Question

prove that a triangle is isosceles if the bisector of the vertical angle bisects the base
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Answer 1

Answer:

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Answer 2

Answer:

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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