Triangles and CongruenceTriangles and Congruence

§§ 5.1 5.1 Classifying Triangles

§§ 5.4 5.4 Congruent Triangles

§§ 5.3 5.3 Geometry in Motion

§§ 5.2 5.2 Angles of a Triangle

§§ 5.6 5.6 ASA and AAS

§§ 5.5 5.5 SSS and SAS

Classifying TrianglesClassifying Triangles

You will learn to identify the parts of triangles and to classify triangles by their parts.

In geometry, a triangle is a figure formed when _____ noncollinear points are connected by segments.

three

Each pair of segments forms an angle of the triangle.

E

D

F

The vertex of each angle is a vertex of the triangle.

Classifying TrianglesClassifying Triangles

Triangles are named by the letters at their vertices.

Triangle DEF, written ______, is shown below.

E

D

F

angle

vertex

side

The sides are:

The vertices are:

The angles are:

In Chapter 3, you classified angles as acute, obtuse, or right.

Triangles can also be classified by their angles.

All triangles have at least two _____ angles.acuteThe third angle is either _____, ______, or _____.obtuseacute right

EF, FD, and DE.

D, E, and F.

E, F, and D.

ΔDEF

Classifying TrianglesClassifying Triangles

TrianglesClassified by

Angles

60°

80°

40°

acutetriangle

3rd angle is

_____acute

obtusetriangle

righttriangle

3rd angle is

______obtuse3rd angle is

____right

17°

43°

120° 30°

60°

Classifying TrianglesClassifying Triangles

TrianglesClassified by

Sides

scalene isosceles equilateral

no___sides

congruent

__________sides

congruent

___sides

congruent

at least two all

Classifying TrianglesClassifying Triangles

leg

The side opposite the vertexangle is called the _____.

The congruent sidesare called legs.

base

leg

The angle formed by the congruent sides is called the

___________.vertex angle

The two angles formed bythe base and one of the

congruent sides are called___________.base angles

Angles of a TriangleAngles of a Triangle

You will learn to use the Angle Sum Theorem. 1) On a piece of paper, draw a triangle.

2) Place a dot close to the center (interior) of the triangle.

3) After marking all of the angles, tear the triangle into three pieces. then rotate them, laying the marked angles next to each other.

4) Make a conjecture about the sum of the angle measures of the triangle.

Angles of a TriangleAngles of a Triangle

Theorem 5-1Angle Sum

Theorem

The sum of the measures of the angles of a triangle is 180.

z°

x°

y°x + y + z = 180

Angles of a TriangleAngles of a Triangle

Theorem 5-2

The acute angles of a right triangle are complementary.

x + y = 90x°

y°

Angles of a TriangleAngles of a Triangle

Theorem 5-3

The measure of each angle of an equiangular triangle is 60.

3x = 180x°

x° x°

x = 60

Geometry in MotionGeometry in Motion

You will learn to identify translations, reflections, androtations and their corresponding parts.

We live in a world of motion.

Geometry helps us define and describe that motion.

In geometry, there are three fundamental types of motion:

__________, _________, and ________.translation reflection rotation

Geometry in MotionGeometry in Motion

In a translation, you slide a figure from one position to another without turning it.

Translations are sometimes called ______.slides

Geometry in MotionGeometry in Motion

line ofreflection

In a reflection, you flip a figure over a line.

Reflections are sometimes called ____.flips

The new figure is a mirror image.

Geometry in MotionGeometry in Motion

30°

In a rotation, you rotate a figure around a fixed point.

Rotations are sometimes called _____.turns

Geometry in MotionGeometry in Motion

Each point onthe original figure is calleda _________.preimage

Its matchingpoint on thecorresponding figure is calledits ______.image

A

B

C

D

E

F

Each point on the preimage can be paired with exactly one point on its image,and each point on the image can be paired with exactly one point on its preimage.

This one-to-one correspondence is an example of a _______.mapping

Geometry in MotionGeometry in Motion

Each point onthe original figure is calleda _________.preimage

Its matchingpoint on thecorresponding figure is calledits ______.image

A

B

C

D

E

F

The symbol → is used to indicate a mapping.

In the figure, ΔABC → ΔDEF. (ΔABC maps to ΔDEF).

In naming the triangles, the order of the vertices indicates the corresponding points.

Geometry in MotionGeometry in Motion

Each point onthe original figure is calleda _________.preimage

Its matchingpoint on thecorresponding figure is calledits ______.image

A

B

C

D

E

F

→Preimage

A

B

C

Image

D

E

F

→→

→Preimage Image

→→

AB DE

BC EF

CA FD

This mapping is called a _____________.transformation

Geometry in MotionGeometry in Motion

When a figure is translated, reflected, or rotated,

the lengths of the sides of the figure DO NOT CHANGE.

Translations, reflections, and rotations are all __________.isometries

An isometry is a movement that does not change the size or shape of thefigure being moved.

The order of the ________ indicates the corresponding parts!

ΔABC ΔXYZ

Congruent TrianglesCongruent Triangles

You will learn to identify corresponding parts of congruenttriangles

If a triangle can be translated, rotated, or reflected onto another triangle, sothat all of the vertices correspond, the triangles are _________________.congruent triangles

The parts of congruent triangles that “match” are called__________________.corresponding parts

vertices

Congruent TrianglesCongruent Triangles

A

C B

F

E D

In the figure, ΔABC ΔFDE.

As in a mapping, the order of the _______ indicates the corresponding parts.

vertices

Congruent Angles Congruent Sides

A FB DC E

AB FDBC DEAC FE

These relationships help define the congruent triangles.

Congruent TrianglesCongruent Triangles

Definition ofCongruen

tTriangles

If the _________________ of two triangles are congruent, thenthe two triangles are congruent.

corresponding parts

If two triangles are _________, then the corresponding partsof the two triangles are congruent.

congruent

Congruent TrianglesCongruent Triangles

ΔRST ΔXYZ. Find the value of n.

T

S

R

Z

XY

40°(2n + 10)°

50°

90°

ΔRST ΔXYZ

S Y

50 = 2n + 10

40 = 2n

20 = n

identify the corresponding parts

corresponding parts are congruent

subtract 10 from both sides

divide both sides by 2

SSS and SASSSS and SAS

You will learn to use the SSS and SAS tests for congruency.

SSS and SASSSS and SAS

1) Draw an acute scalene triangle on a piece of paper. Label its vertices A, B, and C, on the interior of each angle.

A C

B

2) Construct a segment congruent to AC. Label the endpoints of the segment D and E.

D E

F

3) Construct a segment congruent to AB. 4) Construct a segment congruent to CB. 6) Draw DF and EF. 5) Label the intersection F.

This activity suggests the following postulate.

SSS and SASSSS and SAS

Postulate 5-1SSS

Postulate

If three _____ of one triangle are congruent to _____ _____________ sides of another triangle, then the twoTriangles are congruent.

sides threecorresponding

A

B

C R

S

T

If AC RT and AB RS and BC ST

then ΔABC ΔRST

SSS and SASSSS and SAS

In two triangles, ZY FE, XY DE, and XZ DF.

Write a congruence statement for the two triangles.

Z Y F E

X D

Sample Answer:

ΔZXY ΔFDE

SSS and SASSSS and SAS

In a triangle, the angle formed by two given sides is called the____________ of the sides.included angle

A B

C

A is the includedangle of AB and AC

B is the includedangle of BA and BC

C is the includedangle of CA and CB

Using the SSS Postulate, you can show that two triangles are congruent if theircorresponding sides are congruent. You can also show their congruenceby using two sides and the ____________.included angle

SSS and SASSSS and SAS

Postulate 5-2SAS

Postulate

If ________ and the ____________ of one triangle arecongruent to the corresponding sides and included angle ofanother triangle, then the triangles are congruent.

two sides included angle

A

B

C R

S

T

If AC RT and A R and AB RS

then ΔABC ΔRST

SSS and SASSSS and SAS

Determine whether the triangles are congruent by SAS.

If so, write a statement of congruence and tell why they are congruent.

If not, explain your reasoning.

On a piece of paper, write your response to the following:

P

R

Q

F E

D

NO! D is not the included angle for DF and EF.

ASA and AASASA and AAS

You will learn to use the ASA and AAS tests for congruency.

ASA and AASASA and AAS

The side of a triangle that falls between two given angles is called the___________ of the angles.included side It is the one side common to both angles.

A B

CAC is the includedside of A and C

CB is the includedside of C and B

AB is the includedside of A and B

You can show that two triangles are congruent by using _________ and the___________ of the triangles.

two anglesincluded side

R

S

TA

B

C

ASA and AASASA and AAS

Postulate 5-3ASA

Postulate

If _________ and the ___________ of one triangle arecongruent to the corresponding angles and included side ofanother triangle, then the triangles are congruent.

two angles included side

If A R and AC RT and

then ΔABC ΔRST

C T

ASA and AASASA and AAS

A B

C

You can show that two triangles are congruent by using _________ and a______________.

two anglesnonincluded side

CA and CB are the nonincluded sides of A and B

R

S

TA

B

C

ASA and AASASA and AAS

Theorem 5-4AAS

Theorem

If _________ and a ______________ of one triangle arecongruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent.

two angles nonincluded side

If A R and CB TS

then ΔABC ΔRST

C T and

ASA and AASASA and AAS

D

F

E

L

M

N

ΔDEF and ΔLNM have one pair of sides and one pair of angles marked toshow congruence.

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

However, AAS requires the nonincluded sides.

Therefore, D and L must be marked.

If F and M are marked congruent, then FE and MN would be includedsides.