Question:
∆ABC is inscribed inside a circle and there is a point D on the arc BC opposite to A such that BD = CD. If ∠BAC = 70° and ∠ABD = 85°, then find the measure of ∠BCA.
∆ABC is inscribed inside a circle and there is a point D on the arc BC opposite to A such that BD = CD. If ∠BAC = 70° and ∠ABD = 85°, then find the measure of ∠BCA.
Updated On: Jul 22, 2025
- 45°
- 55°
- 35°
- 60°
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The Correct Option is D
Solution and Explanation
The correct option is (D): 60°
The opposite angles in a cyclic quadrilateral are supplementary.
∠BAC + ∠BDC = 180°
∠ABD + ∠ACD = 180°
∠BDC = 110°
∠ACD = 95°
In ∆BCD:
∠BCD = ∠CBD [BD = CD]
∠BCD + ∠CBD + ∠BDC = 180°
∠BCD = 35°
∠ACD = 95°
∠BCA + ∠BCD = 95°
∠BCA = 60°
The opposite angles in a cyclic quadrilateral are supplementary.
∠BAC + ∠BDC = 180°
∠ABD + ∠ACD = 180°
∠BDC = 110°
∠ACD = 95°
In ∆BCD:
∠BCD = ∠CBD [BD = CD]
∠BCD + ∠CBD + ∠BDC = 180°
∠BCD = 35°
∠ACD = 95°
∠BCA + ∠BCD = 95°
∠BCA = 60°
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