1. 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°

2. EXAMPLE 1
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 are right angles, so 1  2.

3. EXAMPLE 2
Prove Theorem 3.10
Prove that if two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Given ED EF
Prove and are complementary.

4. GUIDED PRACTICE
for Examples 1 and 2
Given that ABC  ABD, what can you conclude about and 4? Explain how you know.
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), and are complementary.
ANSWER

5. EXAMPLE 3
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.

6. GUIDED PRACTICE
for Example 3
Use the diagram at the right Is b || a? Explain your reasoning.
4. Is b c? Explain your reasoning.
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.
ANSWER

7. EXAMPLE 4
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?

8. EXAMPLE 4
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 20
50 – 30 3 2 .
The segment SR has a slope of
120 – 110 = 10 15
35 – 50 – 2 3 .
The segment SR is perpendicular to the leg so the distance SR is
(35 – 50)2 + (120 – 110)2
18.0 inches.
d =
The length of SR is about 18.0 inches.

9. GUIDED PRACTICE
for Example 4
Use the graph at the right for Exercises 5 and What is the distance from point A to line c?
6. What is the distance from line c to line d?
5. about 1.3
6. about 2.2
ANSWER

10. 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.
(2, 3); 2.8
ANSWER

11. Daily Homework Quiz 1.
Find m
18°
ANSWER 2.
How do you know that a and b are parallel?
Both are perpendicular to c.
ANSWER

12. Daily Homework Quiz 3.
Find the distance between the two parallel lines.
Round to the nearest tenth.
6.4
ANSWER