Section 5.3Inequalities in One Triangle

The definition of inequality and the properties of inequalities can be applied to the measures of angles and segments, since these are real numbers. Consider Ð1, Ð2, and Ð3 in the figure shown.

By the Exterior Angle Theorem, you know that mÐ1 = mÐ2 + mÐ3.

Since the angle measures are positive numbers, we can also say that

mÐ1 > mÐ2 and mÐ1 > mÐ3

by the definition of inequality.

Example 1: Use the diagram below.

a) Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than mÐ14.

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.

Thus, the measures of 4, 11, 9, 3, 2, 6, and 7 are all less than m14.

b) Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than mÐ5.

By the Exterior Angle Inequality Theorem, m10 > m5 and m16 > m10, so m16 > m5. Since 10 and 12 are vertical angles, m12 > m5. m15 > m12, so m15 > m5. In addition, m17 > m5 + m6, so m17 > m5.

The longest side and largest angle of ∆ABC are opposite each other. Likewise, the shortest side and smallest angle are opposite each other.

Example 2: List the angles of ΔABC in order from smallest to largest.

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.

Example 3: List the sides of ΔABC in order from shortest to longest.

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.

Example 4: HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie?

Theorem 5.10 states that 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. Since X is opposite the longest side, it has the greatest measure.

So, Ebony should tie the ends marked Y and Z.