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3.6 Prove Theorems About Perpendicular Lines
Objective: Find the distance between a point and a line
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What can you conclude if…
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Theorem 3.8
• If 2 lines intersect to form a linear pair of congruent angles, then the lines must be perpendicular.
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What can you conclude if…
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Theorem 3.9
• If 2 lines are ┴ then they form 4 congruent angles.
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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.
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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
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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.
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What can you conclude if…
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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.
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What can you conclude if…
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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.
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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.
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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
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Distance From a Point to a Line
• Length of the perpendicular segment from the point to a line
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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.
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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
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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
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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
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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
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Homework
• 1 – 27, 29 – 31
• Bonus: 28, 35 – 38