a Similar shapes have the same shape. To determine if the triangles have the same shape, you need to find at least two pairs of congruent corresponding angles.
B
b Consider the 50^(∘) angle in △ LMN.
C
c Do two angles in △ ABC match two angles in △ QPR?
A
a Similar, but not congruent.
B
b Neither similar nor congruent.
C
c Congruent
Practice makes perfect
a We will first determine if the triangles are similar. If they are, they have at least two pairs of congruent angles. Using the Triangle Angle Sum Theorem, we can calculate the third angle in △ ABC.
m∠ B +40^(∘)+65^(∘) = 180^(∘) ⇔ m ∠ B = 75^(∘)
At least two pairs of angles in △ ABC and △ XYZ are congruent. Therefore, we know by the AA Similarity condition that these triangles are in fact similar. The known sides in the triangles are both 8 but they are not corresponding sides as they are between different angles.
Therefore, we know that the triangles are not congruent.
b In Part A, we calculated all three angles in △ ABC. Since the 50^(∘) angle in △ LMN does not match an angle in △ ABC, we know these triangles cannot be similar. Therefore, they cannot be congruent either.
c From Part A, we know that two of the angles in △ ABC are 75^(∘) and 40^(∘). With this, we know that two pairs of angles in △ ABC and △ QPR are congruent which means we can claim similarity by the AA Similarity condition. Next, we will determine if the triangles are congruent. For this purpose, we have to identify corresponding sides in these triangles.
As we can see, one pair of corresponding sides are congruent which means we can claim congruence by the ASA Congruence condition.
