorem 3.10.
ANSWER
7. 3.6Guided Practice
Write a plan for a proof for Theorem 3.9, that if two
lines are perpendicular, then they intersect to
form four right angles.
2.
The definition of perpendicular lines implies that
angles formed by the intersecting lines are right
angles.
SAMPLE ANSWER
9. 3.6Example 3
SOLUTION
Lines p and q are both perpendicular to s, so by
Theorem 3.12, p || q. Also, lines s and t are both
perpendicular to q, so by Theroem 3.12, s || t.
Determine which lines, if any, must be
parallel in the diagram. Explain your
reasoning.
10. 3.6Guided Practice
Use the diagram at the right.
yes; Lines Perpendicular to a Transversal Theorem
ANSWER
3. Is b || a? Explain your reasoning.
4. Is b c? Explain your reasoning.
yes; c || d by the Lines Perpendicular to a Transversal
Theorem, therefore b c by the Perpendicular
Transversal Theorem
ANSWER
12. 3.6Example 4
The sculpture
is drawn on a graph where
units are measured in
inches. What is the
approximate distance from S
to PR?
SCULPTURE
13. 3.6Example 4
SOLUTION
The length of SR is about 18.0 inches.
You need to confirm that SR is perpendicular to PR.
Using the points P(30, 80) and R(50, 110), the slope of PR is
110 – 80
=
30
2050 – 30
=
3
2
.
SR has a slope of
120 – 110
=
10
1535 – 50
– =
2–
3
.
(35 – 50)2
+ (120 – 110)2
18.0 inches.d =
SR is perpendicular to PR so the distance SR is
15. 3.6Guided Practice
6. Graph the line y = x + 1. What point on the line is
the shortest distance from the point (4, 1)? What is
the distance? Round to the nearest tenth.
(2, 3); 2.8ANSWER
16. 3.6Exit Slip
1. Find m 3.
18°ANSWER
2. How do you know that a and b are parallel?
Both are perpendicular to c.
ANSWER
17. 3.6
3. Find the distance between
the two parallel lines.
Round to the nearest
tenth.
6.4ANSWER
Exit Slip
