 # Problem of the Week Problem C and Solution Intersecting Triangles

## Problem

$$\triangle ABC$$ and $$\triangle PQR$$ are equilateral triangles with vertices $$B$$ and $$P$$ on line segment $$MN$$. The triangles intersect at two points, $$X$$ and $$Y$$, as shown. If $$\angle NPQ = 75\degree$$ and $$\angle MBA = 65\degree$$, determine the measure of $$\angle CXY$$.

## Solution

In any equilateral triangle, all sides are equal in length and each angle measures $$60\degree$$.

Since $$\triangle ABC$$ and $$\triangle PQR$$ are equilateral, $$\angle ABC = \angle ACB = \angle CAB = \angle QPR = \angle PRQ = \angle RQP = 60\degree$$.

Since the angles in a straight line sum to $$180\degree$$, we have $$180\degree = \angle MBA + \angle ABC + \angle YBP = 65\degree + 60\degree + \angle YBP$$.
Rearranging, we have $$\angle YBP = 180\degree - 65\degree - 60\degree = 55\degree$$.

Similarly, since angles in a straight line sum to $$180\degree$$, we have $$180\degree = \angle NPQ + \angle QPR + \angle YPB = 75\degree + 60\degree + \angle YPB$$.
Rearranging, we have $$\angle YPB = 180\degree - 75\degree - 60\degree = 45\degree$$.

Since the angles in a triangle sum to $$180\degree$$, in $$\triangle BYP$$ we have $$\angle YPB + \angle YBP + \angle BYP = 180\degree$$, and so $$45\degree + 55\degree + \angle BYP = 180\degree$$.
Rearranging, we have $$\angle BYP = 180\degree - 45\degree - 55\degree = 80\degree$$.

When two lines intersect, vertically opposite angles are equal. Since $$\angle XYC$$ and $$\angle BYP$$ are vertically opposite angles, we have $$\angle XYC = \angle BYP = 80\degree$$.

Again, since angles in a triangle sum to $$180\degree$$, in $$\triangle XYC$$ we have $$\angle XYC + \angle XCY + \angle CXY = 180\degree$$. We have already found that $$\angle XYC = 80\degree$$, and since $$\angle XCY = \angle ACB$$, we have $$\angle XCY = 60\degree$$. So, $$\angle XYC + \angle XCY + \angle CXY = 180\degree$$ becomes $$80\degree + 60\degree + \angle CXY = 180\degree$$. Rearranging, we have $$\angle CXY = 180\degree - 80\degree - 60\degree = 40\degree$$.

Therefore, $$\angle CXY = 40\degree$$.