Question

9. Construct Arguments Write a two-column proof for the Angle Bisector Theorem. MP.3

Answer 1

The angle bisector theorems is proved below.

What is the angle bisector theorems?

It should be noted that the angle bisector theorem simply states that an angle bisector of a triangle divides the opposite side into two segments which are proportional to the other sides of the triangle.

The way to proof the theorem is illustrated:

Draw a ray CX parallel to AD and then extend BA to intersect this ray at E.

In triangle CBE, DA is parallel to CE.

BD/DC == BA/AE ......... i

Now we want to prove that AE = AC

Since DA is parallel to CE, we have:

DAB = CEA (corresponding angles) ....... ii

DAC = ACE (alternate interior angles) ...... iii

Since AD is the bisector of BAC, we've DAB = DAC.

From the above, ACE makes and isosceles triangle and since the opposite sides are equal, we've AC = CE.

Substitute AC for AE in equation i

BD/DC = BA/AC

Therefore, the angle bisector theorems is proved.

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