# Question #e3694

Oct 19, 2017

see explanation.

#### Explanation:

a) given that $D C E$ is tangent to the circle, $\implies \angle D C O = {90}^{\circ}$,
$\implies \angle O C E = 180 - 90 = {90}^{\circ}$

b) recall that from the tangent-chord theorem, the angle between a tangent and a chord that meet on a circle, is equal to the inscribed angle on the opposite side of the chord,
$\implies \angle B A C = \angle B C D = {70}^{\circ}$

c) recall that the angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at the circumference,
$\implies \angle B O C = 2 \times \angle B A C = 2 \times 70 = {140}^{\circ}$

d) recall that the sum of the interior angles of a quadrilateral is ${360}^{\circ}$,
$\implies \angle B D C = 360 - \angle B O C - \angle D B O - \angle D C O$
$= 360 - 140 - 90 - 90 = {40}^{\circ}$