TriangleTriangleInequalitInequalit

yy(Triangle (Triangle Inequality Inequality Theorem)Theorem)

ObjectivesObjectives::

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 Triangle Inequality TheoremTheorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.AB + BC > ACAB + AC > BCAC + BC > AB AA

BB

CC

a. 3 ft, 6 ft and 9 ft 3 + 6 > 93 + 6 > 9b. 5 cm, 7 cm and 10 cm 5 + 7 > 105 + 7 > 10 7 + 10 > 57 + 10 > 5 5 + 10 > 75 + 10 > 7c. 4 in, 4 in and 4 in Equilateral: 4 + 4 > 4Equilateral: 4 + 4 > 4

Is it possible for a triangle to have sides with the given lengths?

Explain.

(YES)(YES)

(NO)(NO)

(YES)(YES)

a. 6 ft and 9 ft 9 + 6 > x, x < 159 + 6 > x, x < 15 x + 6 > 9, x > 3x + 6 > 9, x > 3 x + 9 > 6, x > – 3 x + 9 > 6, x > – 3 15 > x > 315 > x > 3b. 5 cm and 10 cm

c. 14 in and 4 in

Solve for the length of an unknown side (XX) of a triangle given the

lengths of the other two sides.The value of x:The value of x:a + b > x > a + b > x > |a - b||a - b|

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 2xb. 3x – 7, 4x and 5x – 6 c. x + 4, 2x – 3 and 3xd. 2x + 5, 4x – 7 and 3x + 1

TRIANGLE TRIANGLE INEQUALITYINEQUALITY(ASIT and SAIT)(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 mB > mC.

SIDE-ANGLE INEQUALITY THEOREM:

If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.angle.If mB > mC, then AC > AB. A

C

B

I. List the sides of each triangle in ascending order.EXAMPLES:

U

I

E

b.

46

P

O

N

a.

59

61

M

E

L

c.70

P A

T

d.

79

42

J R

E

e.

31

73

PO, ON, PN

UE, IE, UI

ME & EL, ML

AT, PT, PA

JR, RE, JE

TRIANGLE INEQUALITYTRIANGLE 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:

a triangle with at least two congruent sides

Parts of an Isosceles :

Base: ACLegs: AB and BCVertex angle: BBase angles: A and

C

AA CC

BB

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.

AA CC

BB

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.

AA CC

BB

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.

AA CC

BB

DD

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.

MM LL

EE

6. AB AC CITT

Prove the following using a two column proof.

1. Given: 1 2 Prove: ABC is

isosceles

1 5 6 23 4B C

A

Statements Reasons 1. 1 2 Given2. 1 & 3, 4 & 2 are vertical angles Def. of VA3. 1 3 and 4 2 VAT4. 2 3 Subs/Trans 5. 4 3 Subs/Trans

7. ABC is isosceles Def. of Isosceles

7. AB AC CITT

Prove the following using a two column proof.

2. Given: 5 6 Prove: ABC is isosceles

1 5 6 23 4B C

A

Statements Reasons1. 5 6 Given2. 5 & 3, 4 & 6 Def. of are linear pairs linear pairs3. m5 = m6 Def. of s4. m5 + m3 = 180 LPP m4 + m6 = 1805. 4 3 Supplement Th. 6. m4 = m3 Def. of s

8. ABC is isosceles Def. of isosceles

8. AC BC CPCTC

1. CD CE, AD BE Given

Prove the following using a two column proof.

3. Given: CD CE, AD BE Prove: ABC is isosceles

D EA B

C

1 23 4

Statements Reasons

2. 1 2 ITT3. m1 = m2 Def. s

5. m1 + m3 = 180 LPP m4 + m2 = 1806. m4 = m3 Supplement Th

4. 1 & 3 are LP s Def. of LP

2 & 4 are LP s

7. ADC BEC SAS

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

1 2

3

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

1 2

3

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

1 2

3

4

Examples: Use the figure on the right to answer nos. 1- 4.1. The m2 = 34.6 and m4 = 51.3,

solve for the m1.2. The m2 = 26.4 and m1 =

131.1, solve for the m3 and m4.

3. The m1 = 4x – 11, m2 = 2x + 1 and m4 = x + 18. Solve for the value of x, m3, m1 and m2.

4. 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

2

3

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

4

1

2

3 CA

B

Statements Reasons1. 4 and 2 are linear pair. Angles 1, 2 and 3 are interior angles of ABC

Given

2. m4 + m2 = 180

LPP

3. m1 + m2 + m3 = 180

TAST

4. m4 + m2 = m1 + m2 + m3

Subs/ Trans

5. m4 = m1 + m3

APE