Question

step statement reason
1 ( mangle mnp = mangle mpn = 65^circ ) given
2 construct altitude ( mq ).
3 ( mangle mqn = mangle mqp = 90^circ ) definition of altitude
4 ( \triangle mqn cong \triangle mqp ) angle - angle - side congruence
5 ( mn = mp ) corresponding sides of congruent figures have equal lengths.
what was the first mistake in lilianas proof?
choose 1 answer:
a ( mangle mpn ) was not given.
b liliana failed to establish the conditions for the angle - angle - side criterion.
c the angle - angle - side criterion does not guarantee congruence.
d sides ( overline{mn} ) and ( overline{mp} ) are not corresponding sides of congruent triangles.

Explanation:

To determine the first mistake, we analyze each option:

• Option A: The problem states "Given" for \( m\angle MNP = m\angle MPN = 65^\circ \), but actually, only \( m\angle MNP \) (or one of the angles) might be given, and \( m\angle MPN \) is assumed without being explicitly given. Wait, no—wait, the first step says both angles are given, but maybe only one is given. Wait, no, let's re - check. The first step's "Given" is incorrect because \( m\angle MPN \) was not given. Wait, no, let's look at the proof steps. Step 1 says \( m\angle MNP=m\angle MPN = 65^\circ \) is given. But in a typical triangle proof, if we are to prove \( MN = MP \), we need to check if both angles are given. If only one angle (e.g., \( m\angle MNP = 65^\circ \)) is given, then assuming \( m\angle MPN = 65^\circ \) is a mistake. But among the options, Option A says \( m\angle MPN \) was not given. Let's check other options:
• Option B: The angle - angle - side (AAS) criterion does require certain conditions (two angles and a non - included side). But the first mistake is in Step 1.
• Option C: The AAS criterion does guarantee congruence, so this is wrong.
• Option D: In \( \triangle MQN\cong\triangle MQP \), \( MN \) and \( MP \) are corresponding sides (since \( \triangle MQN\cong\triangle MQP \), \( MN \) corresponds to \( MP \)), so this is wrong.

So the first mistake is in Step 1 where \( m\angle MPN \) was not given, but Step 1 claims it is given.

Answer:

A. \( m\angle MPN \) was not given.