### Find two unknown angles in a circle related to basic circle properties

Question Sample Titled 'Find two unknown angles in a circle related to basic circle properties'

In the figure, ${A}{B}{C}{D}$ is a circle. ${E}$ is a point lying on ${A}{C}$ such that ${B}{C}={C}{E}$. It is given that ${A}{B}={A}{D}$ , $\angle{A}{D}{B}={54}^{\circ}$ and $\angle{C}{B}{D}={22}^{\circ}$ .

${22}^{\circ}$${54}^{\circ}$${A}$${B}$${C}$${D}$${E}$

 Find $\angle{B}{D}{C}$ and $\angle{A}{B}{E}$ . (5 marks)

 $\angle{A}{B}{D}$ $=\angle{A}{D}{B}={54}^{\circ}$ base ∠s, isos. △ 1M for either one $\angle{A}{C}{B}$ $=\angle{A}{D}{B}={54}^{\circ}$ ∠ in alt. segment $\angle{B}{A}{C}$ $={180}^{\circ}-\angle{A}{B}{D}-{22}^{\circ}-\angle{A}{C}{B}$ ∠ sum of △  $={180}^{\circ}-{54}^{\circ}-{22}^{\circ}-{54}^{\circ}$ 1M  $={50}^{\circ}$ $\angle{B}{D}{C}$ $=\angle{B}{A}{C}$ ∠s in the same segment 1A $={50}^{\circ}$  $\angle{B}{E}{C}$ $=\dfrac{{{180}^{\circ}-{54}^{\circ}}}{{2}}$ base ∠s, isos. △ 1M  $={63}^{\circ}$ $\angle{A}{B}{E}$ $={63}^{\circ}-{50}^{\circ}$ ext. ∠ of △ $={13}^{\circ}$ 1A  Students can use various methods and find out different angles in different orders. Marks for a certain step should be awarded if any relevant and correct concept/technique/theorem had been presented.

 $\angle{A}{B}{D}$ $={54}^{\circ}$ base ∠s, isos. △ 1M ${\left(\angle{B}{D}{C}+{54}^{\circ}\right)}+{\left(\angle{A}{B}{D}+{22}^{\circ}\right)}$ $={180}^{\circ}$ opp. ∠s, cyclic quad. 1M $\angle{B}{D}{C}+{54}^{\circ}+{54}^{\circ}+{22}^{\circ}$ $={180}^{\circ}$ $\angle{B}{D}{C}$ $={50}^{\circ}$ 1A  $\angle{C}{A}{D}={22}^{\circ}$ ∠s in the same segment 1M $\angle{B}{A}{D}={180}^{\circ}-{2}{\left({54}^{\circ}\right)}={72}^{\circ}$ ∠ sum of △ $\angle{B}{A}{E}={72}^{\circ}-{22}^{\circ}={50}^{\circ}$ $\angle{B}{E}{C}=\dfrac{{{180}^{\circ}-{54}^{\circ}}}{{2}}={63}^{\circ}$ base ∠s, isos. △  $\angle{A}{B}{E}$ $={63}^{\circ}-{50}^{\circ}$ ext. ∠ of △ $={13}^{\circ}$ 1A

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