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5.2 Inequalities and Triangles

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.

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5.2 Inequalities and Triangles

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  1. 5.2 Inequalities and Triangles

  2. 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

  3. Inequalities • An inequalitysimply 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).

  4. 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.

  5. 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. By the Exterior Angle Theorem, m1 m3 m4. By the definition of inequality, m1 > m4. 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: Solve Compare m3 to m1. Compare m4 to m1. Compare m5 to m1.

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

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

  8. 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 3m 1 > m 4

  9. 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 .

  10. 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, andm12 > m5, som15 > m5. Answer: Thus, the measures of 10, 16, 12, 15 and 17 are all greater than m5.

  11. 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 Your Turn: Answer:5, 2, 8, 7 Answer:4, 9, 5

  12. 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 .) 2 m 1 > m 2 > m 3 3 1

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

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

  15. Example 3c: Determine the relationship between the measures of RSVand 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

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

  17. 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 AC > BC > CA B C

  18. 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:

  19. 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 Xis opposite the longest side it has the greatest measure. Answer: So, Ebony should tie the ends marked Y and Z.

  20. 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

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

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