SOLUTION: Can you solve this proof for me. I tried every thing and could not prove this triangle congruent. Given: DA is perpendicular to AB, DC is perpendicular to CB, and AB is congruent

Algebra ->  Geometry-proofs -> SOLUTION: Can you solve this proof for me. I tried every thing and could not prove this triangle congruent. Given: DA is perpendicular to AB, DC is perpendicular to CB, and AB is congruent       Log On


   

Question 168787: Can you solve this proof for me. I tried every thing and could not prove this triangle congruent.
Given: DA is perpendicular to AB, DC is perpendicular to CB, and AB is congruent to CB. Prove: angle CBD is congruent to angle ABD
Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
it looks like the triangles are congruent by the hypotenuse-leg postulate.
that states:
-----
The Hypotenuse-Leg Postulate is a rule that you can use with right triangles only. This rule is considered a postulate because it is not based on any other rules, as the theorems discussed above have been. It states if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
-----
based on the information you provided, i sketched the triangles.
it appears that BD is common to both triangles.
if you draw BD horizontal.
then draw BC going up at about a 45 degree angle from BD (not necessarily the real angle - just used for drawing purposes).
then draw BA going down at about a 45 degree angle from BD.
connect C to D forming a right angle at C (given)
connect A to D forming a right angle at A (given)
you have 2 triangles adjacent to each other with a commo side of BD.
angle BCD is right angle (given).
angle BAD is right angle (given).
BC congruent to BA (given).
BD congruent to BD (same line segment)
BA and BC are legs of their respective triangles (BCD ad BAD).
BD is the hypotenuse of triangles BCD and BAD since it is opposite the 90 degree angle of each.
that makes the hypotenuse leg postulate work.
triangles are congruent.
angle CBD is congruent to angle ABD (corresponding angles of congruent triangles are congruent).
-----
information about right triangles being congruent was found at this website:
http://library.thinkquest.org/20991/textonly/geo/crtri.html#LL