If two triangles are similar the ratio of corresponding sides is the same.
A
a
They are similar.
Flowchart: See solution.
B
b
Not similar.
C
c
They are similar.
Flowchart: See solution.
Practice makes perfect
a
If the triangles are similar, they have the same shape. This requires that they have at least two pairs of congruent angles. From the exercise, we already know that one pair of angles are congruent.
∠C ≅ ∠Y
Since we have been given a second angle in both triangles, we can determine the third angle in either triangle by using the Triangle Angle Sum Theorem.
m∠B+42^(∘) + 36^(∘) = 180^(∘) ⇔ m∠B= 102^(∘)
As we can see, the unknown angle in △ ABC is congruent with ∠X in △ ZXY.
Therefore, we can claim that the triangles are similar by the AA Similarity Theorem. Let's show this as a flowchart.
b
Examining the diagram, we see that the triangles have one pair of congruent angles. We also see that one of the triangles is an isosceles triangle. If the triangles are similar, the second triangle would also need to be isosceles. However, since this is not the case, we know that they are not similar.
c
Both triangles are right triangles. Therefore, we can calculate the unknown hypotenuse and leg in the triangle by using the Pythagorean Theorem.
3^2+4^2=(ED)^2 &⇔ ED = 5 [0.5em]
6^2+(UG)^2=10^2 &⇔ UG = 8
If the triangles are similar, the ratio of corresponding sides should be the same. Therefore, let's identify corresponding sides and set up an equation.
If we substitute the given side lengths into the equation, we can check if the ratio of corresponding sides are the same.
5/10? =3/6? =4/8 ⇔ 1/2=1/2=1/2
As we can see, the ratio between corresponding sides is the same and, therefore, the triangles are similar. Let's show this as a flowchart.
