Congruent Triangles Lesson 1.5.1 - Ms. Morgan's Math

Lesson 5: Congruent Triangles

Instruction

Congruent Triangles

Common Core Georgia Performance Standards

MCC9¨C12.G.CO.7

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MCC9¨C12.G.CO.8

MCC9¨C12.G.CO.7

MCC9¨C12.G.CO.8

Essential Questions

1. Why is it important to know how to mark congruence on a diagram?

2. What does it mean if two triangles are congruent?

3. If two triangles have two sides and one angle that are equivalent, can congruence be determined?

NAME:

4. How many equivalent

measuresAND

are needed

to determine if triangles are congruent?

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Lesson

5: Congruent Triangles

WORDS

TO KNOW

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Lesson

1.5.1: Triangle

Congruency

If two angles and the included side of one triangle are

Angle-Side-Angle

(ASA)

Congruence

Warm-Up

1.5.1 Statement congruent to two angles and the included side of another

the two

triangles

congruent.

Cutting mats used in crafting are triangle,

similar tothen

coordinate

planes,

exceptare

that

the axes are labeled with

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inches or centimeters, rather than positive and negative numbers. Juliet is in the middle of a sewing

two angles

haveonthe

measure

congruent

project

and hasangles

laid out two congruent

pieces that

of fabric

hersame

cutting

mat.

congruent sides

two sides that have the same length

congruent triangles

triangles having the same angle measures and side

lengths

corresponding angles

a pair of angles in a similar position

Corresponding Parts of If two or more triangles are proven congruent, then all of

their corresponding parts are congruent as well.

Congruent Triangles

are Congruent (CPCTC)

corresponding sides

the sides of two figures that lie in the same position

relative to the figures

included angle

the angle between two sides

included side

postulate

rigid motion

the side

Piece

1 between two angles of a triangle

a true statement that does not require a proof

Piece

2 to an object that maintains the

a transformation

done

object¡¯s shape and size or its segment lengths and angle

measures

1. What is the series of transformations that has taken place between Piece 1 and Piece 2?

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CCGPS Analytic Geometry Teacher Resource

!!!!!2.!Is!Piece!1!and!Piece!2!congruent?!Explain.!

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Key Concepts

?

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To determine whether two triangles are congruent, you must observe the angle measures and

side

lengths of the triangles.

Introduction

When

a triangle

by amotions,

series of including

rigid motions,

the angles

are images

of each is

If a rigid

motionisortransformed

a series of rigid

translations,

rotations,

or reflections,

other

and are

corresponding

angles.

performed

oncalled

a triangle,

then the transformed

triangle is congruent to the original. When two

triangles

are

congruent,

the

corresponding

angles

haveposition.

the same measures and the corresponding

? Corresponding angles are a pair of angles in a similar

sides have the same lengths. It is possible to determine whether triangles are congruent based on the

? Ifangle

two triangles

any pair

corresponding angles is also congruent.

measuresare

andcongruent,

lengths of then

the sides

of theoftriangles.

? When a triangle is transformed by a series of rigid motions, the sides are also images of each

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other

and are called corresponding sides.

Key Concepts

!

?

Corresponding

sides are

the sides

of triangles

two figures

lie in theyou

same

position

relative

to themeasures

figure. and

? To determine

whether

two

arethat

congruent,

must

observe

the angle

side lengths of the triangles.

? If two triangles are congruent, then any pair of corresponding sides is also congruent.

? When a triangle is transformed by a series of rigid motions, the angles are images of each

? Congruent triangles have three pairs of corresponding angles and three pairs of corresponding

other and are called corresponding angles.

sides, for a total of six pairs of corresponding parts.

? Corresponding angles are a pair of angles in a similar position.

? If two or more triangles are proven congruent, then all of their corresponding parts are

? If two

triangles

congruent,

thenas

any

pair of corresponding

angles

is also congruent.

congruent

as well.

This are

postulate

is known

Corresponding

Parts of

Congruent

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SIMILARITY,

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AND

PROOFS

Triangles

are aCongruent

(CPCTC). Aby

postulate

a truemotions,

statement

? When

triangle is transformed

a series ofis rigid

thethat

sidesdoes

are not

alsorequire

imagesaof each

Lesson 5: proof.

Congruent

Triangles

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other and are called corresponding sides.

Instruction

!

?

? Corresponding

arebethe

sides of two

lie in the same position relative to the figure.

The corresponding

angles andsides

sides can

determined

by thefigures

order ofthat

the letters.

If ABC

to DEF

angles of thethen

two triangles

correspond

in the same order

? isIfcongruent

two triangles

are, the

congruent,

any pair

of corresponding

sides is also congruent.

as

they

are

named.

CCGPS Analytic Geometry Teacher Resource

? Walch Education

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?

?

? symbol

Congruent

triangles

three

pairs of corresponding angles and three pairs of corresponding

Use the

to show

that two have

parts are

corresponding.

sides, for a total of six pairs of corresponding parts.

Angle A

Angle D; they are equivalent.

? If

or more

are proven congruent, then all of their corresponding parts are

Angle

E; theytriangles

are equivalent.

Angle

B two

well.

postulate is known as Corresponding Parts of Congruent

Angle F;asthey

are This

equivalent.

Angle congruent

C

?

?

Triangles

(CPCTC).

A postulate

is a true statement that does not require a

The corresponding

anglesare

are Congruent

used to name the

corresponding

sides.

AB

proof.

DE ; they are equivalent.

BC

EF ; they are equivalent.

AC

DF ; they are equivalent.

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Observe the diagrams of

ABC and

CCGPS Analytic Geometry Teacher Resource

ABC

DEF .

? Walch Education

DEF

A

D

C

B

F

¡ÏA ? ¡ÏD

AB DE

¡ÏB ? ¡ÏE

BC EF

¡ÏC ? ¡ÏF

AC DF

E

? !By observing the angles and sides of two triangles, it is possible to determine if the triangles

are congruent.

?

Two triangles are congruent if the corresponding angles are congruent and corresponding

sides are congruent.

?

Notice the number of tick marks on each side of the triangles in the diagram.

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Instruction

Guided Practice 1.5.1

Example 1

Use corresponding parts to identify the congruent triangles.

R

A

V

M

T

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Lesson 5: Congruent Triangles

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Instruction

Example 2 1. Match the number of tick marks to identify the corresponding

congruent sides.

BDF

HJL

RV and JM each have one tick mark; therefore, they are

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SIMILARITY,

CONGRUENCE,

PROOFS triangles.

Name the corresponding

angles and

and congruent.

sides ofAND

the congruent

corresponding

Lesson 5: Congruent Triangles

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VA and MT each have two tick marks; therefore, they are

corresponding

and congruent.

1. Identify the congruent

angles.

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Instruction

Example 3

RA names

and JTofeach

have three

tick marks;

therefore,

theycorresponding

are

The

the triangles

indicate

the angles

that are

corresponding

and

congruent.

Use constructionand

tools

to determine

if the

congruent.

If they are, name the congruent

congruent.

Begin

withtriangles

the firstare

letter

of each name.

triangles and corresponding angles and sides.

Identify the first set of congruent angles.

B is congruent to

H.

Q

Identify the second set of congruent angles.

D is congruent to

J.

Identify the third set of congruent angles.

F is congruent to

P

L.

R

T

S

2. Identify the congruent sides.

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CCGPS Analytic Geometry Teacher Resource

The names of the triangles indicate the sides that

U are corresponding

and congruent. Begin with the first two letters of each name.

the firsttosetcompare

of congruent

sides.of each side.

1. Identify

Use a compass

the length

is congruent

to HJ .sides, PR and UT .

BD

Begin

with the shortest

Set the sharp

pointset

of of

thecongruent

compass on

point P and extend the pencil of

Identify

the second

sides.

the compass to point R.

DF is congruent to JL .

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SIMILARITY, CONGRUENCE, AND PROOFS

Lesson 5: Congruent Triangles

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Problem-Based Task 1.5.1: Stained Glass Pattern, Part I

Mary creates stained glass art. She is in the planning stages of creating a new piece and has found

a pattern she really likes. Mary studies the pattern to determine which triangles in the pattern are

congruent, so that she can cut the correct size pieces of glass. Pictured below is a portion of the

pattern. Use the pattern and the information that follows to determine which triangles are congruent.

How could Mary use this information to help plan her project?

A

B

E

D

C

F

I

H

G

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ABCD and

CFGH are squares. Each diagonal of a square bisects an opposite pair of angles.

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BEFC and DCHI are rhombuses. The diagonals of a rhombus bisect the opposite pairs

of angles. Remember that opposite pairs of angles are congruent.

?

EC BC

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CCGPS Analytic Geometry Teacher Resource

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6/*5

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SIMILARITY, CONGRUENCE, AND PROOFS

Lesson 5: Congruent Triangles

Practice 1.5.1: Triangle Congruency

Use the diagrams to correctly name each set of congruent triangles according to their

corresponding parts.

D

1.

G

C

B

H

F

2.

P

N

J

T

R

V

3.

F

D

P

NAME:

L AND PROOFS

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Lesson 5: Congruent Triangles

Name the corresponding angles and sides for each pair of congruent triangles.

4.

QRS

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continued

WXY

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CCGPS Analytic Geometry Teacher Resource

5.

AFH

CGJ

6.

LPQ

HJK

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Use a ruler and a protractor or construction tools to determine if the triangles are congruent. If they

are, name the congruent triangles and their corresponding angles and sides.

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