Mathematics
In the figure given alongside, AD = AB = AC, BD is parallel to CA and angle ACB = 65°. Find angle DAC.
Triangles
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Answer
From figure,
In △ ABC,
⇒ AB = AC (Given)
⇒ ∠ABC = ∠ACB = 65° (Angles opposite to equal sides are equal)
By angle sum property of triangle,
⇒ ∠ABC + ∠ACB + ∠BAC = 180°
⇒ 65° + 65° + ∠BAC = 180°
⇒ 130° + ∠BAC = 180°
⇒ ∠BAC = 180° - 130° = 50°.
Given,
BD || CA
∴ ∠DBA = ∠BAC = 50° (Alternate angles are equal)
In △ DAB,
⇒ AD = AB (Given)
⇒ ∠ADB = ∠DBA = 50° (Angles opposite to equal sides are equal)
By angle sum property of triangle,
⇒ ∠ADB + ∠DBA + ∠DAB = 180°
⇒ 50° + 50° + ∠DAB = 180°
⇒ 100° + ∠DAB = 180°
⇒ ∠DAB = 180° - 100° = 80°.
From figure,
⇒ ∠DAC = ∠DAB + ∠BAC = 80° + 50° = 130°.
Hence, ∠DAC = 130°.
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