The triangle 𝐴𝐵𝐶 is isosceles. Two of the angles in 𝐴𝐵𝐶 are labelled 𝑥 and 𝑦. If 𝑥 is equal to 40 degrees, find the three possible values of 𝑦.
Remember an isosceles triangle is a triangle, in which two of its sides and two of its angles are equal. If we sketch triangle 𝐴𝐵𝐶 out, we have two options: either one angle is equal to 40 degrees and the other two are equal or the two equal angles measure 40 degrees and the other one does not.
Now, we do know that angles in a triangle sum to 180 degrees. In our first triangle, we can find the measure of the other two angles by subtracting 40 from 180. 180 minus 40 is 140. Since we know the remaining two angles are equal, we can halve this to find the measure of angle 𝑦. 140 divided by two is 70. So one of the possible values of 𝑦 is 70 degrees.
Alternatively, in our second triangle, we’ve said that the two equal angles are 40 degrees. We can, therefore, subtract 40 and 40 from 180. 40 plus 40 is 80 and 180 minus 80 is 100. So 𝑦 could be equal to 100 degrees.
So we’ve said that 𝑦 could be equal to 70 degrees or 100 degrees. But the question states that there are three possible values of 𝑦. That’s because two of the angles in 𝐴𝐵𝐶 are labelled 𝑥 and 𝑦. In the second triangle, angle 𝑦 could actually be the other angle that’s 40 degrees. The three possible values of 𝑦 then are 100, 70, or 40 degrees.
Explain why we have a special case when 𝑥 is equal to 60 degrees.
An isosceles triangle which has an angle of 60 degrees must actually be an equilateral. If it’s isosceles, we know that two of its angles are equal. So two of them could be 60 and 60. And since angles in a triangle add to 180, the third would automatically have to be 60 degrees.
This makes it an equilateral triangle. All angles in an equilateral triangle measure 60 degrees. When 𝑥 is equal to 60, 𝑦 can also only be 60 degrees since it’s an equilateral triangle and all three angles in this triangle are equal.