Question

12 use the triangles in the diagram below to determine who is correct.
melanie says that (\triangle abccong\triangle rsq). amy says that (\triangle abccong\triangle efg).
who is correct? state the triangle congruence postulate that proves the triangles congruent. explain your reasoning.
melanie is correct.

Explanation:

Step1: Recall the SAS congruence postulate

The Side - Angle - Side (SAS) congruence postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Analyze \(\triangle ABC\) and \(\triangle RSQ\)

In \(\triangle ABC\) and \(\triangle RSQ\):

• \(AC = RQ\) (marked with one tick)
• \(AB=RS\) (marked with two ticks)
• \(\angle A=\angle R\) (the included angles).

By the SAS congruence postulate, \(\triangle ABC\cong\triangle RSQ\).

Step3: Analyze \(\triangle ABC\) and \(\triangle EFG\)

In \(\triangle ABC\) and \(\triangle EFG\), although \(AC = EG\) (marked with one tick) and \(AB = EF\) (marked with two ticks), the angles are not the included angles. So, we cannot use the SAS postulate to prove them congruent.

Answer:

Melanie is correct. The Side - Angle - Side (SAS) congruence postulate proves that \(\triangle ABC\cong\triangle RSQ\) because \(AC = RQ\), \(AB = RS\), and \(\angle A=\angle R\) (two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle). Amy is incorrect as the angles in \(\triangle ABC\) and \(\triangle EFG\) are not the included angles for the given pairs of sides, so we cannot use the SAS postulate to prove their congruence.