If the bisector of the vertical angle of a triangle bisects the base,prove that the triangle is isosceles.

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In triangle ABC,

<BAD=<CAD

BD=CD

AD bisect AC

Extend AD to point E such that CE is parallel to AB. Join CE

Taking transversal AE intersecting AB and CE,

<BAD=<DEC(interior alternate angles)

We know <BAD=<CAD=<DEC

In triangle ACE,

<CAD=<DEC

Therefore,. AC=CE.

In triangles ABD and ECD,

<ADB=<EDC(vertical opposite angles)

<BAD=<CED(interior alternate angles)

BD=CD(given)

By AAS triangle ABD is congruent to ECD

Therefore, AB=EC(CPCT)

We know EC=AC(Already proven)

Therefore, AB = AC and triangle ABC is isosceles.

• 36

AD bisect AC

Extend AD to point E such that CE is parallel to AB. Join CE

Taking transversal AE intersecting AB and CE,

We know

In triangle ACE,

Therefore,. AC=CE.

In triangles ABD and ECD,

BD=CD(given)

By AAS triangle ABD is congruent to ECD

Therefore, AB=EC(CPCT)

We know EC=AC(Already proven)

Therefore, AB = AC and triangle ABC is isosceles.

Posted by Sai Teja(student), on 9/9/12
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