Download 5-6 Inequalities in One Triangle

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.
Using the Converse of the Hinge
Theorem
• What is the range of possible values for x?
What is the range of possible
values for x?