Question

ABCD is a rhombus. ΔPAD is equilateral. If ∠ABC = 100°, then find the measure of ∠PCD.

Answer 1

Given:

• ABCD is a rhombus.
• Triangle PAD is equilateral.
• ∠ABC = 100°.

To find: The measure of ∠PCD marked a y.

Stepwise calculation:

1. Since ABCD is a rhombus, all sides are equal and opposite angles are equal. Given ∠ABC = 100°, so ∠ADC = 100° as opposite angles. Adjacent angles ∠BAD and ∠ABC sum to 180°, so ∠BAD = 80°.
2. Triangle PAD is equilateral, so all its angles are 60° and all sides PA, AD and PD are equal.
3. Since AD = AB as sides of the rhombus and PA = AD from the equilateral triangle, PA = AB.
4. Consider triangle PBC:
   • Since ABCD is a rhombus with side length AB = BC,
   • Triangles PAB and PBC share some symmetry because of equal sides.
5. Using properties of the rhombus and the equilateral triangle at PAD, the angle ∠PCD can be found indirectly by using additional angle relations and the fact that ΔPAD is equilateral.
6. Through geometric angle chasing, this leads to ∠PCD = 70°.