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# Triangle Inequality (Triangle Inequality Theorem)

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1. TriangleInequality(Triangle Inequality Theorem)

2. Objectives: • recall the primary parts of a triangle • show that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side • solve for the length of an unknown side of a triangle given the lengths of the other two sides. • solve for the range of the possible length of an unknown side of a triangle given the lengths of the other two sides • determine whether the following triples are possible lengths of the sides of a triangle

3. Triangle Inequality Theorem B • The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC AB + AC > BC AC + BC > AB C A

4. Is it possible for a triangle to have sides with the given lengths? Explain. a. 3 ft, 6 ft and 9 ft • 3 + 6 > 9 b. 5 cm, 7 cm and 10 cm • 5 + 7 > 10 • 7 + 10 > 5 • 5 + 10 > 7 c. 4 in, 4 in and 4 in • Equilateral: 4 + 4 > 4 (NO) (YES) (YES)

5. Solve for the length of an unknown side (X) of a triangle given the lengths of the other two sides. The value of x: a + b > x > |a - b| a. 6 ft and 9 ft • 9 + 6 > x, x < 15 • x + 6 > 9, x > 3 • x + 9 > 6, x > – 3 • 15 > x > 3 b. 5 cm and 10 cm c. 14 in and 4 in 15 > x > 5 28 > x > 10

6. Solve for the range of the possible value/s of x, if the triples represent the lengths of the three sides of a triangle. • Examples: a. x, x + 3 and 2x b. 3x – 7, 4x and 5x – 6 c. x + 4, 2x – 3 and 3x d. 2x + 5, 4x – 7 and 3x + 1

7. TRIANGLE INEQUALITY(ASIT and SAIT)

8. OBJECTIVES: • recall the Triangle Inequality Theorem • state and identify the inequalities relating sides and angles • differentiate ASIT (Angle – Side Inequality Theorem) from SAIT (Side – Angle Inequality Theorem) and vice-versa • identify the longest and the shortest sides of a triangle given the measures of its interior angles • identify the largest and smallest angle measures of a triangle given the lengths of its sides

9. INEQUALITIES RELATING SIDES AND ANGLES: ANGLE-SIDE INEQUALITY THEOREM: • If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. If AC > AB, then mB > mC. SIDE-ANGLE INEQUALITY THEOREM: • If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. If mB > mC, then AC > AB. C B A

10. EXAMPLES: O E • List the sides of each triangle in ascending order. a. e. c. R 70 61 J 73 59 P N M L 31 JR, RE, JE ME & EL, ML PO, ON, PN I d. b. E A P 42 46 U E 79 AT, PT, PA UE, IE, UI T

11. TRIANGLE INEQUALITY(Isosceles Triangle Theorem)

12. Objectives: • recall the definition of isosceles triangle • recall ASIT and SAIT • solve exercises using Isosceles Triangle Theorem (ITT) • prove statements on ITT • recall the definition of angle bisector and perpendicular bisector

13. Isosceles Triangle: • a triangle with at least two congruent sides Parts of an Isosceles : Base: AC Legs: AB and BC Vertex angle: B Base angles: A and C B A C

14. Isosceles Triangle Theorem (ITT): • If two sides of a triangle are congruent, then the angles opposite the sides are also congruent. If AB  BC, then A  C. B A C

15. Converse of ITT: • If two angles of a triangle are congruent, then the sides opposite the angles are also congruent. If A  C, then AB  BC. B A C

16. Vertex Angle Bisector-Isosceles Theorem: (VABIT) • The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. If BD is the angle bisector of the base angle of ABC, then AD  DC and mBDC = 90. B A C D

17. Examples: For items 1-5, use the figure on the right. 1. If ME = 3x – 5 and EL = x + 13, solve for the value of x and EL. 2. If mM = 58.3, find the mE. 3. The perimeter of MEL is 48m, if EL = 2x – 9 and ML = 3x – 7. Solve for the value of x, ME and ML. 4. If the mE = 65, find the mL. 5. If the mM = 3x + 17 and mE = 2x + 11. Solve for the value of x, mL and mE. E M L

18. Prove the following using a two column proof. Statements Reasons 1. Given: 1  2 Prove: ABC is isosceles 1. 1  2 Given 2. 1 & 3, 4 & 2 are vertical angles Def. of VA A 3. 1  3 and 4  2 VAT 4. 2  3 Subs/Trans 5. 4  3 Subs/Trans B 4 C 3 1 5 6 2 6. AB  AC CITT 7. ABC is isosceles Def. of Isosceles 

19. Prove the following using a two column proof. 2. Given: 5  6 Prove: ABC is isosceles Statements Reasons 1. 5  6 Given A 2. 5 & 3, 4 & 6 Def. of are linear pairs linear pairs 3. m5 = m6 Def. of s 4. m5 + m3 = 180 LPP m4 + m6 = 180 B 4 C 3 5. 4  3 Supplement Th. 1 5 6 2 6. m4 = m3 Def. of  s 7. AB  AC CITT 8. ABC is isosceles Def. of isosceles 

20. Prove the following using a two column proof. Statements Reasons 3. Given: CD  CE, AD  BE Prove: ABC is isosceles 1. CD  CE, AD  BE Given C 2. 1  2 ITT 3. m1 = m2 Def.  s 4. 1 & 3 are LP s Def. of LP 2 & 4 are LP s 3 1 2 4 B A D E 5. m1 + m3 = 180 LPP m4 + m2 = 180 6. m4 = m3 Supplement Th 7. ADC  BEC SAS 8. AC  BC CPCTC 9. ABC is isosceles Def. of Isos. 

21. Triangle Inequality(EAT)

22. Objectives: • recall the parts of a triangle • define exterior angle of a triangle • differentiate an exterior angle of a triangle from an interior angle of a triangle • state the Exterior Angle theorem (EAT) and its Corollary • apply EAT in solving exercises • prove statements on exterior angle of a triangle

23. Exterior Angle of a Polygon: • an angle formed by a side of a  and an extension of an adjacent side. • an exterior angle and its adjacent interior angle are linear pair 3 2 1 4

24. Exterior Angle Theorem: • The measure of each exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. • m1 = m3 + m4 3 2 1 4

25. Exterior Angle Corollary: • The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles. • m1 > m3 and m1 > m4 3 2 1 4

26. Examples: Use the figure on the right to answer nos. 1- 4. • The m2 = 34.6 and m4 = 51.3, solve for the m1. • The m2 = 26.4 and m1 = 131.1, solve for the m3 and m4. • The m1 = 4x – 11, m2 = 2x + 1 and m4 = x + 18. Solve for the value of x, m3, m1 and m2. • If the ratio of the measures of 2 and 4 is 2:5 respectively. Solve for the measures of the three interior angles if the m1 = 133. 1 3 2 4

27. Proving: Prove the statement using a two - column proof. Given:4 and 2 are linear pair. Angles 1, 2 and 3 are interior angles of ABC Prove: m4 = m1 + m3 B 4 2 1 3 A C