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Answers:

The ~ mark means "similar"

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The red angle markings tell us which angles are congruent to one another. We have these two pairs of congruent angles

Because we have two pairs of congruent angles, we use the AA similarity theorem. AA stands for angle angle. We could use three pairs, but two pairs is the minimum needed for similarity statements.

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To figure out what goes in the third answer box, note how the order HGF is given to us. The order is important.

We have H first, G second, F third.

H being first, and it having a single arc, means that whatever has a single arc on the other triangle must be first. That would be H. Unfortunately they reuse angle H which makes things confusing a bit.

G is second in HGF, and G has the double arc marking. Note how M also has this double arc marking. Therefore we have M second in the answer. So far we have HM as part of the third answer.

The only thing left is N. Therefore, we have triangle HMN as the answer for the third box.

Put another way, we have these pairings:

This is why the order is important. So we can see how the angles pair up in the order that they do.

This is why triangle HGF is similar to triangle HMN.

We can write that as triangle HGF ~ triangle HMN

The ~ mark means "similar"

We don't have enough information to prove that the triangles are congruent or not. So we cannot say [tex]\triangle HGF \cong \triangle HMN[/tex]. Your teacher made a typo when using the congruence symbol [tex]\cong[/tex]

Instead, it should be [tex]\triangle HGF \sim \triangle HMN[/tex]

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