Tenth Grade · Math · week 16

Isosceles and equilateral triangle theorems

Apply the Base Angles Theorem and its converse, and the equilateral-equiangular corollary.

An isosceles triangle has at least two congruent sides, called legs, meeting at the vertex angle; the third side is the base, and the two angles touching the base are the base angles. The Isosceles Triangle Theorem (the Base Angles Theorem) says those base angles are always congruent. Its converse is also true: if two angles of a triangle are congruent, the sides opposite them are congruent -- congruent angles and congruent sides come as a package in an isosceles triangle, and either one proves the other.

A quick proof shows why: draw the bisector of the vertex angle down to the base, splitting the triangle into two smaller triangles that share that bisector as a side. Those two triangles are congruent by SAS (the two legs, the bisected angle, and the shared side), and CPCTC hands over the congruent base angles for free. A useful corollary follows immediately: an equilateral triangle is also equiangular, with each angle exactly 60 degrees, since all three sides being congruent forces all three angles to be congruent too, and they still must sum to 180 degrees.

📋 For the teacher
Teach it (10–15 min):
  • Fold a paper isosceles triangle along its line of symmetry (vertex angle to base midpoint) to physically show the two base angles landing exactly on top of each other.
  • Walk through the bisector proof once together, pointing out exactly where SAS and then CPCTC each do their job -- this pattern (auxiliary line, then SAS, then CPCTC) reappears often.
  • Drill the converse direction with its own examples (congruent angles given, congruent sides concluded) so it is not treated as an afterthought.
  • Have them derive "equilateral means equiangular, each angle 60 degrees" themselves from the Base Angles Theorem plus the Triangle Sum Theorem, rather than just being told.
Talk about it:
  • Why does bisecting the vertex angle create two congruent smaller triangles?
  • Why are the Base Angles Theorem and its converse considered two separate facts, even though they are so closely related?
  • Why must every angle of an equilateral triangle be exactly 60 degrees, no matter how big the triangle is?

Mastery looks like: They apply the Base Angles Theorem and its converse to find unknown angles or sides in isosceles triangles, and know every equilateral triangle has three 60-degree angles.

Common stumbles: Assuming the base angles are whichever two angles look like they are "on the bottom" instead of the two opposite the congruent legs; forgetting the converse runs the opposite direction (angles to sides).

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Records attendance + 15 min of Math for this learner.
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