Given: In triangle ABC, <A=135 degree
Prove: <B is not equal to 45 degree ( Write an indirect proof)
The angles add up to 180. If angle B is 45, then that leaves angle C = 0. Not much of a triangle.
To prove that <B is not equal to 45 degrees in triangle ABC using an indirect proof, we start by assuming the opposite, that is, we assume <B = 45 degrees. Then, we will try to derive a contradiction, which would show that our assumption is incorrect. Here is the proof:
1. Assume <B = 45 degrees (Indirect Assumption)
2. The sum of the angles in a triangle is 180 degrees (Triangle Sum Theorem)
3. <A + <B + <C = 180 degrees (Sum of angles in triangle ABC)
4. 135 degrees + 45 degrees + <C = 180 degrees (Substitute given values)
5. 180 degrees + <C = 180 degrees (Simplify)
6. <C = 0 degrees (Subtract 180 degrees from both sides)
7. A triangle must have positive angles (Definition of angles)
8. <C cannot be 0 degrees (Contradiction)
9. Therefore, our initial assumption that <B = 45 degrees must be incorrect.
10. Hence, we conclude that <B is not equal to 45 degrees in triangle ABC. (Proof by contradiction)
By assuming <B to be 45 degrees and reaching a contradiction, we have successfully proven that <B cannot equal 45 degrees in triangle ABC.