Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
The diagonals of a kite intersect at right angles. So, the triangles formed by the diagonals of a kite have 90° angles.
Recall that the by the triangle angle sum theorem, the sum of the measures of the angles in a triangle is 180°. Since each triangle has a right angle, the other two angles in the triangle must be complementary (meaning that their sum is 90°), so that the sum of those two angles and the right angle is 180°. Knowing that the other two angles are complementary, if these angles are given with expressions of a variable, it is possible to find the value of that variable by adding the two expressions together and setting it equal to 90°.
In this problem we have a quadrilateral of four sides and we see that we have two pairs of consecutive congruent sides which tells us that this must be a kite and I also see that I’ve got x. I’ve got one variable and I’m trying to solve for it. To do that we are going to have to use the fact that the diagonals intersect at a right angle.
So I’m going to write that in right now. I know that this angle right here has to 90 degrees which means 2x and the quantity of 3x plus 10 must be complementary, these two have to add up to 90 degrees. So I’m going to write that equation that 90 degrees is equal to the sum if 2x and my other angle which 3x plus 10. So now it just becomes solve for x.
If I combine like terms 2x and 3x is 5x and we have +10 there’s no like terms with 10. So I’m going to subtract x from both sides and I’m going to get 5x equals 80 and so we are going to divide by 5 and x is going to be 16.
So the key to finding x here was realizing that these two angles must be complementary because our diagonals intersect at 90 degree angle.