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CHAPTER 5 SECTION 4 Inequalities for Sides and Angles of a triangle

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Page 1: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

CHAPTER 5SECTION 4

Inequalities for Sides and Angles of a triangle

Page 2: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

WARM-UPDetermine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1) <S is congruent to <V  2) SU is congruent to VT  3) <STU and <VUT are right angles  4) Given: <STU and <UVT are right angles; SU is congruent to VT. Prove: <S is congruent to <V.

ST

U V

Page 3: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

WARM-UPDetermine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1) <S is congruent to <VYes by LA  2) SU is congruent to VTYes by HL  3) <STU and <VUT are right anglesNo 

ST

U V

Page 4: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

WARM-UP4) Given: <STU and <UVT are right angles; SU is congruent to VT. Prove: <S is congruent to <V.

ST

U V

Statements Reasons

<STU and <UVT are right angles; SU is congruent to VT.

Triangle STU and triangle VUT are right triangles

Given

Definition of a right triangle

Congruence of segments is reflexive

TU is congruent to UT

Triangle STU is congruent to triangle VUT

<S is congruent to <V

HL

CPCTC

Page 5: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

VOCABULARYTheorem 5-9- If one side of a triangle is longer that another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. Theorem 5-10- If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

Theorem 5-11- The perpendicular segment from a point to a line is the shortest segment from the point to the line.  Corollary 5-1- The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. 

Page 6: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

Example 1) Given: m<A is greater than m<DProve: BD is greater than AB. C

B

Statements Reasons

m<A + m<E + m<EBA = 180, m<C + m<D + m<CBD = 180

m<EBA = m<EBD

m<A + m<E = 180 - m<EBA,m<C + m<D = 180 - m<CBD

m<A is greater than m<D 

Angle Sum Theorem

Substitution Property of Equality

Subtraction Property of Equality

m<A + m<D = m<C + m<D

Given

Vertical angles are congruent

BD is greater than ABIf an angle of a triangle is

greater than another, the side opposite the greater angle is longer that the side opposite

the lesser angle.

A

E

D

m<C is greater than m<E  Subtraction Property of Equality

Page 7: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

Example 2: Triangle JKL with vertices J(-4, 2), K(4, 3), and L(1, -3). List the angles in order from least measure to greatest measure.

Find the length of each side.The distance formula is d=√((x2 – x1)2 + (y2 – y1)2)

JKd=√((x2 – x1)2 + (y2 – y1)2)d=√((-4 – 4)2 + (2 – 3)2)d=√((-8)2 + (-1)2)d=√(64 + 1)d=√(65)KLd=√((x2 – x1)2 + (y2 – y1)2)d=√((4 – 1)2 + (3 – -3)2)d=√((4 – 1)2 + (3 + 3)2)d=√((3)2 + (6)2)d=√(9 + 36)d=√(45)

JLd=√((x2 – x1)2 + (y2 – y1)2)d=√((-4 – 1)2 + (2 – -3)2)d=√((-1 – 1)2 + (3 + 3)2)d=√((-5)2 + (6)2)d=√(25 + 36)d=√(61)

The shortest side is KL, so the smallest angle is <J. The next shortest side is JL, so the next smallest angle is <K. The greatest side is JK, so the greatest angle is <L.<J, <K, <L

Page 8: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

Example 3: Triangle JKL with vertices J(-2, 4), K(-5, -8), and L(6, 10). List the angles in order from least measure to greatest measure.

Find the length of each side.The distance formula is d=√((x2 – x1)2 + (y2 – y1)2)

JKd=√((x2 – x1)2 + (y2 – y1)2)d=√((-2 – -5)2 + (4 – -8)2)d=√((-2 + 5)2 + (4 + 8)2)d=√((-3)2 + (12)2)d=√(9 + 144)d=√(153)

KLd=√((x2 – x1)2 + (y2 – y1)2)d=√((-5 – 6)2 + (-8 – 10)2)d=√((-11)2 + (-18)2)d=√(121 + 324)d=√(445)

JLd=√((x2 – x1)2 + (y2 – y1)2)d=√((-2 – 6)2 + (4 – 10)2)d=√((-8)2 + (-6)2)d=√(64 + 36)d=√(100)d = 10

The shortest side is JL, so the smallest angle is <K. The next shortest side is JK, so the next smallest angle is <L. The greatest side is KL, so the greatest angle is <J.<K, <J, <L

Page 9: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

Example 4) Refer to the figure.

A) Which is greater, m<CBD or m<CDB?<CDB because it is across from 16 which is greater than 15.  B) Is m<ADB greater that m<DBA?Yes because it is across from a side that is longer.   C)Which is greater, m<CDA or m<CBA?<CDA because it is across from 10 + 16 = 26 and <CBA is across from 8 + 15 = 23.

BA

D C

10

8

15

1612

Page 10: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

Example 5) Refer to the figure.

A) Which side of triangle RTU is the longest?First find all the angles in the triangle.

180 = 110 + m<RUT70 = m<RUT

180 = 30 + 70 + m<TRU180 = 100 + m<TRU80 = m<TRU

The greatest angle is 80 degrees so the longest side is TU.  

UR

T

S

30

70

80 110

B) Name the side of triangle UST that is the longest.Since there can be only one obtuse angle in a triangle, the longest side is across from the obtuse angle.TS is the longest side.

Page 11: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

Example 6) Find the value of x and list the sides of Triangle ABC in order from shortest to longest if <A= 3x +20, <B = 2x + 37, and <C = 4x + 15

The angles of a triangle add up to 180.180 = m<A + m<B + m<C180 = 3x + 20 + 2x + 37+ 4x + 15180 = 9x + 72108 = 9x12 = x

Plug 12 in for x in each equation.

m<A = 3x + 20m<A = 3(12) + 20m<A = 36 + 20m<A = 56

BC, AC, AB

m<B = 2x + 37m<B = 2(12) + 37m<B = 24 + 37m<B = 61

m<C = 4x + 15m<C = 4(12) + 15m<C = 48 + 15m<C = 63

Page 12: Inequalities for Sides and Angles of a triangle. Determine whether triangle STU is congruent to triangle VUT, using the given information. Justify. 1)

Example 7) Find the value of x and list the sides of Triangle ABC in order from shortest to longest if <A= 9x +29, <B = 93 – 5x, and <C = 10x + 2

The angles of a triangle add up to 180.180 = m<A + m<B + m<C180 = 9x + 29 + 93 – 5x + 10x + 2180 = 14x + 12456 = 14x4 = x

Plug 4 in for x in each equation.

m<A = 9x + 29m<A = 9(4) + 29m<A = 36 + 29m<A = 65

AB, BC, AC

m<B = 93 – 5xm<B = 93 – 5(4)m<B = 93 - 20m<B = 73

m<C = 10x + 2m<C = 10(4) + 2m<C = 40 + 2m<C = 42