Congruent triangles have congruent The parts of congruent ......pictures of triangles. You are to...

Congruent triangles have congruent

sides and congruent angles.

The parts of congruent triangles that

“match” are called corresponding parts.

A

B C F

D

E

AB DF

BC FE

AC DE

A D B F

C E

TO PROVE TRIANGLES ARE CONGRUENT YOU DO NOT NEED TO KNOW ALL SIX

DFEABC

Before we start…let’s get a few things straight

A B

C

X Z

Y

INCLUDED ANGLEIt’s stuck in between!

Before we start…let’s get a few things straight

INCLUDED SIDEIt’s stuck in between!

A B

C

A B

C

Overlapping sides are congruent in each

triangle by the REFLEXIVE property

Vertical Angles are congruent

Alt Int Angles are congruent

given parallel lines

Side-Side-Side (SSS) Congruence Postulate

66

4 45 5

All Three sides in one triangle are congruent to all three sides in the

other triangle

Are these triangles congruent?

D

O

G

C

A

T

If so, write the congruence statement.

Side-Angle-Side (SAS) Congruence Postulate

Two sides and the INCLUDED angle

Are these triangles congruent?

If so, write the congruence statement.

C

A

T

H

A

T

Angle-Side-Angle (ASA) Congruence Postulate

Two angles and the INCLUDED side

Are these triangles congruent?

If so, write the congruence statement

B

I

G

T

O

E

Angle-Angle-Side (AAS) Congruence Postulate

Two Angles and One Side that is NOT included

Are these triangles congruent?

If so, write a congruence statement.

T

O

P H

A

T

The following slides will have

pictures of triangles. You are to

determine if the triangles are

congruent. If they are congruent,

then you should write a

congruence statement and state

which postulate you used to

determine congruency.

Δ_____ Δ_____ by ______

Determine if whether the triangles are

congruent. If they are, write a congruency

statement explaining why they are congruent.

ΔJMK ΔLKM by SAS

J K

LM

Determine if whether the triangles are congruent. If they

are, write a congruency statement explaining why they are

congruent.

Ex 4

R

P

S Q

ΔPQS ΔPRS by SAS

Determine if whether the triangles are congruent. If they

are, write a congruency statement explaining why they are

congruent.

Ex 5

R

P

S

Q

ΔPQR ΔSTU by SSST

U

Determine if whether the triangles are congruent. If they

are, write a congruency statement explaining why they are

congruent.R

T

S

ΔRST ΔYZX by SSS

Ex 2

Determine if whether the triangles are

congruent. If they are, write a congruency

statement explaining why they are congruent.

ΔGIH ΔJIK by AAS

G

I

H J

K

Not congruent.Not enough Information to Tell

R

T

S

Determine if whether the triangles are congruent. If they

are, write a congruency statement explaining why they are

congruent.

Ex 3

Determine if whether the triangles are

congruent. If they are, write a congruency

statement explaining why they are congruent.

ΔABC ΔEDC by ASA

B A

C

ED

Determine if whether the triangles are congruent. If they

are, write a congruency statement explaining why they are

congruent.

Ex 6

N

M

R

Not congruent.Not enough Information to Tell

Q

P

Warm up

Are they congruent, if so, tell how.1.

AAS

2.

Not

congruent

3.

Not

congruent

2-Column Proofs

• Going by the facts: definitions, properties, postulates, and theorems

• Numbering the statements and reasons

• Using logical order

Statements Reasons

1.

2.

3..

.

.

1.

2.

3..

.

.

Given: seg WX seg. XY, seg VX

seg ZX,

Prove: Δ VXW Δ ZXY

1 2

W

V

X

Z

Y

Proof

Statements Reasons

1. seg WX seg. XY 1. given

seg. VX seg ZX

2. 1 2 2. Vertical Angles

Congr Theorem

3. Δ VXW Δ ZXY 3. SAS Congr Postul

Given: seg RS seg RQ and seg ST

seg QT

Prove: Δ QRT Δ SRT.

Q

R

S

T

Proof

Statements Reasons

1. Seg RS seg RQ 1. Given

seg ST seg QT

2. Seg RT seg RT 2. Reflexive

Property

3. Δ QRT Δ SRT 3. SSS Congr Postulate

Example• Given that B C, D F, M is the

midpoint of seg DF

• Prove Δ BDM Δ CFM

B

D M

C

F

Proof

Statements Reasons

1. B C, D F 1. Given

2. M is the midpoint of 2. Definition of Midpt

seg DF

3. Seg DM seg FM 3. Reflexive Property

4. Δ QRT Δ SRT 4. AAS Congr Theorem

TRIANGLE

PROPORTIONALITY

THEOREMUsing similarity to find the missing parts of a triangle

TRIANGLE PROPORTIONALITY THEOREM

If a line parallel to one side of a triangle

intersects the other two sides, then it

divides those sides proportionally.

𝑨𝑫

𝑫𝑩=𝑨𝑬

𝑬𝑪

EXAMPLE 1:

Find the missing side length.

𝟖

𝟐𝟎=𝟏𝟖

𝒙

EXAMPLE 2:

Find the missing side length.

𝒙

𝟏𝟓=

𝟒

𝟏𝟎

EXAMPLE 3:

Find the missing side length.

𝒙

𝟏𝟓=

𝟐

𝟏𝟎

ON YOUR OWN:

1) Find the missing side length.

𝒙

𝟗=

𝟗

𝟏𝟓

ON YOUR OWN:

2) Find the missing side length.

𝟏

𝟒=𝟏

𝒙

ON YOUR OWN:

3) Find the missing side length.

𝟖

𝒙=𝟏𝟒

𝟑𝟓

ON YOUR OWN:

4) Find the missing side length.

𝒚

𝟏𝟐=𝟏𝟓

𝟏𝟎

Properties of

Parallelograms

•Both pairs of opposite

sides are parallel

Opposite sides are congruent

Opposite angles are congruent

Consecutive angles are supplementary

ABCD is a parallelogram. Find the lengths

and the angle measures.

1. AD

2. mADC

3. mBCD

B 8 C

A D

6545

E

5

8

110

70

4. Find the value of each variable

in the parallelogram.

4y

x = 5

2y4

2x – 6

y = 30

5. Find the measure of ∠D in the

parallelogram.

∠D = 115°

2x – 1x + 7

2x – 1x + 7

A

B

C

D

How to Prove

Quadrilaterals are

Parallelograms

How do you know if you have one?

1.BOTH pairs of opposite sides are parallel

2.BOTH pairs of opposite sides are congruent

3. BOTH pairs of opposite angles are congruent

4.ONE angle is supplementary to BOTH consecutive angles

5.diagonals BISECT each other

6. ONE pair of opposite sides are CONGRUENT & PARALLEL

60 120

120

6 6

60

50°

130°