Chapter 4 Notes

4.1 – Triangles and Angles

A Triangle Three segments joining three noncollinear points. Each point is a VERTEX of the triangle. Segments are SIDES!

A

B

C

Scalene – No congruent sides

Isosceles – At LEAST 2 congruent sides

Equilateral – All sides congruent

Acute – 3 acute angles

Obtuse – One obtuse angle

Right – One right angle

Equiangular – all angles congruent

A

B

C

adjacent-non means opposite

C opposite is AB Side

BC side theopposite isA

8013m2m1m:Prove

ABC :Given

1 3

TRIANGLE SUM THEOREM

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

2

Exterior Angles Theorem

The measure of an exterior angle of a triangle equals the sum of the two remote interior angles. (remote means nonadjacent)

1 2

3

4

Statement Reason

4m3m1m :Prove

1 angle

exterior with A triangle :Given

Corollary to triangle sum theorem: Acute angles of a right triangle are complementary.

All angles 180, if one is 90, the other two add up to 90, and are complementary

4.2 – Congruence and Triangles

When TWO POLYGONS have the same size and shape, they are called CONGRUENT!

Their vertices and sides must all match up to be congruent.

When two figures are congruent, their corresponding sides and corresponding angles are congruent. Identical twins!

A

C

B

D

E

F

Name all the corresponding parts and sides, then make a congruence statement.

FEAB EDBC DFCA

FA EB DC

FEDABC BCACABEDFDFE

If you notice, the way you name the triangle is important, all the CORRESPONDING SIDES must line

up!

3rd Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the 3rd angles are congruent.

A

B

C D

E

F

FCThen

EBDAIf

,

M

A

HT

G

E

Y Ox3

15

24

45

80 105 y

Find x Find y

• Note, triangles also have the following properties of congruent: Reflexive, symmetric, and transitive.

XYZABC

thenXYZDEF DEF,ABC :Transitive

ABC ,ABC :Symmetric

ABCABC :Reflexive

DEFDEF

4.3 – 4.4 Proving Triangles are Congruent

A

C

B

D

E

FSSS Congruence Postulate – If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

FEDABCThen

DFACDEBCEFABIf

,,

FEDABCThen

DFACDCDEBCIf

,,

A

C

B

D

E

F

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.

Included means IN BETWEEN

A

C

B

D

E

FASA Congruence Postulate –

If two angles and the included side of one triangles are congruent to two angles and the included side of a

second triangle, then the two triangles are congruent.

FEDABCThen

DCDFACFAIf

,,

FEDABCThen

DFACDCEBIf

,,

A

C

B

D

E

F

AAS Congruence Theorem – If two angles and a nonincluded

side of one triangle are congruent to two angles and

the corresponding nonincluded side of a second triangle, then the two trianges

are congruent.

A

C

B

D

E

F

A

C

B

EF

Helpful things for the future!Reflexive sides Reflexive angles

A

B

DC F

E

HG

BCBC EE

When you see shapes sharing a side, you state that fact using the reflexive property of

congruence!

A

CB D

Draw and write down if the triangles are congruent, and by what thrm\post

Proofs! The way I like to think about it to look at all the angles and sides, and don’t be fooled by the picture.

A

B

C

D

E

EDCABC :Prove

BD and AE of

midpoint theis C:Given

BD and AE of

midpoint theis C 1. Given 1.

Tips, label the diagram as you go along.

A

B

D

E

EDBABD :Prove

DE AB,BE AD :Given

DE AB

BE AD 1.

Given 1.

F

E

H

G

ECFEBH :Prove

EC EB,EH EF :Given

B C

EC EB,EH EF 1) Given 1)

What about the angle?

Use SSS Congruence Postulate to show that DEFABC

A

C

B

DE

F

(-3, -2)

(-4, -3)

(-5, 1)

(2, 2)

(5, 4)(1, 3)

AC

BC

AB

DF

EF

DE

A

B

D

E

EDBABD :Prove

DE ||AB,BE ||AD :Given

DE ||AB

BE ||AD 1. Given 1.

Tips, label the diagram as you go along.

U

D

C

K

SKCSDUC

:Prove

KD ;KCDU

KC||DU :Given

KD ;KCDU;KC||DU .1 Given .1

CB

A

D

BDEABC :Prove

BEAC,DE||BC ,BE||AC :Given

E

DECABC :Prove

CEBC,EB :Given

C

B

A

D

E

4.5 – Using Congruent Triangles

A

B

C

D

E

ED||AB :Prove

BD and AE of

midpoint theis C:Given

BD and AE of

midpoint theis C 1. Given 1.

DCBC

ECAC 2.

mdpt of Def 2.

ECDACB 3. VAT 3.ECDACB 4. post SAS 4.

are s' of Parts CorrespCongruent are

TrianglesCongruent ofPart ingCorrespondCPCTC

Some Ideas that may help you.

If they want you to prove something, and you see triangles in the picture, proving triangles to be congruent may be helpful.

If they want parallel lines, look to use parallel line theorems (CAP, AIAT, AEAT, CIAT)

Know definitions (Definition of midpoint, definition of angle bisectors, etc.)

Sometimes you prove one pair of triangles are congruent, and then use that info to prove another pair of triangles are congruent.

A

B

D

E

AD||BE :Prove

DE AB,BE AD :Given

DE AB

BE AD 1.

Given 1.

U

D

C

K

SUS ofmdpt theis C :Prove

KD ;KCDU

KC||DU :Given

KD ;KCDU;KC||DU .1 Given .1

DCAC :Prove

CEBC,EB :Given

C

B

A

D

E

G

N

AL E

LGLN:Prove

LNALGA

ALNALG:Given

EE

LNALGA ALN,ALG Given

You try this classic proof!

A

B

C

D

E12

56

34

DCB ofbisector

angle theis AC :Prove

BEDE 2,1 :Given

4.6 – Isosceles, Equilateral, and Right Triangles.

• Bring book Tuesday

• We will go over what’s going to be on Wednesday’s Quiz at end of Tuesday lesson

Vertex Angle

Base Angles

BASE

LEGS

Remember, definition of isosceles triangles is that AT LEAST two congruent sides.

Base angles theorem – If two sides of a triangle are congruent, then the base angles are congruent.

Converse of Base angles theorem – If base angles are congruent, then the two opposite sides are congruent.

Corollary 1 – An equilateral triangle is also equiangular (Use isosceles triangle theorem multiple times with transitive)

Corollary 2 – An equilateral triangle has three 60 degree angles (Use corollary 1 and angle of triangle equals 180)

Hypotenuse Leg Theorem (HL) – If the hypotenuse and ONE of the legs of a RIGHT triangle are congruent, then the triangles are congruent.

A

CB D

Draw and write down if the triangles are congruent, and by what thrm\post

D

U C K1 2

KUove

anglesrightareand

legsasDKandDU

withtriangleisoscelesanisDUKGiven

:Pr

.21

.

:

DKDU Def of isosceles triangle

CB

A

D

ADCABC :Prove

BDAC ,ADAB :Given

4.7 – Triangles and Coordinate Proof

Given a right triangle with one vertex (-20, -10), and legs of 30 and 40, find two other vertices, then find the length of the hypotenuse.

Given a vertex of a rectangle at the origin, find three other possible vertices if the base is 15 and the height is 10 for a rectangle. Then find the area.

Given the coordinates, prove that the AC is the angle bisector of BCD

B

C D

A

(a, b)(__,__)

(__,__)

(h,k)

(__,__)

(j,__)

(d, k)

(__,__)

(__, k)

(a,__)

Picking convenient variable coordinates, prove that the diagonals of a rectangle are congruent.