The Triangle Inequality Inequalities in Multiple Triangles & Inequalities in Multiple Triangles

Objectives Apply the Triangle Inequality Theorem Recognize and apply properties of inequalities to the measures of angles in a triangle Recognize and apply properties of inequalities to the relationships between angles and sides of triangles

Inequalities An inequality simply shows a relationship between any real numbers a and b such that if a > b then there is a positive number c so a = b + c. All of the algebraic properties for real numbers can be applied to inequalities and measures of angles and segments (i.e. multiplication, division, and transitive).

Example 1: Determine which angle has the greatest measure. Explore Compare the measure of 1 to the measures of 2, 3, 4, and 5. Plan Use properties and theorems of real numbers to compare the angle measures.

Example 1: Solve Compare m3 to m1. By the Exterior Angle Theorem, m1 m3 m4. Since angle measures are positive numbers and from the definition of inequality, m1 > m3. Compare m4 to m1. By the Exterior Angle Theorem, m1 m3 m4. By the definition of inequality, m1 > m4. Compare m5 to m1. Since all right angles are congruent, 4 5. By the definition of congruent angles, m4 m5. By substitution, m1 > m5.

Example 1: Compare m2 to m5. By the Exterior Angle Theorem, m5 m2 m3. By the definition of inequality, m5 > m2. Since we know that m1 > m5, by the Transitive Property, m1 > m2. Examine The results on the previous slides show that m1 > m2, m1 > m3, m1 > m4, and m1 > m5. Therefore, 1 has the greatest measure. Answer: 1 has the greatest measure.

Your Turn: Determine which angle has the greatest measure. Answer: 5 has the greatest measure.

Exterior Angle Inequality Theorem If an  is an exterior  of a ∆, then its measure is greater than the measure of either of its remote interior s. m 1 > m 3 m 1 > m 4

Example 2a: Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m14. By the Exterior Angle Inequality Theorem, m14 > m4, m14 > m11, m14 > m2, and m14 > m4 + m3. Since 11 and 9 are vertical angles, they have equal measure, so m14 > m9. m9 > m6 and m9 > m7, so m14 > m6 and m14 > m7. Answer: Thus, the measures of 4, 11, 9,  3,  2, 6, and 7 are all less than m14 .

Example 2b: Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than m5. By the Exterior Angle Inequality Theorem, m10 > m5, and m16 > m10, so m16 > m5, m17 > m5 + m6, m15 > m12, and m12 > m5, so m15 > m5. Answer: Thus, the measures of 10, 16, 12, 15 and 17 are all greater than m5.

Your Turn: Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. a. all angles whose measures are less than m4 b. all angles whose measures are greater than m8 Answer: 5, 2, 8, 7 Answer: 4, 9, 5

Theorem 5.9 m 1 > m 2 > m 3 If one side of a ∆ is longer than another side, then the  opposite the longer side has a greater measure then the  opposite the shorter side (i.e. the longest side is opposite the largest .) 2 m 1 > m 2 > m 3 1 3

Example 3a: Determine the relationship between the measures of RSU and SUR. Answer: The side opposite RSU is longer than the side opposite SUR, so mRSU > mSUR.

Example 3b: Determine the relationship between the measures of TSV and STV. Answer: The side opposite TSV is shorter than the side opposite STV, so mTSV < mSTV.

Example 3c: Determine the relationship between the measures of RSV and RUV. mRSU > mSUR mUSV > mSUV mRSU + mUSV > mSUR + mSUV mRSV > mRUV Answer: mRSV > mRUV

Your Turn: Determine the relationship between the measures of the given angles. a. ABD, DAB b. AED, EAD c. EAB, EDB Answer: ABD > DAB Answer: AED > EAD Answer: EAB < EDB

Theorem 5.10 AC > BC > CA If one  of a ∆ has a greater measure than another , then the side opposite the greater  is longer than the side opposite the lesser . A AC > BC > CA B C

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?

Example 4: 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. Answer: So, Ebony should tie the ends marked Y and Z.

Your Turn: KITE ASSEMBLY Tanya is following directions for making a kite. She has two congruent triangular pieces of fabric that need to be sewn together along their longest side. The directions say to begin sewing the two pieces of fabric together at their smallest angles. At which two angles should she begin sewing? Answer: A and D

Theorem 5.11 ∆ Inequality Theorem The sum of the lengths of any two sides of a ∆ is greater than the length of the 3rd side. The ∆ Inequality Theorem can be used to determine whether 3 sides can form a triangle or not. d o g d + o > g o + g > d g + d > o

Example 1a: Determine whether the measures and can be lengths of the sides of a triangle. Answer: Because the sum of two measures is not greater than the length of the third side, the sides cannot form a triangle. HINT: If the sum of the two smaller sides is greater than the longest side, then it can form a ∆.

Example 1b: Determine whether the measures 6.8, 7.2, and 5.1 can be lengths of the sides of a triangle. Check each inequality. Answer: All of the inequalities are true, so 6.8, 7.2, and 5.1 can be the lengths of the sides of a triangle.

Your Turn: Determine whether the given measures can be lengths of the sides of a triangle. a. 6, 9, 16 b. 14, 16, 27 Answer: no Answer: yes

Example 2: Multiple-Choice Test Item In and Which measure cannot be PR? A 7 B 9 C 11 D 13

Example 2: Read the Test Item You need to determine which value is not valid. Solve the Test Item Solve each inequality to determine the range of values for PR.

Example 2: Graph the inequalities on the same number line. The range of values that fit all three inequalities is

Example 2: Examine the answer choices. The only value that does not satisfy the compound inequality is 13 since 13 is greater than 12.4. Thus, the answer is choice D. Answer: D

Your Turn: Multiple-Choice Test Item Which measure cannot be XZ? A 4 B 9 C 12 D 16 Answer: D