tex]ABC = 1 rt.angle.~Thanks in advance ! ♡

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TheAnimeGirl

High School

[tex] \underline{ \underline{ \text{question}}} : [/tex]
In the given figure , AP = BP = PC. Prove that [tex] \angle[/tex]ABC = 1 rt.angle.

~Thanks in advance ! ♡

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xKelvin

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Answer:
See Below.
Step-by-step explanation:
In the given figure, AP = BP = PC.
And we want to prove that ∠ABC is a right angle.
Since AP = BP and BP = PC, we can create two isosceles triangles: ΔAPB and ΔCPB.
By the definition of isosceles triangles, in ΔAPB, ∠PAB and ∠PBA are equivalent. Let the measure of each of them be x°.
Likewise, in ΔCPB, ∠PCB and ∠PBC are equivalent.
And since AP = BP = PC, each of the angles∠PCB and ∠PBC will also be equivalent to x°.
And since the sum of the interior angles of a triangle total 180°, we acquire:
$\angle PAB+\angle PBA+\angle PCB+\angle PBC=180$
Since they are all equivalent:
$4x=180$
Hence:
$x=45^\circ$
∠ABC is the sum of ∠PBA and ∠PBC, each of which measures 45°. Hence:
$\angle ABC=\angle PBA+\angle PBC=45+45=90^\circ$
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