Applying Triangle Sum Properties

Applying Triangle Sum Properties

Section 4.1

Triangles Triangles are polygons with three sides.

There are several types of triangle: Scalene Isosceles Equilateral Equiangular Obtuse Acute Right

Scalene Triangles Scalene triangles do not have any congruent

sides.

In other words, no side has the same length.

3cm

8cm

6cm

Isosceles Triangle A triangle with 2 congruent sides.

2 sides of the triangle will have the same length.

2 of the angles will also have the same angle measure.

Equilateral Triangles All sides have the same length

Equiangular Triangles All angles have the same angle measure.

Acute Triangle All angles are acute angles.

Right Triangle Will have one right angle.

Obtuse Angle Will have one obtuse angle.

Exterior Angles vs. Interior Angles Exterior Angles are angles that are on the

outside of a figure.

Interior Angles are angles on the inside of a figure.

Interior or Exterior?



Triangle Sum Theorem (Postulate Sheet) States that the sum of the interior angles is

180.

We will do algebraic problems using this theorem. The sum of the

angles is 180, so

x + 3x + 56= 1804x + 56= 180

4x = 124x = 31

Find the Value for X

2x + 15

3x

2x + 15 + 3x + 90 = 180

5x + 105 = 180

5x = 75

x = 15

Corollary to the Triangle Sum Theorem (Postulate Sheet) Acute angles of a right triangle are

complementary.

3x + 10

5x +16

Exterior Angle Sum Theorem The measure of the exterior angle of a triangle is equal to

the sum of the non-adjacent interior angles of the triangle

88 + 70 = y

158 = y

2x + 40 = x + 72

2x = x + 32 x = 32

Find x and y

3x + 13

46o

8x - 1

2yo

4.1 Apply Congruence and Triangles4.2 Prove Triangles Congruent by SSS, SAS

Objectives:1. To define congruent triangles2. To write a congruent statement3. To prove triangles congruent by SSS, SAS

Congruent Polygons

Congruent Triangles (CPCTC)

Two triangles are congruent congruent triangles triangles if and only if the ccorresponding pparts of those ccongruent ttriangles are ccongruent.

Congruence Statement

When naming two congruent triangles, order is very important.

Example

Which polygon is congruent to ABCDE?ABCDE -?-

Properties of Congruent Triangles

Example

What is the relationship between C and F?

30

30

75

75

E

F

D

A

C

B

Third Angle Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

Congruent Triangles

Checking to see if 3 pairs of corresponding sides are congruent and then to see if 3 pairs of corresponding angles are congruent makes a total of SIX pairs of things, which is a lot! Surely there’s a shorter way!

Congruence Shortcuts?

Will one pair of congruent sides be sufficient? One pair of angles?

Congruence Shortcuts?

Will two congruent parts be sufficient?

Congruent Shortcuts?

Will three congruent parts be sufficient?

And if so….what three parts?

Section 4.3Proving Triangles are Congruents by SSS

Draw any triangle using any 3 size lines For me I use lines of 5, 4, and 3 cm’s. Now use the same lengths and see if you can

make a different triangle.

Now measure both triangles angles and see what you get.

3cm

4cm

5cm

3cm

4cm 5cm90

9053 53

37

37

Are the following triangles congruent? Why?

6 6

10

6 6

10

a. YES, all sidesare equal so SSS

108

9

10

6

9

b. No, all sidesare not equal8 ≠ 6, so failsSSS

Use the SSS Congruence Postulate

Decide whether the congruence statement is true. Explain your reasoning.

NLKL

NMKM

SOLUTION

NLMKLM

LMLM

Given

Given

Reflexive Property

So, by the SSS Congruence Postulate,

NLMKLM

4.4:Prove Triangles Congruent by SAS and HL

Goal:Use sides and angles to prove congruence.

Vocabulary Leg of a right triangle: In a right triangle, a In a right triangle, a

side adjacent to the right angle is called a leg.side adjacent to the right angle is called a leg. Hypotenuse:In a right triangle, the side In a right triangle, the side

opposite the right angle is called the opposite the right angle is called the hypotenuse.hypotenuse.

LegLeg

HypotenuseHypotenuse

Before we start…let’s get a few things straight

INCLUDED SIDE

A B

C

X Z

Y

Angle-Side-Angle (ASA) Congruence Postulate

Two angles and the INCLUDED side

Angle-Angle-Side (AAS) Congruence Postulate

Two Angles and One Side that is NOT included

} Your Only Ways To Prove Triangles Are

Congruent

NO BAD WORDS

Overlapping sides are congruent in

each triangle by the REFLEXIVE property

Vertical Angles

are congruen

t

Alt Int Angles are congruent

given parallel

lines

Things you can mark on a triangle when they aren’t marked.

Ex 1

statement. congruence a Write.

and ,, and In

LE

NLDENDΔLMNΔDEF

DEF NLM

Ex 2

What other pair of angles needs to be marked so that the two triangles are congruent by AAS?

F

D

E

M

L

N

NE

Ex 3

What other pair of angles needs to be marked so that the two triangles are congruent by ASA?

F

D

E

M

L

N

LD

Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.

ΔGIH ΔJIK by AAS

G

I

H J

KEx 4

ΔABC ΔEDC by ASA

B A

C

ED

Ex 5


ΔACB ΔECD by SASB

A

C

E

D

Ex 6


ΔJMK ΔLKM by SAS or ASA

J K

LM

Ex 7


Not possible

K

J

L

T

U

Ex 8


V

Postulate 20:Side-Angle-Side (SAS) Congruence Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

If Side ,

Angle , and

Side ,

then .

UV

U

RS

R

R UW

U

T

T VS WR

Example 1:Use the SAS Congruence Postulate Write a proof.

Given ,

Prove

JN LN KN MN

JKN LMN

21N

J

MK

L

Statements Reasons

1. , 1. Given

JN

K

LN

MN N

2. 1 2 Vertical Angles T2. h em eor

3. 3. SAS Congruence Postul ateJKN LMN

Example 2:Use SAS and properties of shapes

In the diagram, is a rectangle.

What can you conclude about

and ?

ABCD

ABC CDA By the ,

. Opposite sides of a rectangl

Right Angles Congruence Theorem

e are congruent,

so and .AB CD BC

B D

DA

and are congruent by SAS Congruen the ce

Postulate

.

ABC CDA

Checkpoint

In the diagram, , , and pass

through the center of the circle.

Also, 1 2 3 4.

AB CD EF

M

Statements Reaso

Prove

n

t

s

tha .DMY BMY

1. 3 4 1. Given 2. 2. Definition of a DM BM

3. 3. Reflexive Property of

Congruence

MY MY

4. 4. SAS Congruence

Postulate

DMY BMY

Checkpoint

In the diagram, , , and pass

through the center of the circle.

Also, 1 2 3 4.

AB CD EF

M

What can you conclude about and ?AC BD

Because they are vertical angles,

. All points on a circle are the

same distance from the center, so

. By the SAS Congruence

Postulate, . Corresponding parts

of congruent trian

AMC BMD

AM BM CM DM

AMC BMD

gles are congruent, so you

know .AC BD

Theorem 4.5:Hypotenuse-Leg Congruence Theorem If the hypotenuse and a leg of a right

triangle are congruent to the hypotenuse and a leg of a second triangle, then the two triangles are congruentcongruent.

Example 3:Use the Hypotenuse-Leg Theorem Write a proof.

Given ,

,

,

is a bisector of .

Prove

AC EC

AB BD

ED BD

AC BD

ABC EDC

Example 3:Use the Hypotenuse-Leg Theorem

Statements Reasons

H 1. 1. GivenAC EC

2. , 2.

G

iv

en

AB BD

ED BD

3. and are 3. Definition of lines

righ .t angles

B D

4. and 4. Definition of a

right trian are . gles right triang le

ABC EDC

5. is a bisector 5.

Give

n

of .

AC

BD

Example 3:Use the Hypotenuse-Leg Theorem

Statements Reasons

L 6. 6. Definition of segment

bisector

BC DC

7. 7.

HL Congruence

The or em

ABC EDC

Example 4:Choose a postulate or theorem

Gate The entrance to a ranch

has a rectangular gate as shown

in the diagram. You know that

. What postulate

or theorem can you use to

conclude that ?

AFC EFC

ABC EDC

Example 4:Choose a postulate or theorem

You are given that is a rectangle, so and

are . Because opposite sides of a rectangle

are , . You are also given

right angles

con that

, so . The hypotenuse and a

gruent

leg

of

ABDE B D

AB

AFC EFC AC

DE

EC

each triangle is congruent.

HL Congruence TheYou can use the to conclude

th

orem

at .ABC EDC

Using Congruent Triangles: CPCTC

Academic Geometry

Proving Parts of Triangles Congruent

You know how to use SSS, SAS, ASA, and AAS to show that the triangles are congruent.

Once you have triangles congruent, you can make conclusions about their other parts because, by definition, corresponding parts of congruent triangles are congruent. Abbreviated CPCTC

Proving Parts of Triangles CongruentIn an umbrella frame, the stretchers are congruent and they

open to angles of equal measure.

Given SL congruent to SR

<1 congruent <2

Prove that the angles formed by the shaft

and the ribs are congruent

shaft

stretcher

ribl r3 4

1 2

c

s

Proving Parts of Triangles Congruent

Prove <3 congruent <4

Statement Reason

shaft

stretcher

ribl r3 4

1 2

c

s

Proving Parts of Triangles CongruentGiven <Q congruent <R

<QPS congruent <RSP

Prove SQ congruent PR

Statements Reasons

r

p q

s

Proving Parts of Triangles CongruentGiven <DEG and < DEF are right angles.

<EDG congruent <EDF

Prove EF congruent EG

Statements Reasons

d

ef

g

4.7 Isosceles and Equilateral Triangles

Chapter 4Congruent Triangles

4.5 Isosceles and Equilateral Triangles

Isosceles Triangle:

Base

Leg Leg

Vertex Angle

Base Angles

*The Base Angles are Congruent*

Isosceles Triangles Theorem 4-3 Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent

A

B

C

<A = <C

Theorem 4-4 Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent

Isosceles Triangles

A

B

C

Given: <A = <CConclude: AB = CB

Theorem 4-5 The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base

Isosceles Triangles

A

B

C

Given: <ABD = <CBDConclude: AD = DC and

BD is ┴ to AC

D

Equilateral Triangles Corollary: Statement that immediately follows

a theorem

Corollary to Theorem 4-3:If a triangle is equilateral, then the triangleis equiangular

Corollary to Theorem 4-4:If a triangle is equiangular, then the triangle is equilateral

Using Isosceles Triangle TheoremsExplain why ΔRST is isosceles.

R

V

S

W

T

UGiven: <R = <WVS,

VW = SWProve: ΔRST is isosceles

Statement Reason

3. m<R = m<WVS

1. Given1. VW = SW

2. m<WVS = m<S 2. Isosceles Triangle Thm.

3. Given

4. m<S = m<R 4. Transitive Property

5. ΔRST is isosceles 5. Def Isosceles Triangle

Using AlgebraFind the values of x and y:

) )

y°

x°

63°

L

O

N

M

ΔLMN is isosceles

m<L = m< N = 6363°m<LM0 = y = m<NMO

63 + 63 + y + y = 180

y°

126 + 2y = 180- 126 -126

2y = 542 2

y = 27

27 + 63 + x = 180

90 + x = 180-90 -90

x = 90

27°

LandscapingA landscaper uses rectangles and equilateral triangles

for the path around the hexagonal garden. Find the value of x.

x°

Applying Triangle Sum Properties

Documents

triangles angles

congruent parts

congruent shortcuts

congruent statementto

congruent trianglesto

congruent triangleschecking

following triangles

pair of angles