Question

1. name two triangles that are congruent by asa.

choose the correct answer below.
△mnp≅△tuv
△qrs≅△tuv
△mnp≅△qrs

1. determine whether the triangles must be congruent. if so, name the postulate or theorem that best justifies your answer. if not, explain.

choose the correct answer below.
a. the triangles are congruent by asa.
b. the triangles are congruent by aas.
c. the triangles are congruent by sas.
d. the triangles are congruent by sss.
e. the triangles are not congruent.

1. which postulate or theorem could you use to prove △abc≅△def?

choose the correct answer below.
sss postulate
aas theorem
sas postulate
asa postulate

Explanation:

Question 8

To determine which two triangles are congruent by ASA (Angle - Side - Angle), we analyze the given triangles. For $\triangle MNP$ and $\triangle QRS$, we check the angles and the included side. By observing the diagram (the right angles, and the other angles and the included side), we find that $\triangle MNP\cong\triangle QRS$ by ASA. The other options do not satisfy the ASA criteria.

In the given triangle with right angle at $B$ (i.e., $\angle ABD=\angle CBD = 90^{\circ}$), we have $AB = BC$ (since $B$ is the mid - point of $AC$), $BD$ is common to both $\triangle ABD$ and $\triangle CBD$, and $\angle ABD=\angle CBD$. So, by the SAS (Side - Angle - Side) postulate (two sides and the included angle are equal), the two triangles ( $\triangle ABD$ and $\triangle CBD$) are congruent.

To prove $\triangle ABC\cong\triangle DEF$, we look at the given information (from the diagram, we can assume that we have two angles and a non - included side or two angles and the included side? Wait, if we have two angles and the included side, it is ASA, but if we have two angles and a non - included side, it is AAS. But from the diagram (assuming the angles and the side), if we have two angles and a non - included side equal, we use the AAS theorem. Wait, no, let's re - check. If we have two angles and a side that is not between them (AAS) or between them (ASA). But in the given options, if we consider the triangles $\triangle ABC$ and $\triangle DEF$, if we have two angles and a non - included side equal, we use the AAS theorem. Wait, no, the AAS theorem states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent. But let's check the options. If the triangles have two angles and a non - included side equal, AAS is the correct theorem.

Answer:

$\triangle MNP\cong\triangle QRS$

Question 9