3.6 Prove Theorems About Perpendicular Lines

Objective: Find the distance between a point and a line

What can you conclude if…

Theorem 3.8

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

What can you conclude if…

Theorem 3.9

• If 2 lines are ┴ then they form 4 congruent angles.

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 2 are right angles, so 1 2.

GUIDED PRACTICE for Examples 1 and 2

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 linesintersect to form a linear pair of congruent angles (Theorem 3.8), 3 and 4 are complementary.

ANSWER

EXAMPLE 2 Prove Theorem 3.10

Prove that if two sides of two adjacentacute angles are perpendicular, then theangles are complementary.

Given ED EF

Prove 7 and 8 are complementary.

What can you conclude if…

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.

What can you conclude if…

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.

EXAMPLE 3 Draw Conclusions

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.

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.

3. yes; Lines Perpendicular to a Transversal Theorem.4. yes; c || d by the Lines Perpendicular to a TransversalTheorem, therefore b c by the Perpendicular Transversal Theorem.

ANSWER

Distance From a Point to a Line

• Length of the perpendicular segment from the point to a line

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.

The length of SR is about 18.0 inches.

The segment SR is perpendicular to the leg so the distance SR is

(35 – 50)2 + (120 – 110)2 18.0 inches.d =

The segment SR has a slope of 120 – 110 = 1015 35 – 50

– = 2–3

.

Using the points P(30, 80) and R(50, 110), the slope of each leg is 110 – 80 = 30

20 50 – 30= 3

2.

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?

5. about 1.36. about 2.2

ANSWER

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

Daily Homework Quiz

For use after Lesson 3.6

1. Find m 3.

18°ANSWER

2. How do you know that a and b are parallel?

Both are perpendicular to c.

ANSWER

Daily Homework Quiz

For use after Lesson 3.6

3. Find the distance between the two parallel lines.Round to the nearest tenth.

6.4ANSWER

Homework

• 1 – 27, 29 – 31

• Bonus: 28, 35 – 38