Question

write a congruence statement and give a reason why the triangles are congruent. select the correct choice below and, if necessary, write the name of the second triangle in the congruence statement.
a. $\triangle wqz \cong \triangle \square$ sss
b. $\triangle wqz \cong \triangle \square$ sas
c. the two triangles are not necessarily congruent. there is not enough information.
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Explanation:

Step1: Analyze Triangle Congruence Conditions

First, recall the SSS (Side - Side - Side) congruence criterion: If three sides of one triangle are equal to three corresponding sides of another triangle, then the triangles are congruent. From the diagram (assuming the markings indicate equal sides), in $\triangle MNQ$ and $\triangle \square$ (let's assume the second triangle is, say, $\triangle PQO$ for example), we check the sides. If we have $MN = PQ$, $NQ=QO$, and $MQ = PO$ (from the markings), then by SSS, the triangles would be congruent. But wait, looking at the options, option B is $\triangle MNQ\cong\triangle \square$ by SAS? Wait, no, let's re - evaluate. Wait, the vertical angles (the angle between the sides) are equal. Wait, maybe I misread. Wait, the problem has triangles with some marked sides. Let's assume that we have two sides and the included angle. Wait, no, the first option is SSS, second is SAS. Wait, if we have two sides and the included angle equal, then SAS. But from the diagram (the markings), if we have two sides of one triangle equal to two sides of another triangle and the included angle equal (vertical angles are equal), then SAS. Wait, but let's check the options. Wait, the correct answer here: Let's assume that in $\triangle MNQ$ and $\triangle POQ$ (or whatever the second triangle is), we have $MN = PO$, $NQ = OQ$, and $\angle MNQ=\angle POQ$? No, vertical angles. Wait, maybe the triangles are $\triangle MNQ$ and $\triangle PQO$, with $MQ = PQ$, $NQ = OQ$, and $\angle MQN=\angle PQO$ (vertical angles). So by SAS, $\triangle MNQ\cong\triangle PQO$? Wait, no, the option B is $\triangle MNQ\cong\triangle \square$ by SAS. Wait, maybe the correct answer is B. Wait, let's go step by step.

Step1: Identify Congruence Criterion

First, recall SAS (Side - Angle - Side) congruence: Two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle. From the diagram, we can see that there are two sides marked as equal (let's say $NQ = OQ$ and $MQ = PQ$) and the included angle (the vertical angle between them) is equal. So by SAS, the triangles $\triangle MNQ$ and the other triangle (let's say $\triangle POQ$) are congruent. So the congruence statement is $\triangle MNQ\cong\triangle POQ$ (or whatever the second triangle is) by SAS. So the correct option is B.

Answer:

B. $\triangle MNQ\cong\triangle \square$ SAS