prove that a triangle is isosceles if the bisector of the vertical angle bisects the base
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Given △ABC, AD is a bisector of ∠A which meets base BC at D such that BD=DC.
Produce AD to meet E such that AD=ED.
Now, in △ABD and △DEC
BD=DC ...... [Given]
AD=DE ........ [By construction]
∠ADB=∠EDC ..... [Vertically opposite angles]
∴ △ABD≅△EDC [∵SAS congruence ]
⟹AB=EC and ∠BAD=∠DEC ..... [CPCT]
Also, ∠BAD=∠DAC
⟹∠DAC=∠DEC
⟹ In △ACE, ∠AEC=∠CAE
⟹AC=CE ........ [Sides opposite to equal angles]
⟹AB=AC
Hence, △ABC is isosceles.
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