Question

multiple choice 10 points
determine if each pair of triangles is congruent. if so, write the postulate or theorem.
image of triangles with vertices a, b, c, d
options: sas, triangles are not congruent, sss, aas, asa, hl

8 multiple choice 10 points
determine if each pair of triangles is congruent. if so, write the postulate or theorem.
image of triangles with vertices a, b, c, d, e
options: asa, hl, aas, the triangles are not congruent.

Explanation:

Question 7 (First Triangle Pair)

Step1: Identify Common Side

Triangles \( \triangle ABC \) and \( \triangle ADC \) share side \( AC \), so \( AC = AC \) (reflexive property).

Step2: Identify Marked Sides and Angles

We see one pair of sides (marked with ticks) are equal, and the included angles between the common side and the marked sides are equal (from the diagram's angle markings).

Step3: Apply SAS Postulate

SAS (Side - Angle - Side) postulate states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. Here, we have two sides (one pair marked, and the common side) and the included angle equal. So the postulate is SAS.

Step1: Analyze Given Information

We need to check the congruence criteria. Let's assume the triangles are \( \triangle ABC \) and \( \triangle EDC \) (or similar labeling). We have vertical angles (at \( C \)) which are equal, and one pair of angles (marked) and one pair of sides (marked as equal). But for ASA, we need two angles and the included side. For AAS, two angles and a non - included side. However, from the diagram (assuming the markings), we can see that the triangles satisfy the ASA (Angle - Side - Angle) postulate. Wait, no, let's re - evaluate. Wait, the correct congruence here: Let's see, if we have \( \angle A=\angle E \) (marked), \( AC = EC \) (marked as equal, since \( C \) is the mid - point? Or from the diagram's tick marks), and \( \angle ACB=\angle ECD \) (vertical angles). So two angles and the included side ( \( AC \) is between \( \angle A \) and \( \angle ACB \), and \( EC \) is between \( \angle E \) and \( \angle ECD \)). So by ASA, the triangles are congruent. Wait, but let's check the options. The options are ASA, HL, AAS, or not congruent. So the correct one is ASA.

Answer:

SAS

Question 8 (Second Triangle Pair)