Download 5-3 Study Guide and Intervention(continued)

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5-3 Study Guide and Intervention
Inequalities in One Triangle
Angle Inequalities Properties of inequalities, including the Transitive, Addition, and Subtraction Properties of
Inequality, can be used with measures of angles and segments. There is also a Comparison Property of Inequality.
For any real numbers a and b, either a < b, a = b, or a > b.
The Exterior Angle Inequality Theorem can be used to prove this inequality involving an exterior angle.
Exterior Angle
Inequality Theorem
The measure of an exterior angle of a triangle
is greater than the measure of either of its
corresponding remote interior angles.
m∠1 > m∠A,
m∠1 > m∠B
Example : List all angles of EFG with measures that are less than m∠ 1.
The measure of an exterior angle is greater than the measure of either remote
interior angle. So m∠ 3 < m∠ 1 and m∠ 4 < m∠ 1.
Exercises
Use the Exterior Angle Inequality Theorem to list all of the angles that
satisfy the stated condition.
1. measures are less than m∠ 1
2. measures are greater than m∠ 3
3. measures are less than m∠ 1
4. measures are greater than m∠ 1
5. measures are less than m∠ 7
6. measures are greater than m∠ 2
7. measures are greater than m∠ 5
8. measures are less than m∠ 4
9. measures are less than m∠ 1
10. measures are greater than m∠ 4
NAME _____________________________________________ DATE ____________________________ BLOCK ______________
5-3 Study Guide and Intervention (continued)
Inequalities in One Triangle
Angle-Side Relationships When the sides of triangles are not congruent, there
is a relationship between the sides and angles of the triangles.
• If one side of a triangle is longer than another side, then the angle
opposite the longer side has a greater measure than the angle opposite
the shorter side.
• If one angle of a triangle has a greater measure than another angle, then the side
opposite the greater angle is longer than the side opposite the lesser angle.
If AC > AB, then m∠ B > m∠ C.
If m∠ A > m∠ C, then BC > AB.
Example 1: List the angles in order from smallest to
Example 2: List the sides in order from shortest to
largest measure.
longest.
∠ T, ∠ R, ∠ S
,
,
Exercises
List the angles and sides in order from smallest to largest.
1.
2.
3.
4.
5.
6.
8.
9.
7.
NAME _____________________________________________ DATE ____________________________ BLOCK ______________
5-5 Study Guide and Intervention
The Triangle Inequality
The Triangle Inequality If you take three straws of lengths 8 inches, 5 inches, and 1 inch and try to make a triangle
with them, you will find that it is not possible. This illustrates the Triangle Inequality Theorem.
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.
a+b>c
b+c>a
a+c>b
Example : The measures of two sides of a triangle are 5 and 8. Find a range for the length of the third side.
By the Triangle Inequality Theorem, all three of the following inequalities must be true.
5+x>8
8+x>5
5+8>x
x>3
x > –3
13 > x
Therefore x must be between 3 and 13.
Exercises
Is it possible to form a triangle with the given side lengths? If not, explain why not.
1. 3, 4, 6
2. 6, 9, 15
3. 8, 8, 8
4. 2, 4, 5
5. 4, 8, 16
6. 1.5, 2.5, 3
Find the range for the measure of the third side of a triangle given the measures of two sides.
7. 1 cm and 6 cm
9. 1.5 ft and 5.5 ft
8. 12 yd and 18 yd
10. 82 m and 8 m
11. Suppose you have three different positive numbers arranged in order from least to greatest. What single comparison
will let you see if the numbers can be the lengths of the sides of a triangle?
NAME _____________________________________________ DATE ____________________________ BLOCK ______________
5-6 Study Guide and Intervention
Inequalities in Two Triangles
Hinge Theorem The following theorem and its converse involve the relationship between the sides of two triangles and
an angle in each triangle.
Hinge Theorem
If two sides of a triangle are congruent to two sides of
another triangle and the included angle of the first is larger
than the included angle of the second, then the third side
of the first triangle is longer than the third side of the
second triangle.
Converse of the
Hinge Theorem
If two sides of a triangle are congruent to two sides of
another triangle, and the third side in the first is longer
than the third side in the second, then the included angle
in the first triangle is greater than the included angle in the
second triangle.
Example 1 : Compare the measures of
and
Two sides of △HGF are congruent to two sides of
△HEF, and m∠ GHF > m∠ EHF. By the Hinge
Theorem, GF > FE.
.
Example 2 : Compare the measures
of ∠ ABD and ∠ CBD.
Two sides of △ABD are congruent to two sides of
△CBD, and AD > CD. By the Converse of the Hinge
Theorem, m∠ ABD > m∠ CBD.
Exercises
Compare the given measures.
1. MR and RP
2. AD and CD
3. m∠ C and m∠ Z
4. m∠ XYW and m∠ WYZ
Write an inequality for the range of values of x.
5.
6.