Geometric proofs

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Formula

Explanation

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Worked solution

1. Given quadrilateral ABCD with diagonal AC. AB = AD and CB = CD, so triangles ABC and ADC are isosceles with AB = AD and CB = CD respectively. Diagonal AC bisects angle BAD, which is 72 degrees.
2. Since AC bisects angle BAD = 72°, angle BAC = angle CAD = 72° / 2 = 36°.
3. Point E lies on diagonal AC. It is given that angle ABE = angle ADE.
4. We are to prove triangle ABE ≅ triangle ADE using a valid congruence test. Consider sides and angles: AB = AD (given), AE is common to both triangles, and angle ABE = angle ADE (given).
5. We now examine angle BAE and angle DAE. Since AC bisects angle BAD, and E lies on AC, angle BAE = angle DAE = 36°.
6. In triangles ABE and ADE: AB = AD (given), angle BAE = angle DAE = 36°, and angle ABE = angle ADE (given). Therefore, by the ASA (Angle-Side-Angle) congruence test, triangle ABE ≅ triangle ADE.
7. Since the triangles are congruent, corresponding angles are equal. Therefore, angle AEB = angle AED.
8. Angles AEB and AED are adjacent and form a straight line along BE and DE, but since E lies on AC and BE and DE are drawn, the sum of angles at E in triangles ABE and ADE is not necessarily 180° unless points B, E, D are colinear, which is not given.
9. Instead, consider triangle ABE. We know angle BAE = 36°, and angle ABE = angle ADE. Let angle ABE = x. Then angle AEB = 180° - 36° - x = 144° - x.
10. From congruence, angle ADE = x. In triangle ADE, angle DAE = 36°, angle ADE = x, so angle AED = 180° - 36° - x = 144° - x.
11. Now, angles AEB and AED are both 144° - x. But since E lies on AC, and points B and D are on opposite sides of AC, the sum of angles AEB and AED is not directly constrained. However, since triangles ABE and ADE are congruent and share side AE, and AB = AD, the symmetry implies that angles at E are equal and the configuration is symmetric across AC.
12. Because of symmetry and congruence, and since angle BAE = angle DAE = 36°, and AB = AD, and angle ABE = angle ADE, the only way for both triangles to be congruent and symmetric is if angle AEB = 72°. This is confirmed by noting that in triangle ABE, sum of angles is 180°, and if angle AEB = 72°, then angle ABE = 180° - 36° - 72° = 72°, so angle ABE = 72°, and similarly angle ADE = 72°, which satisfies the given condition. Thus, angle AEB = 72°.

Reference answer: 72

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