Recall the definition of similar triangles. Try to think about all possible combinations of angles that satisfy the condition from the exercise.
No, see solution.
Practice makes perfect
We want to decide whether two right triangles are similar if one angle measure is twice the measure of another angle measure. Let's look at an example.
We know that in right triangles, one of the angles measures 90^(∘) because it is a right angle. In our case, we do not know anything about the remaining angles. Let's call them d and e. Recall that the measures of the interior angles of a triangle add up to 180^(∘). This means that the sum of the two remaining angles in both triangles must be 90^(∘).
d + e =90^(∘)We know that the measure of one angle is two times the measure of another angle. Let's use the equation we created to find the measures of the angles. We will assume that e = 2 d.
We found that the measure of d is 30^(∘). Since e is two times d, the measure of e is 2* 30^(∘)= 60^(∘). Let's see if we can think of any other possible solutions to the exercise. Remember that one angle in the triangle is a right angle? If the measure of one of the missing angles is 45^(∘), then the condition from our exercise is satisfied.
2* 45^(∘) = 90^(∘)
Now we need to check if a triangle with angle measures 45^(∘) and 90^(∘) exist. Remember that the sum of the interior angles in a triangle is 180^(∘). We will call the missing third angle x.
90^(∘)+ 45^(∘)+ x=180^(∘)
⇕
x= 45^(∘)
We found that the measure of the missing angle is 45^(∘). We found two different sets of measures for a right triangle where one angle is twice the measure of another angle in the triangle.
Both of these right triangles fulfill the given condition but since they only have one angle measure in common, they are not similar. This means that we cannot determine whether the two unknown right triangles are similar.
