Download Geometry Fall 2016 Lesson 030 _Proving lines are perpendicular

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Lesson Plan #30
Date: Wednesday November 23rd, 2016
Class: Geometry
Topic: How do we prove lines are perpendicular?
Aim: How do we prove lines are perpendicular?
Do your HW the same day
that
this
lesson
is
completed. Start by taking
a blank sheet of paper, and
completing
a
proper
heading at the top of the
paper, as per the rules
sheet given out the 1st day
of class. Then complete the
HW assignment on that
sheet of paper. Have the
HW ready for collection by
the next school day.
HW #30:
Note:
Substitution Postulate for inequalities
If 𝑎, 𝑏 and 𝑐 are real numbers, such that 𝑎 > 𝑏 and 𝑐 = 𝑏, then 𝑎 > 𝑐.
For example 7 > 5 and 𝑥 = 5, therefore 7 > 𝑥.
Postulate
If a, b, c and d are real numbers, such that 𝑎 > 𝑏 and 𝑐 = 𝑑, then 𝑎 + 𝑐 > 𝑏 + 𝑑
For example 7 > 5 and 2 = 2, then 7 + 2 > 5 + 2
Theorem (presented without proof) - The measure of an exterior angle of a triangle is greater than the measure of either
non adjacent interior angle.
Example:
Objectives:
1) Students will be able to prove two lines
perpendicular.
Do Now:
1. Construct a line perpendicular
to AB that goes through
point D.
D
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PROCEDURE:
Write the Aim and Do Now
Get students working!
Take attendance
Give Back HW
Collect HW
Go over the Do Now
Assignment #1: Given: Δ𝐴𝐵𝐶 with < 𝐴𝐷𝐶 ≅ < 𝐵𝐷𝐶
Prove: ̅̅̅̅
𝐶𝐷 ⊥ ̅̅̅̅
𝐴𝐵
Statements
1. < 𝐴𝐷𝐶+< 𝐵𝐷𝐶 = 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒
2. 𝑚 < 𝐴𝐷𝐶 + 𝑚 < 𝐵𝐷𝐶 = 180
3. < 𝐴𝐷𝐶 ≅ < 𝐵𝐷𝐶
4. 𝑚 < 𝐴𝐷𝐶 = 𝑚 < 𝐵𝐷𝐶
5. 𝑚 < 𝐴𝐷𝐶 + 𝑚 < 𝐴𝐷𝐶 = 180
Or
2𝑚 < 𝐴𝐷𝐶 = 180
6. 𝑚 < 𝐴𝐷𝐶 = 90
7. 𝑚 < 𝐵𝐷𝐶 = 90
8. <ADC and <BDC are right angles
9. ̅̅̅̅
𝐶𝐷 ⊥ ̅̅̅̅
𝐴𝐵
Reasons
1.Definition of a linear pair
2.
3. Given
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5. Substitution Postulate ( )
6.
7. Substitution Postulate (
)
9. Definition of perpendicular lines (
)
Theorem: If two intersecting lines form congruent adjacent angles,
the lines are perpendicular.
Assignment #1: For what values of x and y
will the lines L1 and L2 be perpendicular?
Assignment #2:
Statements
Given:
DCF , ACB
AD  BD
AF  BF
Prove:
1) ADF
 BDF
2)
 ADC  BDC
3)
ADC  BDC
4)
DF bisects AB
5)
DF  AB
Reasons
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Theorem: If two points are each equidistant from the endpoints of a line
segment, the points determine the perpendicular bisector of the line segment.
Corollary: Any point on the perpendicular bisector is equidistant from the
endpoints of the line segment.
Summary of ways to prove that lines are perpendicular
1) When two lines intersect, they form right angles.
2) When two lines intersect, they form congruent adjacent angles
3) There are two points on a line segment each of which is equidistant
from the endpoints of the other line segment.
Given:
Prove:
Assignment: Do the proofs below on a separate sheet of paper.
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Sample Test Question:
1)
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If enough time:
1)
2)
3)
4)
5)
Final Summary: What are ways to prove that lines are perpendicular?