Base Angles of Isosceles Triangles

New York State Common Core Math Geometry, Module 1, Lesson 23

Worksheets for Geometry

Student Outcomes

• Students examine two different proof techniques via a familiar theorem.
• Students complete proofs involving properties of an isosceles triangle.

Base Angles of Isosceles Triangles

Classwork

Opening Exercise

Describe the additional piece of information needed for each pair of triangles to satisfy the SAS triangle congruence criteria.

a. Given: 𝐴𝐵 = 𝐷𝐶

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Prove: △ 𝐴𝐵𝐶 ≅△ 𝐷𝐶𝐵

b. Given: 𝐴𝐵 = 𝑅𝑆 𝐴𝐵 ∥ 𝑅𝑆

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Prove: △ 𝐴𝐵𝐶 ≅△ 𝑅𝑆T

• Show Video for Whole Lesson

Exploratory Challenge

Today we examine a geometry fact that we already accept to be true. We are going to prove this known fact in two ways: (1) by using transformations and (2) by using SAS triangle congruence criteria.

Here is isosceles triangle 𝐴𝐵𝐶. We accept that an isosceles triangle, which has (at least) two congruent sides, also has congruent base angles.

Label the congruent angles in the figure.

Now we prove that the base angles of an isosceles triangle are always congruent.

Prove Base Angles of an Isosceles are Congruent: Transformations

Given: Isosceles △ 𝐴𝐵𝐶, with 𝐴𝐵 = 𝐴𝐶
Prove: 𝑚∠𝐵 = 𝑚∠𝐶

Construction: Draw the angle bisector 𝐴𝐷 of ∠𝐴, where 𝐷 is the intersection of the bisector and 𝐵𝐶. We need to show that rigid motions maps point 𝐵 to point 𝐶 and point 𝐶 to point 𝐵.

Let 𝑟 be the reflection through 𝐴𝐷 . Through the reflection, we want to demonstrate two pieces of information that map 𝐵 to point 𝐶 and vice versa: (1) 𝐴𝐵 maps to 𝐴𝐶, and (2) 𝐴𝐵 = 𝐴𝐶.

Since 𝐴 is on the line of reflection, 𝐴𝐷, 𝑟(𝐴) = 𝐴. Reflections preserve angle measures, so the measure of the reflected angle 𝑟(∠𝐵𝐴𝐷) equals the measure of ∠𝐶𝐴𝐷; therefore, 𝑟(𝐴𝐵) = 𝐴𝐶. Reflections also preserve lengths of segments; therefore, the reflection of 𝐴𝐵 still has the same length as 𝐴𝐵. By hypothesis, 𝐴𝐵 = 𝐴𝐶, so the length of the reflection is also equal to 𝐴𝐶. Then 𝑟(𝐵) = 𝐶. Using similar reasoning, we can show that 𝑟(𝐶) = 𝐵.

Reflections map rays to rays, so 𝑟(⃗𝐵𝐴) = 𝐶𝐴 and 𝑟(𝐵𝐶) = 𝐶𝐵. Again, since reflections preserve angle measures, the measure of 𝑟(∠𝐴𝐵𝐶) is equal to the measure of ∠𝐴𝐶𝐵.

We conclude that 𝑚∠𝐵 = 𝑚∠𝐶. Equivalently, we can state that ∠𝐵 ≅ ∠𝐶. In proofs, we can state that “base angles of an isosceles triangle are equal in measure” or that “base angles of an isosceles triangle are congruent.”

Prove Base Angles of an Isosceles are Congruent: SAS

Given: Isosceles △ 𝐴𝐵𝐶, with 𝐴𝐵 = 𝐴𝐶
Prove: ∠𝐵 ≅ ∠𝐶

Construction: Draw the angle bisector 𝐴𝐷 of ∠𝐴, where 𝐷 is the intersection of the bisector and 𝐵𝐶. We are going to use this auxiliary line towards our SAS criteria.

Exercises

1. Given: 𝐽𝐾 = 𝐽𝐿; 𝐽𝑅 bisects 𝐾𝐿
   Prove: 𝐽𝑅 ⊥ 𝐾𝐿 ̅̅̅̅

2. Given: 𝐴𝐵 = 𝐴𝐶, 𝑋𝐵 = 𝑋𝐶
   Prove: 𝐴𝑋 bisects ∠𝐵𝐴𝐶

3. Given: 𝐽𝑋 = 𝐽𝑌, 𝐾𝑋 = 𝐿𝑌
   Prove: △ 𝐽𝐾𝐿 is isosceles

4. Given: △ 𝐴𝐵𝐶, with 𝑚∠𝐶𝐵𝐴 = 𝑚∠𝐵𝐶𝐴
   Prove: 𝐵𝐴 = 𝐶𝐴
   (Converse of base angles of isosceles triangle)
   Hint: Use a transformation.

5. Given: △ 𝐴𝐵𝐶, with 𝑋𝑌is the angle bisector of ∠𝐵𝑌𝐴, and 𝐵𝐶 ∥ 𝑋𝑌
   Prove: 𝑌𝐵 = 𝑌𝐶

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