Download gem 3.6

Warm-Up1Exercises
EXAMPLE
Draw Conclusions
In the diagram, AB BC. What
can you conclude about 1 and
2?
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.
Warm-Up2Exercises
EXAMPLE
Prove Theorem 3.10
Prove that if two sides of two adjacent
acute angles are perpendicular, then the
angles are complementary.
Given
Prove
ED
EF
7 and
8 are complementary.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1 and 2
1. Given that ABC  ABD, what can
you conclude about 3 and 4?
Explain how you know.
ANSWER
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),
3 and 4 are complementary.
Warm-Up3Exercises
EXAMPLE
Draw Conclusions
Determine which lines, if any, must be
parallel in the diagram. Explain your
reasoning.
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.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 3
Use the diagram at the right.
3. Is b || a? Explain your reasoning.
4. Is b
c? Explain your reasoning.
ANSWER
3. yes; Lines Perpendicular to a Transversal Theorem.
4. yes; c || d by the Lines Perpendicular to a Transversal
Theorem, therefore b c by the Perpendicular
Transversal Theorem.
Warm-Up4Exercises
EXAMPLE
Find the distance between two parallel lines
SCULPTURE: The sculpture
on the right is drawn on a
graph where units are
measured in inches. What is
the approximate length of
SR, the depth of a seat?
Warm-Up4Exercises
EXAMPLE
Find the distance between two parallel lines
SOLUTION
You need to find the length of a perpendicular segment
from a back leg to a front leg on one side of the chair.
Using the points P(30, 80) and R(50, 110), the slope of
each leg is 110 – 80 = 30 = 3 .
2
20
50 – 30
The segment SR has a slope of 120 – 110 = – 10 = – 2 .
3
15
35 – 50
The segment SR is perpendicular to the leg so the
distance SR is
d=
(35 – 50)2 + (120 – 110)2
18.0 inches.
The length of SR is about 18.0 inches.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 4
7. 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.
ANSWER
(2, 3); 2.8
Daily
Homework
Quiz
Warm-Up
Exercises
3. Find the distance between the two parallel lines.
Round to the nearest tenth.
ANSWER
6.4
Daily
Homework
Quiz
Warm-Up
Exercises
1. Find m
ANSWER
3.
18°
2. How do you know that a and b are parallel?
ANSWER
Both are perpendicular to c.