Question 747238: I really need help with the statement and reason of this problem.
Given: Line segment BD bisects angle ABC
Angle BAD is congruent to angle BCD
Prove: Triangle ADC is isosceles
So far I have this:
1. Line segment BD bisects angle ABC
Reason: Given
2. Angle BAD is congruent to angle BCD
Reason: Given
3. Line segment AB is congruent to line segment CB
Reason ?
4.Triangle ABC is isosceles
Reason: Def of isosceles triangle
5. Triangle ADC is isosceles
Reason: Triangle ABC is isosceles which makes triangle ADC isosceles; Def of isosceles triangle
I am really unsure about this problem as it just does not see correct but I do not understand how to make it correct.
Answer by timvanswearingen(106)   (Show Source):
You can put this solution on YOUR website! I believe the reason you're looking for on step 3 is the isosceles triangle theorem.
Since angle BAD is congruent to angle BCD, the sides opposite those angles in triangle ABC are congruent.
Thus, AB is congruent to CB.
Assuming this problem looks like a kite, after step 4,
5. let E be the point of intersection between AC and BD.
6. you need to show that triangle ABE is congruent to triangle CBE using ASA (angle side angle).
7. Then AE is congruent to CE by CPCTC. (corresponding parts of congruent triangles are congruent.)
8. Also, angle AEB is congruent to angle CEB by CPCTC
9. Angle CEB is congruent to angle AED and angle AEB is congruent to angle CED (vertical angles)
10. Thus, angle AED is congruent to angle CED (transitive)
11. ED is congruent to ED (reflexive)
12. By SAS (side angle side) triangle AED is congruent to triangle CED.
13. AD is congruent to CD by CPCTC.
14. Thus, triangle ADC is isosceles by the definition of an isosceles triangle.
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