5-5 Indirect Proof

Indirect Reasoning

• In indirect reasoning, all possibilities are considered and then all but one are proved false.– The remaining possibility must be true!

• A proof involving indirect reasoning is an indirect proof.–Also known as “proof by contradiction”.

How to Write an Indirect Proof

1. State as a temporary assumption the opposite (negation) of what you want to prove.

2. Show that this temporary assumption leads to a contradiction.

3. Conclude that the temporary assumption must be false and that what you want to prove must be true.

Writing an Indirect Proof

Given: 7(x + y) = 70 and x ≠ 4.Prove: y ≠ 6

Proof: Assume temporarily that y = 6. Then 7(x + 6) = 70; divide each side by 7 to get x + 6 = 10 and so x = 4. But this contradicts the given statement that x ≠ 4. The temporary assumption that y = 6 led to a contradiction, so we can conclude that y ≠ 6.

5-6 Inequalities in One Triangle

Angle/Side Relationships

• If two sides of a triangle are not congruent, then the larger angles is opposite the longer side.

• If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.

Ordering Sides of a Triangle

• List the sides of TUV in order from shortest to longest.

If mS = 24 and mO = 130, which side of SOX is shortest? Explain.

Side Lengths in Triangles

• For segments to form a triangle, their lengths must be related in a certain way.

• Notice that only one of the sets of segments below can form a triangle.

Triangle Inequality Theorem: The sum of the lengths of ANY two sides of a triangle MUST be greater than the length of the third side.

Using the Triangle Inequality Theorem

• Can a triangle have sides with the given lengths?– 3, 7, 8

– 5, 10, 15

2, 6, 9

4, 6, 9

Finding Possible Side Lengths

• Two sides of a triangle measure 5 and 8. What is the range of possible lengths for the third side?

Two sides of a triangle measure 4 and 7. What is the range of possible lengths for the third side?

5-7 Inequalities in Two Triangles

The Hinge Theorem

• When you close a door, the angle between the door and the frame (at the hinge) gets smaller.

Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle.

Using the Hinge Theorem

• What inequality relates SK to YU?

What inequality relates LN to OQ?

The Converse of the Hinge Theorem

• If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.