Jason B.

asked • 12/26/15

given line ac is congruent to line cb prove triangle abc congruent to triangle bcd

CD looks like its cutting the triangle in half

Marlene S.

tutor
With AC congruent to CB, I'm going to assume these are the sides of the triangle.  From your post, I'm also going to assume that point D is between points A and B and "bisects" angle C.  But without more information these are only assumptions.  Do you have any more information to share with us?  For example, are there any marks on the triangle?  Does the line CD make a right angle with the base AB?  I'm not asking if it looks that way but rather is it marked as such.  Are there any hashmarks on the triangle.  For example if AC is congruent to CB you might see a small strikethrough on those segments indicating congruency.  Are there any others, like AD and BD, or perhaps, congruent angle marks on angles ACD and BCD?  Are you told that point D bisects AB?  Just some thoughts regarding additional information anyone would need to assist you in working this proof.
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12/26/15

1 Expert Answer

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Steven C. answered • 12/28/15

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Proof:
Since ac is congruent to bc, the triangle is isosceles. Thus, when a line bisects angle acb (which is what I'm assuming line cd does), then since the triangle is isosceles, this bisecting ray is a perpendicular bisector, thus the length ab is cut in half by line segment cd. Therefore, we will have ac ≅ bc, ad ≅ bd and cd = cd. Therefore, acd and abc are congruent triangles.
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