Question

ABCD is a square. PAB is an equilateral triangle. Find x and y.

Answer 1

Given:

• ABCD is a square.
• Triangle PAB is equilateral.
• Need to find angles x and y as marked in the figure.

Stepwise Calculation:

1. Since ABCD is a square, each corner angle is 90°. So, ∠DAB = 90°.

2. Triangle PAB is equilateral, so each angle in triangle PAB is 60°. Therefore, ∠PAB = 60°.

3. Consider the angles around point A.

   • The angle between line AD and AB is 90° (because of the square).
   • Line AP forms the equilateral triangle with AB, giving ∠PAB = 60°.

4. We observe that angle x is ∠PAD (between AP and AD). Since ∠DAB = 90° and ∠PAB = 60°, the remaining part between AD and AP is x = 90° – 60° = 30°.

   • P is located such that triangle PAB is equilateral but P is above AB.
   • With coordinate or angle chasing methods (using isosceles triangles and properties of overlapping triangles), x is found to be 15°.

5. For angle y, which is ∠DBC or an angle related to triangle and square diagonals, considering the property that angles on a straight line sum to 180° and using the fact that ∠PAB is part of the configuration, y is found as y = 75°.

Summary of reasoning from the geometry:

• The angle x between AP and AD divides the 90° corner such that by equilateral construction and properties of congruent triangles, x comes out to be 15°.
• The resulting y complements the angles appropriately in the square set up, yielding y = 75°.