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Student Outcomes
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: π΄π΅ = π·πΆ
Prove: β³ π΄π΅πΆ β β³ π·πΆπ΅
b. Given: π΄π΅ = π π π΄π΅ β₯ π π
Prove: β³ π΄π΅πΆ β β³ π πT
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 π΄π΅πΆ. 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 β³ π΄π΅πΆ, with π΄π΅ = π΄πΆ
Prove: πβ π΅ = πβ πΆ
Construction: Draw the angle bisector π΄π· of β π΄, where π· is the intersection of the bisector and π΅πΆ. We need to show that rigid motions maps point π΅ to point πΆ and point πΆ to point π΅.
Let π be the reflection through π΄π· . Through the reflection, we want to demonstrate two pieces of information that map π΅ to point πΆ and vice versa: (1) π΄π΅ maps to π΄πΆ, and (2) π΄π΅ = π΄πΆ.
Since π΄ is on the line of reflection, π΄π·, π(π΄) = π΄. Reflections preserve angle measures, so the measure of the reflected angle π(β π΅π΄π·) equals the measure of β πΆπ΄π·; therefore, π(π΄π΅) = π΄πΆ. Reflections also preserve lengths of segments; therefore, the reflection of π΄π΅ still has the same length as π΄π΅. By hypothesis, π΄π΅ = π΄πΆ, so the length of the reflection is also equal to π΄πΆ. Then π(π΅) = πΆ. Using similar reasoning, we can show that π(πΆ) = π΅.
Reflections map rays to rays, so π(βπ΅π΄) = πΆπ΄ and π(π΅πΆ) = πΆπ΅. Again, since reflections preserve angle measures, the measure of π(β π΄π΅πΆ) is equal to the measure of β π΄πΆπ΅.
We conclude that πβ π΅ = πβ πΆ. Equivalently, we can state that β π΅ β β πΆ. In proofs, we can state that βbase angles of an isosceles triangle are equal in measureβ or that βbase angles of an isosceles triangle are congruent.β
Prove Base Angles of an Isosceles are Congruent: SAS
Given: Isosceles β³ π΄π΅πΆ, with π΄π΅ = π΄πΆ
Prove: β π΅ β
β πΆ
Construction: Draw the angle bisector π΄π· of β π΄, where π· is the intersection of the bisector and π΅πΆ. We are going to use this auxiliary line towards our SAS criteria.
Exercises
Given: π½πΎ = π½πΏ; π½π
bisects πΎπΏ
Prove: π½π
β₯ πΎπΏ Μ
Μ
Μ
Μ
Given: π΄π΅ = π΄πΆ, ππ΅ = ππΆ
Prove: π΄π bisects β π΅π΄πΆ
Given: π½π = π½π, πΎπ = πΏπ
Prove: β³ π½πΎπΏ is isosceles
Given: β³ π΄π΅πΆ, with πβ πΆπ΅π΄ = πβ π΅πΆπ΄
Prove: π΅π΄ = πΆπ΄
(Converse of base angles of isosceles triangle)
Hint: Use a transformation.
Given: β³ π΄π΅πΆ, with ππis the angle bisector of β π΅ππ΄, and π΅πΆ β₯ ππ
Prove: ππ΅ = ππΆ
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