chapter 4 congruent triangles

Chapter 4Congruent Triangles

• Identify the corresponding parts of congruent figures

• Prove two triangles are congruent

• Apply the theorems and corollaries about isosceles triangles

4.1 Congruent Figures

Objectives

• Identify the corresponding parts of congruent figures

What we already know…

• Congruent Segments– Same length

• Congruent Angles– Same degree measure

Congruent Figures

Exactly the same size and shape. Don’t ASSume !

D

C

A

B

EF

Definition of Congruency

Two figures are congruent if corresponding vertices can be matched up so that:

1. All corresponding sides are congruent

2. All corresponding angles are congruent.

What does corresponding mean again?

• Matching

• In the same position

Volunteer

• Draw a large scalene triangle (with a ruler)

• Cut out two congruent triangles that are the same

• Label the Vertices A, B, C and D, E, F

You can slide and rotate the triangles around so that they MATCH up perfectly.

ABC DEF

A

BC

F

E

D

The order in which you name the triangles mattersmatters !

ABC DEF

A

BC

F

E

D

Based on the definition of congruency….

• Three pairs of corresponding angles

• Three pairs of corresponding sides

1. A D

3. C F

2. B E

1. AB DE

3. CA FD

2. BC EF

It is not practical to cut out and move the triangles around

ABC XYZ

• Means that the letters X and A, which appear first, name corresponding vertices and that X A.

• The letters Y and B come next, so – Y B and–XY AB

CAUTION !!

• If the diagram doesn’t show the markings

or

• You don’t have a reason– Shared sides, shared angles, vertical angles,

parallel lines

White Boards

• Suppose TIM BER

IM ___

White Boards

• Suppose TIM BER

IM ER , Why ?

White Boards

• Corresponding Parts of Congruent Triangles are Congruent

White Boards

• Suppose TIM BER

___ R

White Boards

• Suppose TIM BER

M R, Why?

White Boards

• Corresponding Parts of Congruent Triangles are Congruent

White Boards

• Suppose TIM BER

MTI ____

White Boards

• Suppose TIM BER

MTI RBE

White Boards

• If ABC XYZm B = 80m C = 50

Name four congruent angles

White Boards

• If ABC XYZm B = 80m C = 50

A, C , X, Z

White Boards

• If ABC XYZ

Write six congruences that must be correct

White Boards

• If ABC XYZ

1. A X

3. C Z

2. B Y

1. AB XY

3. CA ZX

2. BC YZ

Remote time

A. Always

B. Sometimes

C. Never

D. I don’t know

• An acute triangle is __________ congruent to an obtuse triangle.

A. AlwaysB. SometimesC. NeverD. I don’t know

• A polygon is __________ congruent to itself.


• A right triangle is ___________ congruent to another right triangle.


• If ABC XYZ, A is ____________ congruent to Y.


• If ABC XYZ, B is ____________ congruent to Y.


• If ABC XYZ, AB is ____________ congruent to ZY.


4.2 Some Ways to Prove Triangles Congruent

Objectives

• Learn about ways to prove triangles are congruent

Don’t ASSume

• Triangles cannot be assumed to be congruent because they “look” congruent.

and

• It’s not practical to cut them out and match them up

so,

We must show 6 congruent pairs

• 3 angle pairs and

• 3 pairs of sides

WOW

• That’s a lot of work

Spaghetti Experiment

• Using a small amount of playdough as your “points” put together a 5 inch, 3 inch and 2.5 inch piece of spaghetti to forma triangle.

• Be careful, IT’S SPAGHETTI, and it will break.

• Compare your spaghetti triangle to your neighbors

• Compare your spaghetti triangle to my spaghetti triangle.

We are lucky…..

• There is a shortcut– We don’t have to show

• ALL pairs of angles are congruent and

• ALL pairs of sides are congruent

SSS Postulate

If three sides of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

A

B E

C DF

Patty Paper Practice

5 inches

2.5 inches

3 inches

Volunteer

SAS PostulateIf two sides and the included angle are congruent

to the corresponding parts of another triangle, then the triangles are congruent.

B E

C DF

ASA PostulateIf two angles and the included side of one triangle

are congruent to the corresponding parts of another triangle, then the triangles are congruent.

A

B

C D

E

F

The order of the letters MEAN something

• Is SAS the same as SSA or A$$ ?

Construction 2Given an angle, construct a congruent angle.

Given:

Construct:

Steps:

ABCCDE ABC

Construction 3Given an angle, construct the bisector of the angle

Given:

Construct:

Steps:

ABCbisector of ABC

CAUTION !!

• If the diagram doesn’t show the markings

or

• You don’t have a reason– Shared sides, shared angles, vertical angles,

parallel lines

Remote Time

Can the two triangles be proved congruent? If so, what postulate can be used?

A. SSS Postulate

B. SAS Postulate

C. ASA Postulate

D. Cannot be proved congruent

E. I don’t know

A. SSS PostulateB. SAS PostulateC. ASA PostulateD. Cannot be proved congruentE. I don’t know

White Board

• Decide Whether you can deduce by the SSS, SAS, or ASA Postulate that the two triangles are congruent. If so, write the congruence ( ABC _ _ _ ). If not write not congruent.

D

C

B

A

DBC ABCSSS

A

DC

B

No Congruence

Construction 7Given a point outside a line, construct a line parallel to the given line through the point.

Given:

Construct:

Steps:

line l with point A to l through A

4.3 Using Congruent Triangles

Objectives

• Use congruent triangles to prove other things

Our Goal

• In the last section, our goal was to prove that two triangles are congruent.

The Reason

• If we can show two triangle are congruent, using the SSS, SAS, ASA postulates, then we can use the definition of Congruent Triangles to say other parts of the triangles are congruent. – Corresponding Parts of Congruent Triangles are

Congruent.

This is an abbreviated way to refer to the definition of congruency with respect to triangles.

C orresponding

P arts of

C ongruent

T riangles are

C ongruent

Basic Steps

1. Identify two triangles in which the two segments or angles are corresponding parts.

2. Prove that those two triangles are congruent

3. State that the two parts are congruent using the reason CPCTC.

Given: m 1 = m 2 m 3 = m 4Prove: M is the midpoint of JK

L

MJ K

3 4

21

Given: m 1 = m 2 m 3 = m 4

Prove: M is the midpoint of JK

L

MJ K

3 4

21

LM = LM

m J = m K If 2 ’s of 1 are to 2 ’s of another , then the third ’s are .

Reflexive Property

Given: m 1 = m 2 m 3 = m 4


L

MJ K

3 4

21

JLM KLM ASA

Given: m 1 = m 2 m 3 = m 4


L

MJ K

3 4

21

Statements Reasons

Given: m 1 = m 2 m 3 = m 4


L

MJ K

3 4

21

Statements Reasons

1. m 1 = m 2 m 3 = m 4

1. Given

Given: m 1 = m 2 m 3 = m 4


L

MJ K

3 4

21

Statements Reasons

1. m 1 = m 2 m 3 = m 4

1. Given

5. M is the midpoint of JK

Given: m 1 = m 2 m 3 = m 4


L

MJ K

3 4

21

Statements Reasons

1. m 1 = m 2 m 3 = m 4

1. Given

4. JM = KM


Given: m 1 = m 2 m 3 = m 4


L

MJ K

3 4

21

Statements Reasons

1. m 1 = m 2 m 3 = m 4

1. Given

3. JLM KLM

4. JM = KM 4. CPCTC


Given: m 1 = m 2 m 3 = m 4


L

MJ K

3 4

21

Statements Reasons

1. m 1 = m 2 m 3 = m 4

1. Given

2. LM = LM 2. Reflexive Property

3. JLM KLM 3. ASA Postulate

4. JM = KM 4. CPCTC


Given: m 1 = m 2 m 3 = m 4


L

MJ K

3 4

21

Statements Reasons

1. m 1 = m 2 m 3 = m 4

1. Given


3. JLM KLM 3. ASA Postulate

4. JM = KM 4. CPCTC

5. M is the midpoint of JK 5. Definition of midpoint

Given: m 1 = m 2 m 3 = m 4


L

MJ K

3 4

21

Statements Reasons

1. m 1 = m 2 m 3 = m 4

1. Given


3. JLM KLM 3. Postulate

4. JM = KM 4. CPCTC

5. M is the midpoint of JK 5. Definition of midpoint

Given: MK OK;KJ bisects MKO;

Prove: JK bisects MJO

3 4 Definition of bisector

JK JK Reflexive Property

MKJ OKJ SAS Postulate

K

O

J

M

1

2

34



Statements Reasons

K

O

J

M

1

2

34

K

O

J

M

1

2

34



Statements Reasons

1. MK OK;

KJ bisects MKO

1. Given

2. 3 4 2. Def of bisector

3. JK JK 3. Reflexive Property

K

O

J

M

1

2

34

K

O

J

M

1

2

34



Statements Reasons

1. MK OK;

KJ bisects MKO

1. Given



6. JK bisects MJO 6.

K

O

J

M

1

2

34

K

O

J

M

1

2

34



Statements Reasons

1. MK OK;

KJ bisects MKO

1. Given



5. 1 2 5. CPCTC


K

O

J

M

1

2

34

K

O

J

M

1

2

34



Statements Reasons

1. MK OK;

KJ bisects MKO

1. Given



4. MKJ OKJ 4. SAS Postulate

5. 1 2 5. CPCTC


K

O

J

M

1

2

34

K

O

J

M

1

2

34



Statements Reasons

1. MK OK;

KJ bisects MKO

1. Given



4. MKJ OKJ 4. SAS Postulate

5. 1 2 5. CPCTC

6. JK bisects MJO 6. Def of bisector

K

O

J

M

1

2

34

K

O

J

M

1

2

34

4.4 The Isosceles Triangle Theorem

Objectives

• Apply the theorems and corollaries about isosceles triangles

Isosceles TriangleBy definition, it is a triangle with two

congruent sides called legs.X

Y Z

Base

Base Angles

Legs Vertex Angle

Experiment - Goal

• Discover Properties of an Isosceles Triangle

Supplies

• Blank sheet of paper

• Ruler

• Pencil

• Scissors

Procedure

1. Fold a sheet of paper in half.

Procedure

2. Draw a line with the ruler going from the folded edge (very important) to the corner of the non folded edge.

Folded edge

Procedure

3. Cut on the red line

Cut here

Procedure

4. Open and lay flat. You will have a triangle

Procedure

5. Label the triangleP

Q

SR

Procedure6. Since PRQ fits exactly over PSQ

(because that’s the way we cut it),

PRQ PSQ P

Q

SR

Procedure7. What conclusions can you make? Be

careful not to ASSume anything. P

Q

SR

Conclusions

P

Q

SR

1. PRS PSR

2. PQ bisects RPS

3. PQ bisects RS

4. PQ RS at Q

5. PR PS

These conclusions are actually

• Theorems and corollaries

TheoremThe base angles of an isosceles triangle are

congruent.

A

B

C

Corollary• An equilateral triangle is also equiangular.

Corollary• An equilateral triangle has angles that

measure 60.

Corollary• The bisector of the vertex angle of an isosceles

triangle is the perpendicular bisector of the base.

TheoremIf two angles of a triangle are congruent, then

it is isosceles.

A

B

C

Corollary

• An equiangular triangle is also equilateral.

White Board Practice

• Find the value of x

30º

xº

x = 75º


• Find the value of x2x - 4

x + 5

2x + 2


• Find the value of x

56 º 62 º

x

4142

x = 42

4.5 Other Methods of Proving Triangles Congruent

Objectives

• Learn two new ways to prove triangles are congruent

Proving Triangles We can already prove triangles are congruent by the ASA, SSS

and SAS. There are two other ways to prove them congruent…

AAS TheoremIf two angles and a non-included side of one

triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

A

B

C D

E

F

The Right Triangle

leg

leg

hypotenuse

right angle

A

B

C

acute angles

HL TheoremIf the hypotenuse and leg of one right triangle are

congruent to the corresponding parts of another right triangle, then the triangles are congruent.

A

B

C D

E

F

Five Ways to Prove ’s

All Triangles:ASA SSS SAS AASRight Triangles Only:HL

White Board Practice• State which of the congruence methods can

be used to prove the triangles congruent. You may choose more than one answer.

SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem

SSS PostulateSAS PostulateASA PostulateAAS TheoremHL Theorem

4.6 Using More than One Pair of Congruent Triangles

Objectives

• Construct a proof using more than one pair of congruent triangles.

• Sometimes two triangles that you want to prove congruent have common parts with two other triangles that you can easily prove congruent.

More Than One Pair of ’s

Given: X is the midpt of AF & CD

Prove: X is the midpt of BE

A

B

C

D

E

F

X

Lecture 7 (4-7)

Objectives

• Define altitudes, medians and perpendicular bisectors.

Median of a TriangleA segment connecting a vertex to the

midpoint of the opposite side.

midpoint

vertex

Median of a TriangleEach triangle has three Medians

midpoint

vertex

Median of a Triangle• Notice that the three medians will meet

at one point.

If they do not meet, then you are not drawing the segments well.

Altitude of a TriangleA segment drawn from a vertex

perpendicular to the opposite side.

vertex

perpendicular

Altitude of a TriangleEach Triangle has three altitudes

vertex

perpendicular

Altitude of a TriangleEach triangle has three altitudes

vertexperpendicular

Altitude of a TriangleNotice that the three altitudes will meet at

one point.

If they do not meet, then you are not drawing the segments well.

Special Cases - AltitudesObtuse Triangles: Two of the altitudes are drawn

outside the triangle. Extend the sides of the triangle

Special Cases - AltitudesRight Triangles: Two of the altitudes are

already drawn for you.

Perpendicular BisectorA segment (line or ray) that is perpendicular to and

passes through the midpoint of another segment.

Must put the perpendicular and congruent markings !

Angle BisectorA ray that cuts an angle into two

congruent angles.

TheoremIf a point lies on the perpendicular bisector of a segment of a

segment, then the point is equidistant from the endpoints.

TheoremIf a point is equidistant from the endpoints of a segment, then

the point lies on the perpendicular bisector of the segment.

Remember

• When you measure distance from a point to a line, you have to make a perpendicular line.

TheoremIf a point lies on the bisector of an angle then the

point is equidistant from the sides of the angle.

Construction 10Given a triangle, circumscribe a circle about the triangle.

Given:

Construct:

Steps:

ABC circumscribed about R ABC

Construction 11Given a triangle, inscribe a circle within the triangle.

Given:

Construct:

Steps:

ABC inscribed within R ABC

Remote Time

A. Always

B. Sometimes

C. Never

D. I don’t know

• An altitude is _____________ perpendicular to the opposite side.


• A median is ___________ perpendicular to the opposite side.


• An altitude is ______________ a perpendicular bisector.


• An angle bisector is _______________ perpendicular to the opposite side.


• A perpendicular bisector of a segment is ___________ equidistant from the endpoints of the segment.


chapter 4 congruent triangles

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