Question

a side - side - side congruent triangle theorem
b side - angle - side congruent triangle theorem
c angle - angle - side congruent triangle theorem
d angle - side - angle congruent triangle theorem
e hypotenuse leg congruent triangle theorem
f not congruent

Explanation:

Step1: Analyze the given information

We have two triangles \( \triangle ONP\) and \( \triangle QNP\). We know that \( \angle O=\angle Q\) (given as angles with double - arc markings), \( \angle ONP=\angle QNP\) (given as angles with single - arc markings), and \( NP = NP\) (common side).

Step2: Recall the congruent triangle theorems

• Side - Side - Side (SSS): Requires three pairs of equal sides. Here we don't have information about three pairs of sides.
• Side - Angle - Side (SAS): Requires two pairs of sides and the included angle. We don't have information about two pairs of sides.
• Angle - Angle - Side (AAS): If two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle, then the triangles are congruent. In \( \triangle ONP\) and \( \triangle QNP\), \( \angle O=\angle Q\), \( \angle ONP=\angle QNP\), and \( NP\) (a non - included side for these angles) is common.
• Angle - Side - Angle (ASA): Requires two angles and the included side. Here \( NP\) is not the included side for \( \angle O\) and \( \angle ONP\) (similarly for \( \angle Q\) and \( \angle QNP\)).
• Hypotenuse - Leg (HL): Applies to right - triangles. There is no indication that these are right - triangles.

Answer:

C. Angle - Angle - Side Congruent Triangle Theorem