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