3.6 Prove Theorems About Perpendicular Lines

Objectives

• Recognize relationships within lines

• Prove that two lines are parallel based on given information

Theorems

• Theorem 3.8 If 2 lines intersect to form a linear pair of s, then the lines are .

• Theorem 3.9If 2 lines are , then they intersect to form 4 right s.

• Theorem 3.10If 2 sides of 2 adjacent acute s are , then the s are complementary.

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.

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.

YOUR TURN

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

Theorems

• Theorem 3.11( Transversal Theorem)If a transversal is to one or two || lines, then it is to the other.

• Theorem 3.12 (Lines to a Transversal Theorem)In a plane, if 2 lines are to the same line, then they are || 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.

YOUR TURN

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

The distance from a line to a point not on the line is the length of the segment ┴ to the line from the point.

l

A

Distance Between Parallel Lines

• Two lines in a plane are || if they are equidistant everywhere.

• To verify if two lines are equidistant find the distance between the two || lines by calculating the distance between one of the lines and any point on the other line.

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?

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.

YOUR TURN

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

YOUR TURN

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

Assignment

Geometry:Pg. 194 – 197 #2 – 10, 13 – 24, 26, 31, 35 – 38