Question 201873: Have a triangle (JKL) with J as point of triangle with a line going down to line K L and the point on the line where it meets is M .the angles created at M are right angles. The given is that angle KJM is equal to angle LJM and line JM bisects line KL. How do I write a 2 column proof for tringle JKL as an isosceles.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! If I understand you correctly, your givens are:
1. angle JMK and angle JML are right angles.
2. angle KJM is congruent to angle LJM
3. line JM bisects line KL.
start of note 1:
I drew triangle KJL with K on the lower left, J on top, and L on the lower right.
end of note 1:
from this you want to prove that triangle KJL is an isosceles triangle.
first you want to prove that triangle KJM and triangle LJM are congruent.
this would lead to angle JKM and angle JLM being equal.
in the proof below, the statement is in the left hand column and the reason is in the right hand column (in parentheses)
start of example 1:
1. i will prove this for you (i think i know how).
the statement above is i will prove this for you which is in the left hand column. the reason is i think i know how which is shown in parentheses which means it will be in the right hand column.
end of example 1:
start of proof:
1. angle KMJ and angle LMJ are right angles (given).
2. angle KMJ is congruent to angle LMJ (all right angles are congruent).
3. segment JM bisects segment KL at M (given).
4. segment KM is congruent to ML (segment bisector creates 2 congruent segments).
5. triangle KJM is congruent to triangle LJM (SAS).
start of note for 5:
the congruent segments are JM to JM and KM to ML
the congruent angle between them is angle KMJ to angle LMJ
end of note for 5:
6. angle MKJ congruent to angle MLJ (corresponding angles of congruent triangles are congruent).
7. triangle KJL is an isosceles triangle (if 2 corresponding angles of a triangle are congruent, then the triangle is isosceles).
end of proof:
start of note 2:
since you stated that segment JM bisects segment KL at a right angle, I did not need the fact that angle KJM is congruent to angle LJM because I had enough to prove triangles KJM and LJM were congruent without it.
end of note 2:
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