In the given figure, AP = BP = PC.
prove that Angle ABC is a right angle.
Answers
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Given
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- BP = BC = AP
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To Prove
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- ∠ABC = 90°
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Proof
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In △BPC
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BP = BC
∠PBC = ∠PCB [ Angles opposite to equal sides are also equal ]
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Let ∠PBC and ∠PCB be x
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In △APB
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PA = PB
∠PBA = ∠PAB [ Angles opposite to equal sides are also equal ]
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Let ∠PBA = ∠PAB be y
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In △ABC
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➞ ∠A + ∠B + ∠C = 180° [ Angle Sum Property ]
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➞ ∠PAB + ∠PBA + ∠PBC + ∠PCB = 180°
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➞ y + y + x + x = 180°
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➞ 2x + 2y = 180°
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➞ 2 ( x + y ) = 180°
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➞ x + y = 180° / 2
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➞ x + y = 90°
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So, ∠PBA + ∠PBC = 90°
⠀ ∠ABC = 90°
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Hence, ∠ABC is a right angle.
Given
⠀⠀⠀⠀⠀⠀
- BP = BC = AP
⠀⠀⠀⠀⠀⠀
To Prove
⠀⠀⠀⠀⠀⠀
- ∠ABC = 90°
⠀⠀⠀⠀⠀⠀
Proof
⠀⠀⠀⠀⠀⠀
In △BPC
⠀⠀⠀⠀⠀⠀
BP = BC
∠PBC = ∠PCB [ Angles opposite to equal sides are also equal ]
⠀⠀⠀⠀⠀⠀
Let ∠PBC and ∠PCB be x
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In △APB
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PA = PB
∠PBA = ∠PAB [ Angles opposite to equal sides are also equal ]
⠀⠀⠀⠀⠀⠀
Let ∠PBA = ∠PAB be y
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In △ABC
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➞ ∠A + ∠B + ∠C = 180° [ Angle Sum Property ]
⠀⠀⠀⠀⠀⠀
➞ ∠PAB + ∠PBA + ∠PBC + ∠PCB = 180°
⠀⠀⠀⠀⠀⠀
➞ y + y + x + x = 180°
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➞ 2x + 2y = 180°
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➞ 2 ( x + y ) = 180°
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➞ x + y = 180° / 2
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➞ x + y = 90°
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So, ∠PBA + ∠PBC = 90°
⠀ ∠ABC = 90°
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