Question

In the circle below, AB is a diameter. Suppose stack CB equals 32 degrees and m angle B C D equals 52 degrees.

Answer 1

The 32° measure of arc CB, and m<BCD which is 52° gives m<BAC and m<ACB as 16° and 90° respectively, from which we have;

• m<CBA = 74°

• m<ACD = 38°

Which circle theory can be used to find the required angles?

First part;

Given;

Angle subtended by arc CB = 32°

m<BCD = 52°

Based on circle theory, we have;

• Angle subtended by an arc at the center of a circle is twice the angle subtended at the circumference

Therefore;

Arc CB = 2 × m<BAC

Which gives;

Arc CB = 32° = 2 × m<BAC

m<BAC = 32° ÷ 2 = 16°

Angle subtended by the diameter at the circumference is 90°.

Therefore;

m<ACB = 90°

In triangle ∆ABC, we have;

m<ACB + m<BAC + m<CBA = 180°

m<CBA = 180° - (m<ACB + m<BAC)

Therefore;

m<CBA = 180° - (90° + 16°)

m<CBA = 180° - (90° + 16°) = 74°

• m<CBA = 74°

Second part;

The arc subtending m<ACD is AB which is also the diameter.

Angle formed by the diameter, which is a straight line = 180°

Therefore;

Angle subtended at the center by arc AB = 180°

Angle subtended at the circumference, m<ACB is therefore;

m<ACB = 180° ÷ 2 = 90°

A

m<ACB = m<ACD + m<BCD (Angle addition property)

Therefore;

90° = m<ACD + 52°

m<ACD = 90° - 52° = 38°

• m<ACD = 38°

Learn more about relationships of the arc of a circle in geometry here:

brainly.com/question/3123587

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