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In Fig., AB divides ∠DAC in the ratio 1: 3 and AB =DB. Determine the value of x.

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Given,

AB divides ∠DAC in the ratio 1: 3

∠DAB: ∠BAC = 1: 3

∠DAC + ∠EAC = 180°

∠DAC + 108° = 180°

∠DAC = 180° – 108°

= 72°

∠DAB = \(\frac{1}{4}\) x 72° = 18°

∠BAC = \(\frac{3}{4}\) x 72° = 54°

In ΔADB

∠DAB + ∠ADB + ∠ABD = 180°

18° + 18° + ∠ABD = 180°

36° + ∠ABD = 180°

∠ABD = 180° – 36°

= 144°

∠ABD + ∠ABC = 180°(Linear pair)

144° + ∠ABC = 180°

∠ABC = 180° – 144°

= 36°

In ΔABC

∠BAC + ∠ABC + ∠ACB = 180°

54° + 36° + x = 180°

90° + x = 180°

x = 180° – 90°

= 90°

Thus, x = 90°

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