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5.2 Inequalities and Triangles

5.2 Inequalities and Triangles. What you’ll learn: To recognize and apply properties of inequalities to the measure of angles of a triangle. To recognize and apply properties of inequalities to the relationships between angles and sides of a triangle. Definition of Inequality.

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5.2 Inequalities and Triangles

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  1. 5.2 Inequalities and Triangles What you’ll learn: To recognize and apply properties of inequalities to the measure of angles of a triangle. To recognize and apply properties of inequalities to the relationships between angles and sides of a triangle.

  2. Definition of Inequality For any real numbers a and b, a>b iff there is a positive number c such that a=b+c.

  3. Properties if InequalitiesFor all numbers a, b, and c Comparison Property a<b, a=b, or a>b Transitive Property: if a<b and b<c, then a<c if a>b and b>c, then a>c Addition/Subt: If a>b then a+c>b+c and a-c>b-c If a<b, then a+c<b+c and a-c<b-c Mult./Div: If c>0 and a<b, then ac<bc and If c>0 and a>b then ac>bc and If c<0 and a<b then ac>bc and If c<0 and a>b then ac<bc and

  4. Theorems Theorem 5.8 Exterior Angle Inequality TheoremIf an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles. 1>2, 1>3 Theorem 5.9 If one side of a triangle is longer than another side, then the side opposite the longer side has a greater measure than the angle opposite the shorter side. If BC>AB, then 1>3 2 1 3 A 1 C B 2 3

  5. Theorem 5.10 If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. If C>A, then AB>BC. B A C

  6. Examples Which angle has the greatest measure? 1>3 1>4 4=5 1>5 2<90 1>2 Use the exterior angle inequality to list all of the angles that satisfy the stated condition. All angles whose measure are less than m14 4,11,9,3,2,6,7 All angles whose measures are greater than m15 10,16,15,17,12 5 17 14 4 16 4 5 15 3 6 11 1 10 3 2 12 9 2 1 7 8

  7. Determine the relationship between the measures of the given angles RSU____SUR TSV____STV RSV____RUV R 7 S 9 T 5 8 3 4 V U 6

  8. Given: JMJL, JLKLProve: m1>m2 L 2 • JMJL, JLKL • KLJK MJLM • 1>LKJ LJK>2 • 1>LKJ • Given • Isos. Triangle theorem • Ext. angle ineq. thm • substitution 1 M K J

  9. Homeworkp. 25218-42 even46-50 even

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