5-3 Inequalities in One Triangle

You found the relationship between the angle measures of a triangle.

• Recognize and apply properties of inequalities to the measures of the angles of a triangle.

• Recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle.

Page 344

The definition of inequality and the properties of inequalities can be applied to the measures of angles and segments, since these are real numbers.

1

2

3

Since 11 and 9 are vertical angles, they have equal measure, so m14 > m9. m9 > m6 and m9 > m7, so m14 > m6 and m14 > m7.

By the Exterior Angle Inequality Theorem, m14 > m4 and m14 > m11. In addition, m14 > m2 and m14 > m4 + m3, so m14 > m4 and m14 > m3.

A.

B.

C.

D.

The Littlest Angle

1. Draw a scalene triangle.2. Measure the three sides.3. Measure the three angles.4. What relationship do you see

between the sides and the angles?

A

B

C

Name the largest and smallest angles

911

12

A

BC

Name the largest and smallest anglesD

E F

5.6

5.5

110°

List the angles of ΔABC in order from smallest to largest.

Answer: C, A, B

The sides from the shortest to longest are AB, BC, and AC. The angles opposite these sides are C, A, and B, respectively. So, according to the Angle-Side Relationship, the angles from smallest to largest are C, A, B.

Name the longest and shortest segments

A

B C

130°

30°

20°

Name the longest and shortest segmentsV

W X

49°

List the sides of ΔABC in order from shortest to longest.

Answer: AC, AB, BC

The angles from smallest to largest are B, C, and A. The sides opposite these angles are AC, AB, and BC, respectively. So, the sides from shortest to longest are AC, AB, BC.

Theorems

If two sides of a triangle have unequal lengths, then the measure of the angle opposite the longer side is greater than the measure of the angle opposite the shorter side.

If two angles of a triangle have unequal measures, then the side opposite the large angle is longer than the side opposite the smaller angle

5-3 Assignment• Page 348, 8-12, 14-19, 22-

28 even