The figure shows two isosceles triangles. If 180 minus 𝑧 equals five 𝑦 and 𝑦 equals 22, what is the value of 𝑥?
First, we know that the figure is not drawn to scale. So we’ll need to rely on the given information: that 180 minus 𝑧 equals five 𝑦 and
that 𝑦 equals 22. We know that these two triangles are isosceles. And isosceles triangles have two sides that are equal. Here’s our first isosceles triangle and our second isosceles triangle. Here’s where we need to be careful. It’s really tempting to say that 𝑦 is equal to 𝑧. This is because when two lines intersect, opposite angles are equal. However, the line segments that form each side of these isosceles triangles are not
part of the same line. We are not dealing with intersecting lines. We are dealing with two isosceles triangles that share a vertex. And so, we’ll rely on the information we were given that 𝑦 is 22 degrees. And we can plug that in to the other equation we were given to solve for 𝑧.
Five times 22 equals 110. To solve the equation, we need to get 𝑧 by itself. So we subtract 180 from both sides of the equation. On the left, we have negative 𝑧. And on the right, 110 minus 180 equals negative 70. Now, we multiply through by negative one which gives us positive 𝑧 equals positive
70. This means 𝑧 equals 70 degrees.
But we’re primarily interested in the value of 𝑥. And so, we’ll need to take a few more steps. We need to think about what else we know about isosceles triangles. We’ve already said they have two equal sides. But it also has two equal angles. The equal angles are the angles opposite the two equal sides. In our isosceles triangle highlighted in yellow, these two angles are equal to each
other. We could say that each of these angles measure 𝑎 degrees. And then, we could say that 𝑎 plus 𝑎 plus 70 must equal 180. This is because angles in a triangle must sum to 180. First, we combine 𝑎 plus 𝑎 which is two 𝑎. Two 𝑎 plus 70 equals 180.
To solve for 𝑎, we subtract 70 from both sides which gives two 𝑎 equals 110. And then, we divide both sides of the equation by two. 𝑎 divided by two equals 𝑎. 110 divided by two equals 55. This tells us that the angles inside our yellow isosceles triangle will be equal to
55, 55, 70. We also see that the angle 𝑥 and a 55-degree angle make up a straight line. And so, we say that 55 plus 𝑥 equals 180 because we know that a straight line
measures 180 degrees. To solve for 𝑥, we subtract 55 from both sides. 180 minus 55 equals 125. So we say that 𝑥 equals 125. 𝑥 degrees equals 125 degrees.