4.3-4.6 Proving Triangles Congruent

Warm up:

Are the triangles congruent? If so, write a congruence statement and justify your

answer.

Proving Triangles Congruent…

• How can you prove sides congruent? (things to look for)

• How can you prove angles congruent?

Given Shared side(reflexive POE)Midpoints Segment Addition PropertySegment bisector Transitive POEothers?

Given Shared angle(reflexive POE)//→Alt. Int <s, . . . Angle Addition PropertyAngle bisector Vertical AnglesRight Angles(┴) Transitive POE

Now you try!

GIVEN:

R M

MN = RS

MO = RT

PROVE:

ΔMNO ΔRST

N

M O

S

R T

Now you try!

GIVEN:

R M

MN = RS

MO = RT

PROVE:

ΔMNO ΔRST

N

M O

S

R TSTEP 1 – DRAW IT AND MARK IT!

Now you try!

GIVEN:

R M

MN = RS

MO = RT

PROVE:

ΔMNO ΔRST

N

M O

S

R TSTEP 1 – DRAW IT AND MARK IT!

STEP 2 – CAN YOU PROVE THE Δs =?

HOW?

Now you try!

GIVEN:

R M

MN = RS

MO = RT

PROVE:

ΔMNO ΔRST

N

M O

S

R TYES, BY SAS FROM THE GIVENS!

REAL LIFE EXAMPLES

Bridges – Golden Gate, Brooklyn Bridge, New River Bridge . . . .

Real Life

Real Life

Types of Proofs

Traditional two-column: This looks like a T-chart and has the statements on the left and reasons on the right.

Types of Proofs

Flow Chart: Starts from a “base line” and all information flows from the given. Great for visual learners.

Paragraph: Write it out! Tell me what you’re doing!

Helpful Hints with Proofs…

• ALWAYS mark the given in your picture.

• Use different colors in your picture to see the parts better.

• ALWAYS look for a _______________________ which

uses the __________________ property.

• ALWAYS look for ______________ lines to prove mostly

that _____________________________________.

• ALWAYS look for ____________ angles which are always

___________.

common side/anglecommon side/angle

reflexivereflexive

parallelparallel

alternate interior angles are congruentalternate interior angles are congruent

verticalvertical

congruentcongruent

Given: PQ PS; QR SR; 1 2

Prove: 3 4

Statements1. PQ PS; QR SR;

1 2

2. PR PR

3. ∆QPR ∆SPR

4. 3 4

Reasons1. Given

2. Reflexive Property

3. SAS Postulate

4. CPCTC

Given: WO ZO; XO YO

Prove: ∆WXO ∆ZYO Statements1. WO ZO; XO YO

2. WOX ZOY

3. ∆WXO ∆ZYO

Reasons1. Given

2. Vertical angles are .

3. SAS Postulate

Proof Practice

Given: PSU PTR; SU TR

Prove: SP TPHINT: draw the triangles separately!

Proof Practice…

1. 1. PSU PSU PTR; SU PTR; SU TR TR 1. given1. given

2. <P 2. <P <P <P 2. Reflexive POE2. Reflexive POE

3. 3. ∆∆SUP SUP ∆∆TRP TRP 3. AAS Theorem3. AAS Theorem

4. SP 4. SP TP TP 4. CPCTC4. CPCTC

Proof Practice…

PSU PTR SU TR <P <P

∆SUP ∆TRP

SP TP

CPCTCCPCTC

AAS Theorem

Practice

Name the included side for 1 and 5.

Name a pair of angles in which DE is not included.

If 6 10, and DC VC, then

∆ DCA ∆ _______, by _________.

DCDC

<8, <9, for example<8, <9, for example

VCEVCE ASAASA

More Proofs…Using 2 Column • Given: PQ RQ; S

is midpoint of PR.• Prove: P R

QS is an auxiliary line P

Q

RS

1. PQ PQ RQ; S is midpoint of PR RQ; S is midpoint of PR 1. givengiven

2. PS PS SR SR 2. Def midpointDef midpoint3. QS QS QS QS 3. Reflexive POEReflexive POE4. ∆∆PQS PQS ∆∆RQSRQS 4. SSS PostulateSSS Postulate

5. P P RR 5. CPCTCCPCTC

More Proofs…Using Flow Chart • Given: PQ RQ;

S is midpoint of PR.

• Prove: P R

P

Q

RS

PQ RQ S is midpoint of PR

PS SR

QS QS

∆PQS ∆RQS

P P RR

CPCTC

SSS Postulate

More Proofs…Using Paragraph • Given: PQ RQ; S is midpoint of PR.• Prove: P R

QS is an __________________.

We are given that ____________ and ___________________. Because S is the midpoint, we know that __________because of _____________.

We drew in QS so that we can use the reflexive property to prove that

_________. We now have enough information to prove that ∆PQS ∆RQS

by ____________. Therefore <P <R by __________________.

P

Q

RS

auxiliary lineauxiliary line

PQ PQ RQ RQ S is midpoint of PRS is midpoint of PR

PS PS SR SR def. of midptdef. of midpt

QS QS QSQS

SSS Post. SSS Post. CPCTCCPCTC

Do the following proofs in whatever way

you feel comfortable

Given: AB EB; DEC B

Prove: ∆ABE is equilateral

Group Work Time:• Group 4.3-4.6 proof

practice WS• Group presentations

• Next Class• Group presentations• More group practice

work