Question

which theorem, term, or corollary is represented by the picture? the bold lines in the pictures represent the hypothesis of the theorem or corollary.
the angles are bold; the sides are not bold.
cpctc
converse to the isosceles triangle theorem
corollary to the isosceles triangle theorem
corollary 2 of the isosceles triangle theorem

1. choose the correct answer.

which theorem, term, or corollary is represented by the picture? the bold lines in the pictures represent the hypothesis of the theorem or corollary.
the angles are bold; the sides are not bold.
cpctc
isosceles triangle theorem
corollary to the isosceles triangle theorem
corollary 2 of the isosceles triangle theorem

Explanation:

• CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is a concept related to congruent triangles, not directly about the relationship between angles and sides in a single triangle as shown in the picture.
• The Isosceles Triangle Theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent.
• The Converse to the Isosceles Triangle Theorem states that if two angles of a triangle are congruent, then the sides opposite those angles are congruent. In the first picture, we have two congruent angles (bold - hypothesis), and the theorem would lead to the sides opposite them being congruent. In the second picture, if we assume the converse (since we start with angles), the converse of the isosceles triangle theorem is about using angle congruence (given as hypothesis) to get side congruence.
• Corollaries to the Isosceles Triangle Theorem usually build on the main theorem (either the basic theorem or its converse) with additional specific conditions (like equilateral - equiangular relationships for corollaries 1 and 2). For example, Corollary 1: A triangle is equilateral if and only if it is equiangular. Corollary 2: Each angle of an equilateral triangle measures \(60^{\circ}\).

Answer:

For the first set of options: Converse to the Isosceles Triangle Theorem
For the second set of options: Isosceles Triangle Theorem