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In the figure (2) given below, prove that

(i) x + y = 90°

(ii) z = 90°

(iii) AB = BC.

In the figure (2) given below, prove that x + y = 90°, z = 90°, AB = BC. Triangles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

Triangles

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Answer

(i) From figure,

In the figure (2) given below, prove that x + y = 90°, z = 90°, AB = BC. Triangles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 9.

∠ACB = x (Alternate angles)

In △ABC,

⇒ x + (y + y) + ∠ACB = 180°

⇒ x + 2y + x = 180°

⇒ 2x + 2y = 180°

⇒ x + y = 90°.

Hence, proved that x + y = 90°.

(ii) Now in △BCD,

⇒ y + z + ∠BCD = 180° (Sum of all angles in a triangle is 180°)

⇒ y + z + x = 180°

⇒ 90° + z = 180° [∵ x + y = 90°]

⇒ z = 90°.

Hence, proved that z = 90°.

(iii) In △ABC,

∠ACB = ∠BAC = x

∴ AB = BC (As sides opposite to equal angles are equal.)

Hence, proved that AB = BC.

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