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Standard VIII

Mathematics

Congruency Postulates

In an isoscel...

Question

In an isosceles triangle $A B C$ , with $A B = A C$ , the bisectors of $∠ B$ and $∠ C$ intersect each other $O$ . Join $A$ to $O$ . Show that $A O$ bisects $∠ A$

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Solution

To prove:
$∠ O A B = ∠ O A C$
$O B = O C$ ............(1)
Also, in $Δ A B O$ and $Δ A C O$ , we have
$A B = A C$ (Given)
$A O = A O$ (Common)
$O B = O C$ (From(1))
$∴ Δ A B O ≅ Δ A C O$ (SSS congruence rule)
$\Rightarrow ∠ O A B = ∠ O A C$ (CPCT)
Hence proved

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Question 1 (ii)
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In an isosceles triangle ABC, with AB=AC, the bisectors of ∠B and ∠C intersect each other O. Join A to O. Show that AO bisects ∠A

To prove:∠OAB=∠OACOB=OC ............(1)Also, in ΔABO and ΔACO, we haveAB=AC (Given)AO=AO (Common)OB=OC (From(1))∴ΔABO≅ΔACO (SSS congruence rule)⇒∠OAB=∠OAC (CPCT)Hence proved

In an isosceles triangle $A B C$ , with $A B = A C$ , the bisectors of $∠ B$ and $∠ C$ intersect each other $O$ . Join $A$ to $O$ . Show that $A O$ bisects $∠ A$

To prove:
$∠ O A B = ∠ O A C$
$O B = O C$ ............(1)
Also, in $Δ A B O$ and $Δ A C O$ , we have
$A B = A C$ (Given)
$A O = A O$ (Common)
$O B = O C$ (From(1))
$∴ Δ A B O ≅ Δ A C O$ (SSS congruence rule)
$\Rightarrow ∠ O A B = ∠ O A C$ (CPCT)
Hence proved