Chapter 4: Congruent Triangles. 4-1 Congruent Figures Congruent- when two figures have the same size and shape

Chapter 4: Chapter 4: Congruent TrianglesCongruent Triangles

4-1 Congruent Figures4-1 Congruent Figures

CongruentCongruent- - when two figures have the same when two figures have the same size and shapesize and shape

A

B C FE

D

4-1 Continued4-1 Continued

Congruent triangles-Congruent triangles- two triangles are congruent two triangles are congruent if and only if their vertices can be matched up if and only if their vertices can be matched up so that the corresponding parts (angles and so that the corresponding parts (angles and sides) of the triangle are congruentsides) of the triangle are congruent

1.1. Their corresponding angles are congruent Their corresponding angles are congruent because congruent triangles have the same because congruent triangles have the same shape.shape.

2.2. Their corresponding sides are congruent Their corresponding sides are congruent because congruent triangles have the same because congruent triangles have the same size.size.


Congruent parts of triangles are marked alike.Congruent parts of triangles are marked alike. Congruent triangles must be named in the Congruent triangles must be named in the

same order of congruency.same order of congruency.

S

N U

A Y

R

SUN RAY


When justifying statements by use of the When justifying statements by use of the definition of congruent triangles, use this definition of congruent triangles, use this wording:wording:

Corresponding parts of congruent Corresponding parts of congruent triangles are congruent, which is written:triangles are congruent, which is written:

Corr. Parts of s are .Corr. Parts of s are .


Congruent polygons-Congruent polygons- two polygons are two polygons are congruent if and only if their vertices can be congruent if and only if their vertices can be matched up so that their corresponding parts matched up so that their corresponding parts are congruentare congruent

A

H

G

F

C

D

E

B ABFGH BCDEF

4-2 Some Ways to Prove 4-2 Some Ways to Prove Triangles CongruentTriangles Congruent

Proving triangles Proving triangles congruent with only congruent with only three corresponding three corresponding parts. parts.

1. 1. Side Side Side Side Side Side Postulate (SSS)-Postulate (SSS)- if if three sides of one three sides of one triangle are congruent triangle are congruent to three sides of another to three sides of another triangle, then the triangle, then the triangles are congruenttriangles are congruent

G

O B TE

L

GOB LET by the SSS Postulate


Side Angle Side Side Angle Side Postulate (SAS)-Postulate (SAS)- if two if two sides and the included sides and the included angle of one triangle are angle of one triangle are congruent to two sides congruent to two sides and the included angle and the included angle of another triangle, then of another triangle, then the triangles are the triangles are congruentcongruent

J

E N KA

P

JEN PAK by the SAS Postulate


Angle Side Angle Angle Side Angle Postulate (ASA)- Postulate (ASA)- if two if two angles and the included angles and the included side of one triangle are side of one triangle are congruent to two angles congruent to two angles and the included side of and the included side of another triangle, then another triangle, then the triangles are the triangles are congruentcongruent

C

A R YL

O

CAR OLY by the ASA Postulate

Proof of ASA PostulateProof of ASA Postulate

Given:

Prove:

Statement Reason

1. Given2. Definition of a midpoint3. Given4. If two lines are perpendicular then they form congruent adjacentangles.5. Reflexive property of congruence6. SAS postulate

M J

T

E

1. E is the midpoint of

2.

3.

4.

5.

6.

E is the midpoint of

4-3 Using Congruent 4-3 Using Congruent TrianglesTriangles

Learning how to extract information on Learning how to extract information on segments or angles once it is shown that segments or angles once it is shown that they are corresponding parts of congruent they are corresponding parts of congruent triangles…triangles…


Given: AB and CD bisect each other at M

Prove: AD BC

Statement Reason

1. Given

2. Definition of a bisector of a segment

3. Definition of a midpoint

4. Vertical angles are congruent

5. SAS Postulate

6. Corresponding parts of congruent triangles are congruent

7. If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

A

B

C

D

M

1. and

2. M is the midpoint of

and of

3. ;

4.

5.

6.

7. ll

bisect each other at M

ll


A line and a plane are perpendicular if and only if A line and a plane are perpendicular if and only if they intersect and the line is perpendicular to they intersect and the line is perpendicular to all lines in the plane that pass through the point all lines in the plane that pass through the point of intersection.of intersection.

P

X

O


Given: PO plane X;

AO BO

Prove: PA

Statement Reason

1. Given

2. Definition of a line perpendicular to a plane.

3. Definition of perpendicular lines

4. Defintion of congruent angles

5. Given

6. Reflexive Property

7. SAS postulate

8. Corresponding parts of congruent angles are congruent.

X

P

OA B

plane X

;

3. m = 90; m = 90

5.

6.

7.

8.

1.

2.

4.


To prove two segments or two angles are To prove two segments or two angles are congruent:congruent:

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

2.) Prove that the triangles are 2.) Prove that the triangles are congruent.congruent.

3.) State that the two parts are 3.) State that the two parts are congruent, using this reasoncongruent, using this reason

Corr. Parts of s are .Corr. Parts of s are .

4-4 The Isosceles Triangle 4-4 The Isosceles Triangle TheoremsTheorems

Legs-Legs- the congruent sides the congruent sides of a triangleof a triangle

Base-Base- the non-congruent the non-congruent side of a triangleside of a triangle

Base angles-Base angles- the angles the angles at the base of the at the base of the triangletriangle

Vertex angle- Vertex angle- the angle the angle opposite the base of the opposite the base of the isosceles triangleisosceles triangle

Vertex angle

Leg Leg

Base angles

Base


The Isosceles Triangle Theorem- The Isosceles Triangle Theorem- if two sides if two sides of a triangle are congruent, then the angles of a triangle are congruent, then the angles opposite those sides are congruentopposite those sides are congruent

CD B

A


Corollary 1- Corollary 1- an equilateral triangle is also an equilateral triangle is also equiangularequiangular

Corollary 2-Corollary 2- anan equilateral triangle has three equilateral triangle has three 60 degree angles60 degree angles

Corollary 3- Corollary 3- The bisector of the vertex angle The bisector of the vertex angle of an isosceles triangle is perpendicular to of an isosceles triangle is perpendicular to the base at its midpointthe base at its midpoint


Theorem 4-2Theorem 4-2

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

Corollary- Corollary- an equilateral an equilateral triangle is also triangle is also equilateralequilateral

* * Theorem 4-2 is the converse of Theorem 4-2 is the converse of Theorem 4-1, and the corollary of Theorem 4-1, and the corollary of Theorem 4-2 is the converse of Theorem 4-2 is the converse of Corollary 1 of Theorem 4-1.Corollary 1 of Theorem 4-1.

CD B

A

4-5 Other Methods of 4-5 Other Methods of Proving Triangles Proving Triangles

CongruentCongruentAngle Angle Side Theorem (AAS)- Angle Angle Side Theorem (AAS)- if two if two

angles and a non-included side of one triangle angles and a non-included side of one triangle are congruent to the corresponding parts of are congruent to the corresponding parts of another triangle, then the triangles are another triangle, then the triangles are congruentcongruent

A

E C OU

Y


Hypotenuse- Hypotenuse- the side opposite the right angle the side opposite the right angle in a right trianglein a right triangle

Legs- Legs- the other two sides of the trianglethe other two sides of the triangle

hypotenuse

leg

leg


Hypotenuse Leg Theorem- Hypotenuse Leg Theorem- if the hypotenuse if the hypotenuse and a leg of one right triangle are congruent to and a leg of one right triangle are congruent to the corresponding parts of another right the corresponding parts of another right triangle, then the triangles are congruenttriangle, then the triangles are congruent

O

TC GB

A

N


Leg-Leg Method- Leg-Leg Method- if two legs of one right triangle if two legs of one right triangle are congruent to the two legs of another right are congruent to the two legs of another right triangle, then the triangles are congruenttriangle, then the triangles are congruent

Hypotenuse-Acute Angle Method- Hypotenuse-Acute Angle Method- if the if the hypotenuse and an acute angle of one right hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the acute angle of another right triangle, then the triangles are congruenttriangles are congruent

Leg-Acute Angle Method- Leg-Acute Angle Method- If a leg and an acute If a leg and an acute angle of one right triangle are congruent of the angle of one right triangle are congruent of the corresponding parts in another right triangle, corresponding parts in another right triangle, then the triangles are congruent.then the triangles are congruent.

4-6 Using More than One 4-6 Using More than One Pair of Congruent TrianglesPair of Congruent Triangles

Given:

Prove:

Statement Reason

1. Given2. Reflexive property3. ASA postulate4. Corresponding parts of congruent angles are congruent.5. Reflexive property6. SAS postulate (1, 4, 5)7. Corresponding parts of congruent angles are congruent. 8. If two lines form congruentadjacent angles, then the lines are perpendicular.

1. ;

2.

3.

4.

5.

6.

7.

8.

6

5

4

3

2

1 O

B

D

AC

4-7 Medians, Altitudes, and 4-7 Medians, Altitudes, and Perpendicular BisectorsPerpendicular Bisectors

Median- Median- a segment from a vertex to the a segment from a vertex to the midpoint of the opposite side in a trianglemidpoint of the opposite side in a triangle

B

A C

CA

B

CA

B


Altitude-Altitude- the perpendicular segment from a vertex to a the perpendicular segment from a vertex to a line that contains the opposite sideline that contains the opposite side

In an acute triangle, the three altitudes are all inside the right triangle.

CA

B

CA

B

CA

B

4-7 Continued4-7 ContinuedA

B CCB

A

CB

A

In a right triangle, two of the altitudes are parts of the triangle. They are the legs of the right triangle. The third altitude is inside the triangle.

In an obtuse triangle, two of the altitudes are outside the triangle.

A

B CCB

A

CB

A

K

J

L


Perpendicular bisector- a line (or ray or segment) that is perpendicular to the segment at its midpoint

p

NM O



If a point lies on the perpendicular bisector of a If a point lies on the perpendicular bisector of a segment, then the point is equidistant segment, then the point is equidistant fromfrom the the endpoints of the segment.endpoints of the segment.

tL

I

JK


Theorem 4-6Theorem 4-6If a point is equidistant from the endpoints of a If a point is equidistant from the endpoints of a segment, then the point lies on the segment, then the point lies on the perpendicular bisector of the segment.perpendicular bisector of the segment.

*Theorem 4-6 is the converse of Theorem 4-5.*Theorem 4-6 is the converse of Theorem 4-5.

21

L

I

JK


The distance from a point to a line (or plane) The distance from a point to a line (or plane) is defined to be the length of the is defined to be the length of the perpendicular segment from the point to perpendicular segment from the point to the line (or plane).the line (or plane).

t

A D

B

R


Theorem 4-7Theorem 4-7If a point lies on the If a point lies on the bisector of an angle, bisector of an angle, then the point is then the point is equidistant from the equidistant from the sides of the angle.sides of the angle.

B

E

D

A

C

FG



If a point is equidistant If a point is equidistant from the sides of an from the sides of an angle, then the point angle, then the point lies on the bisector of lies on the bisector of the angle.the angle.

* Theorem 4-8 is the * Theorem 4-8 is the converse of Theorem 4-converse of Theorem 4-7. 7.

H

I

J

L

K

NM

The EndThe End(Thank God!)(Thank God!)

Documents

Chapter 4: Congruent Triangles. 4-1 Congruent Figures Congruent- when two figures have the same size and shape