Question

ABCD is a parallelogram. ∠C = 110°. E is a point on DC so that AD = ED. Find ∠AEC.

Answer 1

Given:

• ABCD is a parallelogram.
• ∠C = 110°.
• E is a point on DC such that AD = ED.

Stepwise calculation:

1. Since ABCD is a parallelogram, opposite angles are equal. So ∠A = ∠C = 110°.
2. Adjacent angles in a parallelogram sum to 180°, so ∠D = 180° – 110° = 70°.
3. Since AD = ED, triangle ADE is isosceles with AD = ED.
4. ∠ADE = ∠DEA because of the isosceles triangle property.
5. Angle ∠ADC whole angle at D = ∠ADE + ∠EDC = 70° and since ∠EDC = ∠C = 110°, point E lies on DC such that DE = AD.
6. To find ∠AEC, we use the exterior angle theorem in triangle AED.
   • ∠AEC = 180° – ∠DEA since E, A, C are points in a polygon.
7. From the isosceles triangle ADE, calculate ∠ADE and ∠DEA
   = `(180^circ - 70^circ)/2`
   = 55°
8. Therefore, ∠AEC = 180° – 55° = 125°.

∠AEC = 125°.