Question

question 2

which of the following statements about \\(\delta vme\\) and \\(\delta mvl\\) is true?

a the triangles are congruent by aas.
b the triangles are congruent by sas.
c the triangles are not congruent.
d the triangles are congruent by asa.

Explanation:

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<reasoning>

Identify given congruent angles

The diagram shows two triangles, \(\Delta VME\) and \(\Delta MVL\), sharing a common side \(MV\).
Looking at the angle markings:

• \(\angle EMV\) has a single arc marking.
• \(\angle LVM\) has a single arc marking.

Therefore, \(\angle EMV \cong \angle LVM\).

• \(\angle EVM\) has a double arc marking.
• \(\angle LMV\) has a double arc marking.

Therefore, \(\angle EVM \cong \angle LMV\).

Identify the shared side

Both triangles share the side \(MV\).
By the Reflexive Property of Congruence:
\(MV \cong VM\).

Apply congruence postulate

We have:

1. An angle: \(\angle EMV \cong \angle LVM\)
2. An included side: \(MV \cong VM\)
3. Another angle: \(\angle EVM \cong \angle LMV\)

Since the congruent side is directly between the two pairs of congruent angles, we use the Angle-Side-Angle (ASA) congruence postulate.
Thus, \(\Delta VME \cong \Delta MVL\) by ASA.
</reasoning>

<answer>
<mcq-option>a The triangles are congruent by AAS.</mcq-option>
<mcq-option>b The triangles are congruent by SAS.</mcq-option>
<mcq-option>c The triangles are not congruent.</mcq-option>
<mcq-correct>d The triangles are congruent by ASA.</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify given congruent angles

The diagram shows two triangles, \(\Delta VME\) and \(\Delta MVL\), sharing a common side \(MV\).
Looking at the angle markings:

• \(\angle EMV\) has a single arc marking.
• \(\angle LVM\) has a single arc marking.

Therefore, \(\angle EMV \cong \angle LVM\).

• \(\angle EVM\) has a double arc marking.
• \(\angle LMV\) has a double arc marking.

Therefore, \(\angle EVM \cong \angle LMV\).

Identify the shared side

Both triangles share the side \(MV\).
By the Reflexive Property of Congruence:
\(MV \cong VM\).

Apply congruence postulate

We have:

1. An angle: \(\angle EMV \cong \angle LVM\)
2. An included side: \(MV \cong VM\)
3. Another angle: \(\angle EVM \cong \angle LMV\)

Since the congruent side is directly between the two pairs of congruent angles, we use the Angle-Side-Angle (ASA) congruence postulate.
Thus, \(\Delta VME \cong \Delta MVL\) by ASA.
</reasoning>

<answer>
<mcq-option>a The triangles are congruent by AAS.</mcq-option>
<mcq-option>b The triangles are congruent by SAS.</mcq-option>
<mcq-option>c The triangles are not congruent.</mcq-option>
<mcq-correct>d The triangles are congruent by ASA.</mcq-correct>
</answer>

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