Transcript 3.6 Prove Theorems About Perpendicular Lines

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
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.
EXAMPLE 2
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.
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
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.
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.
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.
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
d=
(35 – 50)2 + (120 – 110)2
18.0 inches.
The length of SR is about 18.0 inches.
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
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
Daily Homework Quiz
1. Find m
ANSWER
For use after Lesson 3.6
3.
18°
2. How do you know that a and b are parallel?
ANSWER
Both are perpendicular to c.
Daily Homework Quiz
For use after Lesson 3.6
3. Find the distance between the two parallel lines.
Round to the nearest tenth.
ANSWER
6.4
Homework
• 1 – 27, 29 – 31
• Bonus: 28, 35 – 38