Download 5-6 Inequalities in One Triangle

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 mS = 24 and mO = 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
hinge) gets smaller.
the
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.