Question

1. determine whether the two triangles are congruent. if they are, justify your answer using one of the triangle congruence postulates. if not, explain why not.

a.
b.

Explanation:

Step1: Recall triangle - congruence postulates

There are SSS (Side - Side - Side), SAS (Side - Angle - Side), ASA (Angle - Side - Angle), AAS (Angle - Angle - Side), and HL (Hypotenuse - Leg for right - triangles).

Step2: Analyze part a

In \(\triangle QRS\) and \(\triangle VUW\), we have \(QS = VU=2.5\), \(\angle R = \angle V = 36.5^{\circ}\), but \(\angle Q = 72^{\circ}\) and \(\angle W = 71.5^{\circ}\). The third - angle of \(\triangle QRS\) is \(180^{\circ}-72^{\circ}-36.5^{\circ}=71.5^{\circ}\) and the third - angle of \(\triangle VUW\) is \(180^{\circ}-71.5^{\circ}-36.5^{\circ}=72^{\circ}\). However, the corresponding angles are not in the correct order for any of the congruence postulates. So, \(\triangle QRS\) and \(\triangle VUW\) are not congruent.

Step3: Analyze part b

In \(\triangle ABC\) and \(\triangle ADC\), we have \(AC = AC\) (common side), \(\angle BAC=\angle DAC\) (given), and \(\angle BCA=\angle DCA\) (given). By the ASA (Angle - Side - Angle) congruence postulate, \(\triangle ABC\cong\triangle ADC\).

Answer:

a. The two triangles are not congruent. The corresponding sides and angles do not match up according to any of the triangle - congruence postulates.
b. The two triangles \(\triangle ABC\) and \(\triangle ADC\) are congruent by the ASA congruence postulate.