Unit 6
7 marks · HardIn the diagram, $O$ is the centre of the circle. $AD$ and $BC$ are chords that intersect at $E$. $AB$ is parallel to $CD$. $\angle BAD = 30^\circ$, $\angle DCA = 52^\circ$.
Find $\angle EAB$ and justify your answer using a circle theorem.
Find $\angle BCD$ and give a reason for your answer.
Let set $S = \{x \mid x$ is an integer angle less than $70^\circ$ occurring at vertex $C$ or $A$ in this diagram $\}$. Describe $S$ using set-builder notation.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
Show answer
(a) 30^\circ; angle in the same segment as $\angle BAD$
(b) 52^\circ; corresponding angles since $AB$ || $CD$
(c) {x \in \mathbb{Z} \mid x = 30 \text{ or } x = 52, x < 70}
Common mistakes: incorrect circle theorem applied; missing justification
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Identifies correct angle using a circle theorem.
A1 (a) · 1 mark · Gives $30^\circ$ with valid justification.
M2 (b) · 1 mark · Identifies corresponding angles for parallel chords.
A2 (b) · 1 mark · Gives $52^\circ$ with clear reason.
M3 (c) · 1 mark · Identifies the relevant angles at vertex $C$ and $A$ under $70^\circ$.
A3 (c) · 2 marks · Writes correct set-builder notation for $S$ with both elements.
Worked solution
a) $\angle EAB$ lies in the same segment as $\angle BAD$, so $\angle EAB = 30^\circ$ (angles in the same segment are equal).
b) $\angle BCD$ is equal to $\angle DCA = 52^\circ$ since $AB$ is parallel to $CD$, so the alternate angle or corresponding angle property applies.
c) The integer angles less than $70^\circ$ present at $C$ or $A$ are $30^\circ$ and $52^\circ$. \ So $S = \{x \in \mathbb{Z}\,|\, x = 30 \text{ or } x = 52,\ x < 70\}$.