Questions: Isosceles Triangle Theorem

Question 1

Triangle PQR has PQ = PR. Angles are labeled ∠P (the angle at vertex P), ∠Q (at vertex Q), and ∠R (at vertex R). Which pair of angles must be congruent?

• A. ∠P and ∠Q — the vertex angle and one base angle are always equal
• B. ∠Q and ∠R — the angles opposite the two equal sides are congruent
• C. ∠P and ∠R — the vertex angle equals the opposite base angle
• D. All three angles — equal sides always force all angles to be equal

Answer: (B)

The Isosceles Triangle Theorem states that angles opposite congruent sides are congruent. Since PQ = PR, the angles opposite those sides are ∠R (opposite PQ) and ∠Q (opposite PR). Both are base angles; ∠P is the vertex angle between the two equal sides. The common mistake is thinking the vertex angle is a base angle or that all three must be equal (that's only the equilateral case).

Question 2

The standard proof of the Isosceles Triangle Theorem draws the angle bisector from the vertex angle. Which congruence criterion establishes that the two resulting triangles are congruent?

• A. SSS — the two legs, the two halves of the base, and the bisector are all paired
• B. ASA — the bisected vertex angle, bisector, and a base angle are paired
• C. SAS — the two legs, the two halves of the bisected vertex angle, and the shared bisector
• D. AAS — two angles and a non-included side are equal in both triangles

Answer: (C)

The angle bisector from the vertex divides the isosceles triangle into two smaller triangles. For SAS, you need two sides and the included angle: (1) the two legs are given as congruent (the definition of isosceles), (2) the bisector creates two equal halves of the vertex angle (included between the leg and the bisector), and (3) the bisector itself is shared — equal to itself by the reflexive property. This gives Side-Angle-Side, establishing congruence. CPCTC then delivers the base angles as corresponding parts.

Question 3

In an isosceles triangle drawn with the vertex angle at the top, the 'base angles' are typically the two angles at the geometric bottom of the figure.

• T. True
• F. False

Answer: False

This is the most common identification error. 'Base angles' are defined by their relationship to the congruent sides — they are the angles opposite the two equal legs — not by their geometric position in a diagram. If the triangle is tilted, inverted, or oriented sideways, the base angles are still the two equal ones regardless of where they appear in the figure. The term 'base' is a conceptual label, not a positional one.

Question 4

If two angles of a triangle are equal, then the sides opposite those angles must also be equal.

• T. True
• F. False

Answer: True

This is the converse of the Isosceles Triangle Theorem, and it is also true. The converse establishes that the relationship between equal sides and equal opposite angles works in both directions. If you know angle information (two angles are equal), you can conclude side information (the sides opposite them are equal). This bidirectional relationship makes isosceles triangles the class where legs and base angles always come in matched pairs.

Question 5

Outline the key steps of the proof of the Isosceles Triangle Theorem. What role does each of SAS and CPCTC play?

Model answer: Draw the angle bisector from the vertex angle to the opposite side. This creates two smaller triangles. Apply SAS: the two legs of the original isosceles triangle are congruent (given), the angle bisector creates two equal halves of the vertex angle (included angle), and the bisector itself is shared by both smaller triangles (reflexive property). SAS establishes that the two smaller triangles are congruent. Then CPCTC (Corresponding Parts of Congruent Triangles are Congruent) delivers the conclusion: the base angles are corresponding parts of the now-proven congruent triangles, so they must be equal.

SAS does the structural work of establishing congruence; CPCTC does the logical work of extracting a conclusion about parts from the whole congruence. The clever move in this proof is creating a self-comparison — using the angle bisector to split one triangle into two that share a side and have known angle and side relationships — rather than comparing two external triangles.