Answer:
CAO = 21°
Step-by-step explanation:
In a circle, the angle of tangency and the inscribed angle subtended by the same arc are equal in measures
In circle O
∵ DAE is a tangent to circle O at point A
∵ AB is a chord
∴ ∠BAE is an angle of tangency subtended by the arc AB
∵ ∠BCA is an inscribed angle subtended by the arc AB
→ By using the rule above
∴ m∠BAE = m∠BCA
∵ m∠BAE = 53°
∴ m∠BCA = 53°
In ΔBOC
∵ OB and OC are radii
∴ OB = OC
∴ ΔBOC is an isosceles Δ
→ That means its base angles are equal
∴ m∠OBC = m∠OCB
∵ m∠OBC = 32°
∴ m∠OCB = 32°
∵ m∠BCA = m∠BCO + m∠OCA
∴ 53 = 32 + m∠OCA
→ Subtract 32 from both sides
∴ 21 = m∠OCA
∴ m∠OCA = 21°
In ΔAOC
∵ OA and OC are radii
∴ OA = OC
∴ ΔAOC is an isosceles Δ
→ That means its base angles are equal
∴ m∠CAO = m∠OCA
∵ m∠OCA = 21°
∴ m∠CAO = 21°
∴ CAO = 21°
