Download Geometry Fall 2012 Lesson 028 _Proving lines are perpendicular

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Lesson Plan #28
Date: Thursday November 1st, 2012
Class: Geometry
Topic: How do we prove lines are perpendicular?
Aim: How do we prove lines are perpendicular?
HW #28:
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:
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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
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.
Theorem: Any point on the perpendicular bisector is equidistant from
the endpoints of the line segment.
Theorem : If two points are each equidistant
from the endpoints of a line segment, the points
determine the perpendicular bisector 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:
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Sample Test Question:
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