Answers:
Angle 1 is 40 degrees
Angle 2 is 90 degrees
Angle 3 is 50 degrees
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Explanations:
Triangle ABC is a reflection of triangle ADC over the line AC. Doing this reflection forms the kite ABCD. Or you can think of it as gluing two congruent triangles together (ABC and ADC). We can prove the two triangles ABC and ADC congruent by the SSS congruence property. Then we use CPCTC (corresponding parts of congruent triangles are congruent) to prove that angle 1 is 40.
Let E be the intersection of the two diagonals. Through similar reasoning as earlier, we can prove that triangle AEB is congruent to triangle AED, so that leads to angle 2 being 90 degrees because this angle adds to its counterpart to get 180. Both unknown angles are x each meaning x+x = 180 leads to x = 90. One property of kites is that the diagonals are perpendicular.
Triangle AED is a right triangle as discussed above. Therefore, the angle 3 and the 40 degree angle are complementary. If angle 3 is y, then y+40 = 90 giving y = 90-40 = 50, which is why angle 3 is 50 degrees.
