Download geometry module 1 lesson 23 base angles of isosceles triangles

GEOMETRY
MODULE 1 LESSON 23
BASE ANGLES OF ISOSCELES TRIANGLES
OPENING EXERCISE
Describe the additional piece of information needed for each pair of triangles to satisfy the SAS
triangle congruence criteria.

Given: 𝐴𝐵 = 𝐷𝐶
Additional Info Necessary:
We need information to show congruent angles.
Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐶𝐵

̅̅̅̅ ∥ 𝑅𝑆
̅̅̅̅
Given: 𝐴𝐵 = 𝑅𝑆 and 𝐴𝐵
Additional Info Necessary:
We need information to show congruent bases.
Prove: ∆𝐴𝐵𝐶 ≅ ∆𝑅𝑆𝑇
DISCUSSION
Consider the isosceles triangle. We accept that an isosceles triangle, which
has (at least) two congruent sides, also has congruent base angles.
We can prove that 𝑚∠𝐵 = 𝑚∠𝐶 in two ways: Transformations and SAS.
What transformation can we use to map ∠𝐵 onto ∠𝐶?
Reflection on the line of symmetry.
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Let’s prove ∠𝐵 ≅ ∠𝐶 through SAS. To help with our proof, we
construct an angle bisector.
STEP
JUSTIFICATION
𝐴𝐵 = 𝐴𝐶
Given
𝐴𝐷 = 𝐴𝐷
Reflexive Property
𝑚∠𝐴𝐵𝐷 = 𝑚∠𝐴𝐶𝐷
Definition of Angle Bisector
∆𝐴𝐵𝐷 ≅ ∆𝐴𝐶𝐷
SAS
∠𝐵 ≅ ∠𝐶
Corresponding angles of congruent triangles are
congruent.
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PRACTICE
1. Given: 𝐽𝐾 = 𝐽𝐿 and ̅̅̅
𝐽𝑅 bisects ̅̅̅̅
𝐾𝐿.
̅̅̅ ⊥ 𝐾𝐿
̅̅̅̅
Prove: 𝐽𝑅
STEP
JUSTIFICATION
̅̅̅ bisects 𝐾𝐿
̅̅̅̅.
𝐽𝐾 = 𝐽𝐿 and 𝐽𝑅
Given
∠𝐾 = ∠𝐿
Base angles of an isosceles triangle are
congruent.
𝐾𝑅 = 𝐾𝐿
Definition of segment bisector
∆𝐽𝑅𝐾 ≅ ∆𝐽𝑅𝐿
SAS
∠𝐽𝑅𝐾 ≅ ∠𝐽𝑅𝐿
Corresponding angles of congruent triangles
are congruent.
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𝑚∠𝐽𝑅𝐾 + 𝑚∠𝐽𝑅𝐿 = 180°
Linear pairs are supplementary.
2(𝑚∠𝐽𝑅𝐾) = 180°
Substitution
𝑚∠𝐽𝑅𝐾 = 90°
Division
̅̅̅
𝐽𝑅 ⊥ ̅̅̅̅
𝐾𝐿
Definition of perpendicular lines
3
2. Given: 𝐴𝐵 = 𝐴𝐶, 𝑋𝐵 = 𝑋𝐶.
̅̅̅̅ bisects ∠𝐵𝐴𝐶
Prove: 𝐴𝑋
1
2
STEP
JUSTIFICATION
𝐴𝐵 = 𝐴𝐶
Given
𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐴𝐶𝐵
Base angles of an isosceles triangle are equal in
measure.
𝑋𝐵 = 𝑋𝐶
3
4
𝑚∠𝑋𝐵𝐶 = 𝑚∠𝑋𝐶𝐵
Given
Base angles of an isosceles triangle are equal in
measure.
5
𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐴𝐵𝑋 + 𝑚∠𝑋𝐵𝐶
Angle Addition
𝑚∠𝐴𝐶𝐵 = 𝑚∠𝐴𝐶𝑋 + 𝑚∠𝑋𝐶𝐵
6
𝑚∠𝐴𝐵𝑋 = 𝑚∠𝐴𝐵𝐶 − 𝑚∠𝑋𝐵𝐶
Subtraction
𝑚∠𝐴𝐶𝑋 = 𝑚∠𝐴𝐶𝐵 − 𝑚∠𝑋𝐶𝐵
7
𝑚∠𝐴𝐵𝑋 = 𝑚∠𝐴𝐵𝐶 − 𝑚∠𝑋𝐵𝐶
Substitution from lines 2 and 4
𝑚∠𝐴𝐶𝑋 = 𝑚∠𝐴𝐵𝐶 − 𝑚∠𝑋𝐵𝐶
8
𝑚∠𝐴𝐵𝑋 = 𝑚∠𝐴𝐶𝑋
Transitive
9
∆𝐴𝐵𝑋 ≅ ∆𝐴𝐶𝑋
SAS
10
Corresponding angles of congruent triangles are
∠𝐵𝐴𝑋 ≅ ∠𝐶𝐴𝑋
congruent.
11
̅̅̅̅
𝐴𝑋 bisects ∠𝐵𝐴𝐶
Definition of Angle Bisector
SUMMARY
Today’s lesson used properties of isosceles triangles to complete proofs.
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