Angles in a composite figure

A B C D is a square and D C E is an isosceles triangle with D E and D C equal. Angle D C E is 65 degrees. Find angle A D E. Try it first. Pause if you like. The square and the triangle share the line D C. So D and C are corners of both shapes. Angle A D E starts on the square's side D A and finishes on the triangle's side D E. One part of it is the square's corner and the rest is the triangle's. The triangle's part is not marked, so we work inside the triangle first. D E and D C are the same length, so the two angles facing them are the same size. Angle D E C is 65 degrees, the same as angle D C E. The three angles in triangle D C E add up to one hundred and eighty degrees. So the angle at D is one hundred and eighty minus 65 minus 65, which is 50 degrees. Now back to angle A D E. It is two angles at D, one from the square and one from the triangle. So angle A D E is 90 plus 50, which is one hundred and forty degrees.

Your turn. P Q R S is a square and Q R T is a triangle. Angle P Q T is one hundred and thirty seven degrees and angle Q T R is 62 degrees. Find angle Q R T. Try it first. Pause if you like. Angle P Q T crosses the line Q R, so the square's right angle is inside it. The triangle's angle at Q is one hundred and thirty seven minus 90, which is 47 degrees. The three angles in triangle Q R T add up to one hundred and eighty degrees. So angle Q R T is one hundred and eighty minus 62 minus 47, which is 71 degrees. The one hundred and thirty seven degrees at Q crosses the line Q R. Inside it, 90 degrees is the right angle of the square. And 47 degrees is the triangle's. One corner point, one angle from each shape.