Mathematics
In the following figure, O is the centre of the circle and AB is a diameter.
C and D are the points on the circumference of the circle such that ∠CAB = 35° and ∠ABD = 55°. Find the measures of angles CAD and CBD.
Circles
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Answer
We know that,
Angle in a semi-circle is a right angle.
⇒ ∠ACB = ∠ADB = 90°
In ΔACB, using angle sum property
⇒ ∠CAB + ∠CBA + ∠ACB = 180°
⇒ 35° + ∠CBA + 90° = 180°
⇒ 125° + ∠CBA = 180°
⇒ ∠CBA = 180° - 125°
⇒ ∠CBA = 55°
From figure,
⇒ ∠CBD = ∠CBA + ∠ABD = 55° + 55° = 110°.
Similarly, in ΔADB, using angle sum property
⇒ ∠DAB + ∠DBA + ∠ADB = 180°
⇒ ∠DAB + 55° + 90° = 180°
⇒ ∠DAB + 145° = 180°
⇒ ∠DAB = 180° - 145°
⇒ ∠DAB = 35°
From figure,
⇒ ∠CAD = ∠CAB + ∠DAB = 35° + 35° = 70°.
Hence, angles CAD = 70° and angle CBD = 110°.
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