a If two triangles are similar, they need at least two pairs of congruent angles. If they are congruent, at least one pair of corresponding sides has to be congruent.
B
b Use the congruence between the triangles to find DF. To determine AC, consider using the Law of Sines.
A
a See solution.
B
b AC≈ 9.4 units
DF= 20 units
Practice makes perfect
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 △ ABC and one side in △ EDF.
m∠ A+39^(∘)+25^(∘) = 180^(∘) ⇕ m∠ A = 116^(∘)
We can use the Triangle Angle Sum Theorem again to find the measure of ∠ F.
39^(∘)+116^(∘) + m∠ F = 180^(∘) ⇕ m∠ F = 25^(∘)
As we can see, △ ABC and △ EDF 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.
b From Part A, we know that the triangles are congruent. This means corresponding sides have equal measures.
We know that BC=20 and since BC=DF, we can claim that DF=20 as well. Finally, to find AC we can use the Law of Sines.
