Geometry Mrs. Spitz Fall, 2004

5.5 Inequalities in One Triangle

Objectives:

• Use triangle measurements to decide which side is longest or which angle is largest.

• Use the Triangle Inequality

Assignment

pp. 298-300 #1-25, 34

Objective 1: Comparing Measurements of a Triangle

• In activity 5.5, you may have discovered a relationship between the positions of the longest and shortest sides of a triangle and the position of its angles.

The diagrams illustrate Thms. 5.10

and 5.11.

longest side smallest angle largest angle shortest side

Theorem 5.10

If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

A 3 m  A B > m  C 5 C

Theorem 5.11

If one ANGLE of a triangle is larger than another ANGLE, then the D 60 ° 40 ° SIDE opposite larger angle is the F longer than the side opposite the smaller angle.

EF > DF You can write the measurements of a triangle in order from least to greatest. E

Ex. 1: Writing Measurements in Order from Least to Greatest

Write the measurements of the triangles from least to greatest.

a. m  G < m  H < m  J H JH < JG < GH 45 ° J 100° 35 ° G

Ex. 1: Writing Measurements in Order from Least to Greatest

Write the measurements of the triangles from least to greatest.

Q 8 5 7 b. QP < PR < QR m  R < m  Q < m  P P R

Paragraph Proof – Theorem 5.10

A Given►AC > AB Prove ►m  ABC > m  C 2 D 1 3 B Use the Ruler Postulate to locate a point D on AC such that DA = BA. Then draw the segment BD. In the isosceles triangle ∆ABD,  1 ≅  2. Because m  ABC = m  1+m  3, it follows that C m  ABC > m  1. Substituting m  2 for m  1 produces m  ABC > m  2. Because m  2 = m  3 + m  C, m  2 > m  C. Finally because m  ABC > m  2 and m  2 > m  C, you can conclude that m  ABC > m  C.

NOTE:

The proof of 5.10 in the slide previous uses the fact that  2 is an exterior angle for m  ∆BDC, so its measure is the sum of the measures of the two nonadjacent interior angles. Then 2 must be greater than the measure of either nonadjacent interior angle. This result is stated in Theorem 5.12

Theorem 5.12-Exterior Angle Inequality

• The measure of an exterior angle of a triangle is greater than the measure of either of the two non adjacent interior angles.

• m  1 > m  A and m  1 > m  B A 1 C B

Ex. 2: Using Theorem 5.10

• DIRECTOR’S CHAIR. In the director’s chair shown, AB ≅ AC and BC > AB. What can you conclude about the angles in ∆ABC?

A B C

Ex. 2: Using Theorem 5.10

Solution

• Because AB ∆ABC is isosceles, so  B ≅  ≅ AC, C. Therefore, m  B = m  C. Because BC>AB, m  A > m  C by Theorem 5.10. By substitution, m  A > m  B. In addition, you can conclude that m  A >60 ° , m  B< 60 ° , and m  C < 60 ° .

B A C

Objective 2: Using the Triangle Inequality

• Not every group of three segments can be used to form a triangle. The lengths of the segments must fit a certain relationship.

Ex. 3: Constructing a Triangle

a. 2 cm, 2 cm, 5 cm b. 3 cm, 2 cm, 5 cm c. 4 cm, 2 cm, 5 cm Solution: Try drawing triangles with the given side lengths. Only group (c) is possible. The sum of the first and second lengths must be greater than the third length.

Ex. 3: Constructing a Triangle

a. 2 cm, 2 cm, 5 cm b. 3 cm, 2 cm, 5 cm c.

4 cm, 2 cm, 5 cm 2 5 2 A 3 5 C D 2 B A 4 5 D 2 B

Theorem 5.13: Triangle Inequality

• The sum of the lengths of any two sides of a Triangle is greater than the length of the third side.

AB + BC > AC AC + BC > AB AB + AC > BC C B A

Ex. 4: Finding Possible Side Lengths

• A triangle has one side of 10 cm and another of 14 cm. Describe the possible lengths of the third side • SOLUTION: Let x represent the length of the third side. Using the Triangle Inequality, you can write and solve inequalities.

x + 10 > 14 x > 4 10 + 14 > x 24 > x ►So, the length of the third side must be greater than 4 cm and less than 24 cm.

#24 - homework

x + 2 A • Solve the inequality: AB + AC > BC.

B 3x - 2 x + 3 C (x + 2) +(x + 3) > 3x – 2 2x + 5 > 3x – 2 5 > x – 2 7 > x

5. Geography

AB + BC > AC MC + CG > MG 99 + 165 > x 264 > x 99 miles Masbate x + 99 < 165 x < 66 Cadiz 165 miles 66 < x < 264 Guiuan