Recall the definition of similar triangles. How can we use angle measures to check if triangles are similar?
Yes, see solution.
Practice makes perfect
We want to know whether the given triangles are similar. Let's give names to the vertices to make it easier to work through the exercise. will name one of the triangles ABC and the second one EDC.
Recall that two triangles are similar if and only if all their angles are congruent. We will use this fact to check if △ ABC is similar to △ EDC. First, we need to find at least two angle measures in both triangles. Let's start with △ ABC.
Notice that both triangles are right triangles. This means that one of their angles is a right angle with a measure of 90^(∘). We are also given that the second angle in △ ABC is 51^(∘). We can use the property of the interior angles of a triangle to find the measure of the third angle.
Sum of Interior Angles of a Triangle
The measures of the interior angles in a triangle add up to 180^(∘).
With this in mind, let's find the measure of the third angle in △ ABC. We will mark the missing measure angle as x.
x+90^(∘)+51^(∘)=180^(∘)
⇕
x= 39^(∘)
We found that the measure of the missing angle in △ ABC is 39^(∘). Now let's find the measures of the angles in the other triangle, starting with ∠ ECD. Notice that ∠ ACB, ∠ BCD, and ∠ ECD form a straight line. Because of this, their measures need to add up to 180^(∘). We can use this fact to find measure of ∠ ECD. Let's mark the missing angle measure as y.
y+ 39^(∘)+ 102^(∘)=180^(∘)
⇕
y= 39^(∘)
We found that the measure of ∠ ECD is 39^(∘). Let's update our image.
We can see that two angles in △ ABC, 90^(∘) and 39^(∘), are congruent to two angles in △ EDC. Because of this, the third angles are also congruent. This means that the triangles given in the exercise are similar!
