Question

b, c, d and e are points on a circle.
ab is the tangent at b to the circle.
ab is parallel to ed.
angle abe = 73°
work out the size of angle dce.
give a reason for each stage of your working.
∠ded = 73° (z pattern)
∠cde = 73° (a.s.t)

Explanation:

Step1: Find angle BED (Alternate Angles)

Since \( AB \parallel ED \) and \( BE \) is a transversal, by the Alternate Angles Theorem, \( \angle ABE = \angle BED \). Given \( \angle ABE = 73^\circ \), so \( \angle BED = 73^\circ \).

Step2: Find angle DCE (Alternate Segment Theorem)

The Alternate Segment Theorem states that the angle between the tangent and a chord is equal to the angle in the alternate segment. Also, angles in the same segment are equal. Since \( \angle BED \) and \( \angle DCE \) are angles in the same segment (or related by the alternate segment theorem's implication for parallel lines and cyclic quadrilaterals), \( \angle DCE=\angle BED \). So \( \angle DCE = 73^\circ \).

Answer:

The size of angle \( DCE \) is \( \boldsymbol{73^\circ} \).