b For each triangle, order the sides from shortest to longest. Then, find the ratios of sides in the same position within each ordering.
C
c For each triangle, order the sides from shortest to longest. Then, find the ratios of sides in the same position within each ordering.
A
a △ ABC ~ △ FED, AA~
B
b △ ACB ~ △ MLK, SSS~
C
c Not similar, there is no zoom factor.
Practice makes perfect
a We want to decide whether the following two triangles are similar.
Let's notice that the only interior angle of △ ABC that we do not know the measure of, is ∠ C. To find m ∠ C, let's use the fact that in a triangle, the sum of interior angles equals 180^(∘).
m ∠ C + 43^(∘) + 20^(∘) = 180^(∘)
Let's solve this equation for m ∠ C.
We can add this measure to our diagram. Let's also look for any pairs of congruent angles between our two triangles.
We have two pairs of congruent angles: ∠ A ≅ ∠ F and ∠ C ≅ ∠ D. Therefore, our triangles are similar by the Angle-Angle Similarity Theorem.
△ A B C ~ △ F E D
b We want to decide whether the following two triangles are similar.
Notice that we know all sides of the two triangles. If the triangles are similar and we order the sides of each triangle from shortest to longest, then the corresponding sides are in the same place in each ordering.
△ ACB: & 5, 10, 13
△ MLK: & 9, 18, 23.4
The triangles are similar if the ratios of the corresponding sides are equal. Let's check whether that is the case. Note that we will put the sides of △ ACB in the denominator, since the decimal expansions of the ratios might be finite.
Ratio
Value
9/5
1.8
18/10
1.8
23.4/13
1.8
As we can see, for each pair of corresponding sides the ratio is equal to 1.8 Therefore, the triangles are similar by the Side-Side-Side Similarity Theorem.
△ ACB ~ △ MLK
c We want to decide whether the following two triangles are similar.
Notice that we know all sides of the two triangles. If the triangles are similar and we order the sides of each triangle from shortest to longest, then the corresponding sides are in the same place in each ordering.
△ ACB: & 5, 10, 13
△ ZYX: & 4, 9, 12
The triangles are similar if the ratios of the corresponding sides are equal. Let's check whether that is the case. Note that we will put the sides of △ ACB in the denominator, since the decimal expansions of the ratios might be finite.
Ratio
Value
4/5
0.8
9/10
0.9
12/13
0.923076...
As we can see, for each pair of corresponding sides the ratio is different. Therefore, the triangles are not similar as there is no zoom factor.
