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 AB.
We can calculate the length of BC and AC the same way.
Side
Endpoints
Distance Formula
Length
AB
A( - 4, 6), B( - 2, 1)
sqrt(( - 2-( - 4))^2+( 1- 6)^2)
≈ 5.4
BC
B( - 2, 1), C( 5, 6)
sqrt(( 5-( - 2))^2+( 6- 1)^2)
≈ 8.6
AC
A( - 4, 6), C( 5, 6)
sqrt(( 5-( - 4))^2+( 6- 6)^2)
9
Let's now gather the information that we have found.
AB=5.4
BC=8.6
AC=9
Comparing the values, we can order the lengths from least to greatest.
AB< BC< AC
The last thing we need to do is to find the angles opposite to the sides.
As we can see, side AB is opposite to ∠ C, BC is opposite to ∠ A, and AC is opposite to ∠ B. Therefore, we can order the angles from smallest to greatest as follows.
∠ C, ∠ A, ∠ B
