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Core Connections Geometry, 2013 View details

1. Section 7.1

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Exercise 6 Page 400

a We are given two triangles and want to justify the relationship between them. In both triangles we know two angles. We have also been given two sides in

△ A B C

and one side in

△ E D F .

If two triangles have at least two pairs of congruent angles, they are similar. Using the Triangle Angle Sum Theorem, we can find the measure of

∠ A .

m ∠ A + 3 9^{\circ} + 2 5^{\circ} = 18 0^{\circ} ⇕ m ∠ A = 11 6^{\circ}

We can use the Triangle Angle Sum Theorem again to find the measure of

∠ F .

3 9^{\circ} + 11 6^{\circ} + m ∠ F = 18 0^{\circ} ⇕ m ∠ F = 2 5^{\circ}

As we can see,

△ A B C

and

△ E D F

have two pairs of congruent angles,

∠ A

and

∠ E,

and

∠ C

and

∠ D .

Therefore, the triangles are similar by the AA (Angle-Angle) Similarity Theorem. To determine if the triangles are congruent, we must identify pairs of corresponding sides of the two triangles.

From the diagram we see that two pairs of corresponding angles — $∠ A$ and $∠ E,$ and $∠ B$ and $∠ F$ — and the side between them are congruent. Therefore, we can claim that the triangles are congruent by the ASA (Angle-Side-Angle) Congruence Theorem. Let's show this as a flowchart.