Aim: SAS – Triangle Congruence Course: Applied Geometry Do Now: Aim: Are there any shortcuts to...

Aim: SAS – Triangle Congruence Course: Applied Geometry

Do Now:

Aim: Are there any shortcuts to prove triangles are congruent?

In triangle ABC, the measure of angle B is twice the measure of angle A and an exterior angle at vertex C measures 120o. Find the measure of angle A.


Is ABCDE the exact same size and shape as STUVW?

Congruence

C

B

A

E D W

S

T

U

V

How would you prove that it is?

Measure to compare.

Measure what?

5 sides 5 angles

If the 5 side pairs and 5 angle pairs measure the same, then the two polygons are exactly the same.


Corresponding Parts

CORRESPONDING PARTS

IF

AB BC CD DE EA

ST TU UV VW WS

THEN THE POLYGONS ARE

CONGRUENT

ARE CONGUENT

A B C D E

S T U V W

Corresponding Parts – pairs of segments or angles that are in similar positions in two or more polygons.

C

B

A

E D W

S

T

U

V


Congruence Definitions & Postulates

Two polygons are congruent if and only if1. corresponding angles are .2. corresponding sides are .

Corresponding parts of congruent polygonsare congruent.

True for all polygons,triangles our focus.

CPCPC

CPCTC

Corresponding Parts of Congruent Trianglesare Congruent.


Model Problem

Hexagon ABCDEF hexagon STUVWX. Find the value of the variables?

AB

C

DE

F

ST

U

VW

X

10

X

8

2y

120

AB and ST are corresponding sides

x = 10

x = 1200

F & X are corresponding ’s

ED and WV are corresponding sides2y = 8 y = 4

AB

C

DE

F

ST

U

VW

X

10

X

8

2y

120


Corresponding Parts.

A B

C G

HI

Is ABC the exact same size and shape as GHI?

How would you prove that it is?

Measure corresponding sides and angles.

What are the corresponding sides? angles?

AC GIAB IHBC GH

A I B H C G


Side-Angle-Side

I. SAS = SAS

Two triangles are congruent if the two sides of one triangle and the included angle are equal in measure to the two sides and the included angle of the other triangle.S represents a side of the triangle and

A represents an angle.A

B B’C C’

A’

If CA = C'A', A =A', BA = B'A', then ABC = A'B'C'

If SAS SAS , then the triangles are congruent


Model Problem

Each pair of triangles has a pair of congruent angles. What pairs of sides must be congruent to satisfy the SAS postulate?

E

D

A B

C

;CE and EB AE and ED

A

C

F

G

H

B

;GH and BC FG and AB


Model Problem

Each pair of triangles is congruent by SAS. List the given congruent angles and sides for each pair of triangles.

A

C

B

E

F

D

; ,AB DE BC EF B E

; ,DE DG DF DF

EDF GDF

G

D F

E


Do Now:

Aim: Are there any shortcuts to prove triangles are congruent?

Is the given information sufficient to prove congruent triangles?

A B

C F

ED

SAS = SAS Two triangles are congruent if the two sides of one triangle and the included angle are equal in measure to the two sides and the included angle of the other triangle.


Side-Angle-Side

Is the given information sufficient to prove congruent triangles?

D E

FB

A C

A B

C F

ED

A

B

D

CA B

CD


Side-Angle-Side

Given that C is the midpoint of AD and AD bisects BE, prove that ABC CDA. A

B D

E

C

• C is the midpoint of AD means that CA CD.

• BCA DCE because vertical angles are congruent.

• AD bisects BE means that BE is cut in to congruent segments resulting in BC CE.

The two triangles are congruent because of SAS SAS

(S S)

(A A)

(S S)


Side-Angle-Side

In ABC, AC BC and CD bisects ACB. Explain how ACD BCD

A D B

C


Side-Angle-Side

In ABC is isosceles. CD is a median. Explain why ADC BDC.

A D B

C


Sketch 12 – Shortcut #1

SAS SASSAS SAS

B

A

C

Copied 2 sides and included angle:

AB A’B’, BC B’C’, B B’

Copied 2 sides and included angle:

AB A’B’, BC B’C’, B B’

B’

A’

C’

B’

A’

C’

Shortcut for proving congruence in triangles:

Measurements showed: ABC A’B’C’ABC A’B’C’


The Product Rule

Aim: SAS – Triangle Congruence Course: Applied Geometry Do Now: Aim: Are there any shortcuts to...

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