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Warm-Up Exercises
Lesson 3.6, For use with pages 190-197
1. What is the distance between the points (2, 3)
and (5, 7)?
ANSWER
2. If m
5
DBC = 90°, what is m
ANSWER
90°
ABD?
Warm-Up Exercises
Theorems
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-Up Exercises
Theorem
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-Up Exercises
Theorem
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
18.0 inches.
(35 – 50)2 + (120 –
110)2
The length of SR is about 18.0 inches.
d=
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 4
Use the graph at the right for Exercises
5 and 6.
5. What is the distance from point A to
line c?
6. What is the distance from line c to
line d?
ANSWER
5. about 1.3
6. about 2.2
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