If one side of a triangle is longer than another side, then the angle opposite the longer side has greater measure than the angle opposite the shorter side.
According to this theorem, to order the angles we need to know the length of the triangle's sides. Let's find them by substituting the coordinates of the endpoints into the Distance Formula. We can start with XY.
We can calculate the length of YZ and XZ the same way.
Side
Endpoints
Distance Formula
Length
XY
X( - 3, - 2), Y( 3, 2)
sqrt(( 3-( - 3))^2+( 2-( - 2))^2)
≈ 7.2
YZ
Y( 3, 2), Z( - 3, - 6)
sqrt(( - 3- 3)^2+( - 6- 2)^2)
10
XZ
X( - 3, - 2), Z( - 3, - 6)
sqrt(( - 3-( - 3))^2+( - 6-( - 2))^2)
4
Let's now gather the information that we have found.
XY &=7.2
YZ &=10
XZ &=4
Comparing the values, we can order the lengths from least to greatest.
XZ< XY< YZ
The last thing we need to do is to find the angles opposite to the sides.
As we can see, side XY is opposite to ∠ Z, YZ is opposite to ∠ X, and XZ is opposite to ∠ Y. Therefore, we can order the angles from smallest to greatest as follows.
∠ Y, ∠ Z, ∠ X
