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In the figure (ii) given below, ABC is an isosceles triangle with AB = AC. If ∠ABC = 50°, find ∠BDC and ∠BEC.

In the figure (ii) given below, ABC is an isosceles triangle with AB = AC. If ∠ABC = 50°, find ∠BDC and ∠BEC. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

Circles

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Answer

Since, AB = AC.

Hence, in △ABC,

∠ACB = ∠ABC = 50°.

Since sum of angles in a triangle = 180°.

In △ABC,

⇒ ∠ABC + ∠ACB + ∠BAC = 180°
⇒ 50° + 50° + ∠BAC = 180°
⇒ ∠BAC + 100° = 180°
⇒ ∠BAC = 180° - 100° = 80°.

From figure,

∠BDC = ∠BAC = 80° (∵ angles in same segment are equal.)

In cyclic quadrilateral sum of opposite angles = 180°,

Hence in BDCE,

⇒ ∠BDC + ∠BEC = 180°
⇒ 80° + ∠BEC = 180°
⇒ ∠BEC = 180° - 80° = 100°.

Hence, the value of ∠BDC = 80° and ∠BEC = 100°.

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