Triangle Inequality (Triangle Inequality Theorem)

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 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)

The value of x: a + b > x > |a - b| 15 > x > 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

Examples: a. x, x + 3 and 2x b. 3x – 7, 4x and 5x – 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

TRIANGLE INEQUALITY (ASIT and SAIT)

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 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 A B

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

TRIANGLE INEQUALITY (Isosceles Triangle Theorem)

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: B 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 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 mBDC = 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 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

Prove the following using a two column proof. 1. Given: 1  2 Prove: ABC is isosceles Statements Reasons 1. 1  2 Given A 2. 1 & 3, 4 & 2 are vertical angles Def. of VA 3. 1  3 and 4  2 VAT 4. 2  3 Subs/Trans B 3 4 C 5. 4  3 Subs/Trans 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. m5 = m6 Def. of  s 4. m5 + m3 = 180 LPP m4 + m6 = 180 B 3 4 C 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 

Prove the following using a two column proof. 3. Given: CD  CE, AD  BE Prove: ABC is isosceles Statements Reasons 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 A B 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. 

Triangle Inequality (EAT)

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 1 2 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. m1 = m3 + m4 3 1 2 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. m1 > m3 and m1 > m4 3 1 2 4

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

Proving: Prove the statement using a two - column proof. Statements Reasons 1. 4 and 2 are linear pair. Angles 1, 2 and 3 are interior angles of ABC Given 2. m4 + m2 = 180 LPP 3. m1 + m2 + m3 = 180 TAST 4. m4 + m2 = m1 + m2 + m3 Subs/ Trans 5. m4 = m1 + m3 APE 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