1. 3.6 Prove Theorems About Perpendicular Lines
Objective: Find the distance between a point and a line

2. What can you conclude if…

3. Theorem 3.8
If 2 lines intersect to form a linear pair of congruent angles, then the lines must be perpendicular.

4. What can you conclude if…

5. Theorem 3.9
If 2 lines are ┴ then they form 4 congruent angles.

6. 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.

7. 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

8. 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.

9. What can you conclude if…

10. Theorem 3.11 Perpendicular Transversal Theorem:
If a transversal is perpendicular to one of 2 parallel lines, then it’s perpendicular to both of them.

11. What can you conclude if…

12. Theorem 3.12 Lines Perpendicular to a Transversal Theorem:
If 2 lines are perpendicular to the same line, then they are parallel to each other.

13. 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.

14. 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

15. Distance From a Point to a Line
Length of the perpendicular segment from the point to a line

16. 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.

17. 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

18. 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

19. Daily Homework Quiz
For use after Lesson 3.6 1.
Find m
18°
ANSWER 2.
How do you know that a and b are parallel?
Both are perpendicular to c.
ANSWER

20. Daily Homework Quiz
For use after Lesson 3.6 3.
Find the distance between the two parallel lines.
Round to the nearest tenth.
6.4
ANSWER

21. Homework
1 – 27, 29 – 31
Bonus: 28, 35 – 38