Geometry / Unit 6: Relationships in Triangles
6.6 Indirect Proof and the Hinge Theorem
Notes
An indirect proof assumes the opposite of what you want to prove, then shows that leads to a contradiction.
- Step 1: assume the conclusion is false. Step 2: reason to a contradiction. Step 3: conclude the original statement is true.
- Hinge Theorem: two triangles have two pairs of congruent sides. The triangle with the larger included angle has the longer third side. The converse also holds.
- Example: two doors of equal width; the one opened wider has a larger gap from frame to edge.
Practice
To prove "$\angle A$ is not a right angle" indirectly, what do you assume first?
Assume the opposite of the conclusion.
In $\triangle ABC$ and $\triangle DEF$, $AB = DE$, $AC = DF$, $m\angle A = 50^\circ$ and $m\angle D = 65^\circ$. Which is true?
The larger included angle, $\angle D$, is opposite the longer side, $\overline{EF}$.
Two triangles share two pairs of congruent sides. The third sides are 9 and 12. Which statement is true about the included angles?
By the converse of the Hinge Theorem, the longer third side faces the larger included angle.
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