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## Triangle Inequality (Triangle Inequality Theorem)

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**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**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**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)**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**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**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**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 mB > mC. SIDE-ANGLE INEQUALITY THEOREM: • If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. If mB > mC, then AC > AB. C B A**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**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**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**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**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**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 mBDC = 90. B A C D**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 mM = 58.3, find the mE. 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 mE = 65, find the mL. 5. If the mM = 3x + 17 and mE = 2x + 11. Solve for the value of x, mL and mE. E M L**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 **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. m5 = m6 Def. of s 4. m5 + m3 = 180 LPP m4 + m6 = 180 B 4 C 3 5. 4 3 Supplement Th. 1 5 6 2 6. m4 = m3 Def. of s 7. AB AC CITT 8. ABC is isosceles Def. of isosceles **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. m1 = m2 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. m1 + m3 = 180 LPP m4 + m2 = 180 6. m4 = m3 Supplement Th 7. ADC BEC SAS 8. AC BC CPCTC 9. ABC is isosceles Def. of Isos. **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**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**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. • m1 = m3 + m4 3 2 1 4**Exterior Angle Corollary:**• The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles. • m1 > m3 and m1 > m4 3 2 1 4**Examples: Use the figure on the right to answer nos. 1- 4.**• The m2 = 34.6 and m4 = 51.3, solve for the m1. • The m2 = 26.4 and m1 = 131.1, solve for the m3 and m4. • The m1 = 4x – 11, m2 = 2x + 1 and m4 = x + 18. Solve for the value of x, m3, m1 and m2. • 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 m1 = 133. 1 3 2 4**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: m4 = m1 + m3 B 4 2 1 3 A C