5.2 Inequalities and Triangles

Objectives 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).

Determine which angle has the greatest measure. ExploreCompare the measure of  1 to the measures of  2,  3,  4, and  5. PlanUse 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:

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. Compare m  2 to m  5. ExamineThe 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. Example 1:

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

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.  1 > m  3 m  1 > m  4 m  1 > m  3 m  1 > m  4

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 2a:

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. Example 2b:

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 Your Turn:

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

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 3a:

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 3b:

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

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 Your Turn:

Theorem 5.10 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 BC AC > BC > CA AC > BC > CA

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. Example 4:

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 Your Turn:

Assignment Geometry: Pg. 251 # 4 – 50 evens Pre-AP Geometry: Pg. 252 # 4 – 34, 38 – 42, 46 evens