Indirect Proofs
Prove a triangle cannot have 2 right angles.
1) Given triangle ABC.
2) Assume angle A and angle B are both right angles is true by one of two possibilities (it is either true or false, so we assume it is true)
This is my first time to do indirect proofs, so if someone can talk me through it so I can do the next problems by myself I would be greatful.
To prove that a triangle cannot have two right angles, you can use an indirect proof, also known as a proof by contradiction. Here's how you can proceed with your indirect proof:
1) Start by assuming the opposite of what you want to prove. So, in this case, assume that a triangle can have two right angles.
2) Let's call the triangle ABC, and assume that angle A and angle B are both right angles.
3) Since angle A and angle B are both right angles, each measures 90 degrees. Therefore, the sum of angle A and angle B would be 90 + 90 = 180 degrees.
4) However, according to the angle sum property of triangles, the sum of the interior angles of a triangle is always 180 degrees.
5) If angle A and angle B both measure 90 degrees, then the third angle, angle C, would have to measure 0 degrees in order to make the sum of the angles 180 degrees.
6) But having an angle measure of 0 degrees means that angle C would not exist, as it would be a degenerate triangle.
7) Therefore, our assumption that a triangle can have two right angles leads to a contradiction.
8) Since our assumption leads to a contradiction, it must be false.
9) Consequently, we can conclude that a triangle cannot have two right angles.
By following this proof, you have shown that a triangle cannot have two right angles. Keep practicing with more examples to strengthen your understanding of indirect proofs.
The sum of the angles of a triangle must be 180 degrees. If two angles are 90 degrees, the third angle must be zero.
You need three non-zero angles to make a triangle.
Therefore a triangle cannot have two right angles.
QED