1 The sum of 2 sides of the triangle greater than the other side? Ordering the angles of a triangle? Ordering the sides of a triangle? SAS Inequality SSS Inequality PROBLEM 1 PROBLEM 2 PROBLEM 5PROBLEM 6 PROBLEM 3 PROBLEM 7 STANDARD 6 PROBLEM 4 END SHOW PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

2 STANDARD 6: Students know and are able to use the Triangle Inequality Theorem. Los estudiantes conocen y son capaces de usar el Teorema de Desigualdad del Triángulo. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

3 The sum of the lengths of any two sides of a triangle is greater than the third side >15 or 17>15 STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

4 The sum of the lengths of any two sides of a triangle is greater than the third side >12 or 20>12 STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

5 The sum of the lengths of any two sides of a triangle is greater than the third side >5 or 27>5 STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

6 The sum of the lengths of any two sides of a triangle is greater than the third side >15 or 17> >12 or 20> >5 or 27>5 STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

7 The measures of two sides of a triangle are 15 and 8. Between what two numbers is the third side. X 15+8 > X 15+X > 8 8+X > 15 STANDARD 6 23 > X X < X > X > -7 8+X > X > x xxx X | 7<X< The third side will be any value between 7 and PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

8 If a triangle has sides of measure x, x+4, 3x-5, find all possible values of x (X+4)+(3X-5) > X (X+4 )+X > (3X-5) X X+4 3X-5 STANDARD 6 4X -1 >X -4X -1 >-3X -3.3 <X X>.3 2X +4 > 3X-5 -2X 4 > X > X X < 9 Sign (>) changes when dividing by (-3) x xxx (3X-5) +X > (X+4 ) 4X – 5 > X +4 -X -X 3X – 5 > X > 9 3 X > X | 3<X< 9 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

9 If one side of a triangle is the longest then A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

10 If one side of a triangle is the longest then The opposite angle to this side is the largest A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

11 And the angle opposite to the shortest side A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

12 And the angle opposite to the shortest side is the smallest A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

13 And the angle opposite to the shortest side is the smallest A B C The opposite angle to this side is the largest If one side of a triangle is the longest then m B > m C > m A STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

14 In STU, ST=37-X, TU=2X-16, SU=X+13. The perimeter of the triangle is 90. List the angles in order from smallest to largest. S T U 37-X 2X-16 X+13 =90 STANDARD 6 37-X +2X-16 +X – –X +2X +X = X = X = 56 2 X=28 ST=37-X Substituting X: =37 – ( ) 28 = 9 9 TU=2X-16 = 2( ) = = 40 SU=X + 13 = ( ) = is the longest side and it is opposite to T So T is the largest 9 is the shortest side and it is opposite to U so then U is the smallest. Then: m U < m S < m T The perimeter is 90, so: PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

15 If one angle of a triangle is the largest then A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

16 The opposite side to this angle is the longest A B C If one angle of a triangle is the largest then STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

17 And the side opposite to the smallest angle A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

18 And the side opposite to the smallest angle is the shortest A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

19 A B C The opposite side to this angle is the longest If one angle of a triangle is the largest then And the side opposite to the smallest angle is the shortest AC > AB> BC STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

20 25° 92° 33° D C A B What is the shortest side in the figure below? STANDARD 6 180°- 92°-33°= 55° Finding missing angles: 55° 180°- 90°-25°= 65° 65° So, which angle’s measure is the smallest? 25° So, the opposite side to this angle is DC and it is the shortest side in the figure. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved 63° 27° E

21 In JKL, m J=12x+11, m K=9x+3, m L=7x+26. List the sides in order from longest to shortest. m J + m K +m L = 180° STANDARD 6 (12x+11)+(9x+3)+(7x+26)=180° 12x+11 9x+3 28X + 40 = 180° X = 140° 28 X = 5 Finding the angles: Adding the interior angles in the triangle: m J =12x + 11 =12( ) = = 71° m K=9x+3 =9( ) = = 48° m L =7x+26 = 7( )+26 5 = = 61° 7x+26 61° 71° 48° The largest angle is J and opposite segment LK is the longest side. Then: LK > KJ> JL K L J The smallest angle is K and opposite segment JL is the shortest side. PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

22 If two sides of a triangle are congruent to two sides in another triangle K L M A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

23 If two sides of a triangle are congruent to two sides in another triangle And the included angle between the sides in one triangle is larger than K L M A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

24 If two sides of a triangle are congruent to two sides in another triangle The included angle between the sides of the other triangle And the included angle between the sides in one triangle is larger than K L M A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

25 If two sides of a triangle are congruent to two sides in another triangle And the included angle between the sides in one triangle is larger than The included angle between the sides of the other triangle Then the opposite side to the largest angle is also larger: K L M A B C AC > KM by SAS Inequality STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

26 If two sides of a triangle are congruent to two sides in another triangle K L M A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

27 If two sides of a triangle are congruent to two sides in another triangle And the third side is larger in one than in the other K L M A B C STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

28 If two sides of a triangle are congruent to two sides in another triangle Then the included angle opposite to the larger K L M A B C And the third side is larger in triangle than in the other STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

29 If two sides of a triangle are congruent to two sides in another triangle Then the included angle opposite to the larger is greater than the angle opposite to the shorter. K L M A B C And the third side is larger in one triangle than in the other STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

30 If two sides of a triangle are congruent to two sides in another triangle Then the included angle opposite to the larger is greater than the angle opposite to the shorter: K L M A B C And the third side is larger in one triangle than in the other m B > m L by SSS Inequality STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved

31 Write an inequality or pair of inequalities to describe the possible values of x ° 88 9 (3x+5)° STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved 14

32 Write an inequality or pair of inequalities to describe the possible values of x. STANDARD ° 88 9 (3x+5)° PRESENTATION CREATED BY SIMON PEREZ. All rights reserved 14

33 Write an inequality or pair of inequalities to describe the possible values of x ° 88 9 (3x+5)° 115 > 3x+5 by SSS inequality STANDARD 6 PRESENTATION CREATED BY SIMON PEREZ. All rights reserved 14