Question : In the given figure, $\angle D B C=65^{\circ}, \angle B A C=35^{\circ}$ and $\mathrm{AB}=\mathrm{BC}$, then the measure of $\angle \mathrm{ECD}$ is equal to:
Option 1: 65°
Option 2: 50°
Option 3: 55°
Option 4: 45°
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Correct Answer: 45°
Solution :
$\angle$ DBC = 65$^\circ$
$\angle$ BAC = 35$^\circ$
AB = BC
Concept:
The sum of opposite angles in a cyclic quadrilateral is 180$^\circ$.
Angles made by the same chord on the same side of the circle are equal.
If two sides of a triangle are equal, it is an isosceles triangle and thus has two equal angles
Using the concept of the angle made by the same chord,
$\angle$ DBC = $\angle$ DAC
⇒ $\angle$ DAC = 65$^\circ$
$\triangle$ ABC is an isosceles triangle,
$\angle$ BAC = $\angle$ ACB
⇒ $\angle$ ACB = 35$^\circ$
Using the concept of cyclic quadrilateral,
$\angle$ DAB + $\angle$ DCB = 180$^\circ$
⇒ ($\angle$ DAC + $\angle$ BAC) + ($\angle$ DCA + $\angle$ ACB) = 180$^\circ$
⇒ (65$^\circ$ + 35$^\circ$) + ($\angle$ DCA + 35$^\circ$) = 180$^\circ$
⇒ $\angle$ DCA = 180$^\circ$ - (65$^\circ$ + 70$^\circ$)
⇒ $\angle$ DCA = 45$^\circ$
Since $\angle$ DCA = $\angle$ ECD
⇒ $\angle$ ECD = 45$^\circ$
$\therefore$ The measure of $\angle$ ECD is 45$^\circ$.
Hence, the correct answer is 45$^\circ$.
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