Question

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consider the following proof. what theorem is being proved?
given: isosceles \\( \triangle a b c \\) with \\( \overline{a b} \cong \overline{b c} \\)
prove: \\( \angle a \cong \angle c \\)
theorem table
(2 point)
the base angles theorem
the triangle inequality theorem
the triangle sum theorem
the sss congruence theorem

Explanation:

Step1: Analyze the given proof

The proof starts with an isosceles triangle \( \triangle ABC\) (\(AB\cong BC\)) and aims to show \( \angle A\cong\angle C\).

Step2: Recall theorem definitions

• Base Angles Theorem: In an isosceles triangle, the angles opposite the congruent sides are congruent.
• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
• Triangle Sum Theorem: The sum of the interior angles of a triangle is \(180^{\circ}\).
• SSS Congruence Theorem: If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Since the proof is about showing the angles (\( \angle A\) and \( \angle C\)) opposite the congruent sides (\(AB\) and \(BC\)) in an isosceles triangle are congruent.

Answer:

the Base Angles Theorem