Question

In circle A shown below, Segment BD is a diameter and the measure of Arc CB is 36°: Points B, C, D lie on Circle A; line segment BD is the diameter of circle A; measure of arc CB is 36 degrees. What is the measure of ∠DBC?

Answer 1

Answer:

The measure of ∠DBC is 72°.

Step-by-step explanation:

Given information: BD is a diameter, A is center of circle, Arc CB=36°.

According the angled inscribed in a semicircle theorem, the angle inscribed in a semicircle is a right angle.

$\angle BCD=90^{\circ}$

According to the central angle theorem, the angle inscribed on the circle is half of its central angle.

$\angle BDC=\frac{\angle BAC}{2}=\frac{36^{\circ}}{2}=18^{\circ}$

By angle sum property, the sum of interior angles of a triangle is 180 degrees.

$\angle BDC+\angle BCD+\angle DBC=180$

$18^{\circ}+90^{\circ}+\angle CBD=180^{\circ}$

$108^{\circ}+\angle CBD=180^{\circ}$

$\angle CBD=180^{\circ}-108^{\circ}$

$\angle CBD=72^{\circ}$

Therefore the measure of ∠DBC is 72°.

Answer 2

The measure of ∠DBC is 36°

In the circle, BD is the diameter and BC is the minor chord on the circle, due to which ∠BCD = 90°, in triangle BDC

Then, 
Given arc BD measure is 54° i.e ∠BDC is 54°

The, from property of triangle, sum of all angles is 180°, we can find the third angle i.e. ∠DBC.

Thus, ∠DBC = 180°- 90°- 54° 
        ∠DBC = 36°