Proving Triangles are Congruent: SSS, SAS Section 5.2

Objectives:

• Show triangles are congruent using SSS and SAS.

Key Vocabulary

• • Included angle proof

Postulates

• • 12 SSS Congruence Postulate 13 SAS Congruence Postulate

Review

•

Congruent Triangles

▫ Triangles that are the same shape and size

are congruent.

▫ Each triangle has three sides and three

angles.

▫ If all 6 parts (

3 corresponding sides

and

3 corresponding angles

congruent.

) are congruent….then the triangles are

Review

•

CPCTC – Corresponding Parts of Congruent Triangles are Congruent

•

Be sure to label proper markings (i.e. if



D

 

L,



V

  

P, Δs with



W

 

M, DV



WD



LP, VW



write ΔDVW



PM, and ML then we must ΔLPM)

So, to prove Δs



we must prove ALL sides & ALL



s are



?

Fortunately, NO!

 There are some shortcuts…  Two of which follow!!!

Postulate 12: Side-Side-Side Congruence Postulate

SSS Congruence

- If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

If

MN

  

SQ

then,

MNP



QRS

Remember!

Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts.

∆ PMN is adjacent to ∆ PON, PN ≅ PN by the Reflexive Property.

Example 1: Using SSS to Prove Triangle Congruence

Use SSS to explain why ∆ABC



∆DBC.

It is given that

AC



DC

and that

AB

 Reflexive Property of Congruence,

BC

Therefore ∆

ABC

 ∆

DBC

by SSS.

DB

. By the 

BC

.

Your Turn

Use SSS to explain why ∆

ABC

 ∆

CDA

.

It is given that

AB



CD

and

BC



DA

.

By the Reflexive Property of Congruence,

AC

So ∆

ABC

 ∆

CDA

by SSS.



CA

.

Example #2 – SSS – Coordinate Geometry Use the SSS Postulate to show the two triangles are congruent. Find the length of each side.

AC = 5 BC = 7 AB = MO = 5 2 5 7 NO = 7 74 MN = 5 2 7 V

ABC

 V

MNO

74

Example 3

Does the diagram give enough information to show that the triangles are congruent? Explain.

SOLUTION From the diagram you know that

HJ



LJ

and

HK



LK

.

By the Reflexive Property, you know that

JK



JK

.

ANSWER Yes, enough information is given. Because corresponding sides are congruent, you can use the SSS Congruence Postulate to conclude that ∆

HJK

 ∆

LJK

.

Definition : Included Angle

  The angle between two sides in a figure.

∠ B is included between:

BA

&

BC

INCLUDED ANGLE

C Y A B X Z Angle in between two consecutive sides

Use the diagram. Name the included angle between the pair of sides given.

∠ MTR ∠ RTQ ∠ MRT ∠ Q

The 2

nd

Congruence Short Cut

• • It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent.

SAS

Postulate 13: Side-Angle-Side Congruence Postulate

SAS 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.

If

PQ

 

XY

,

X

then,

PQS



WXY

Caution The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.

S S A S S A

Example 4: Engineering Application

The diagram shows part of the support structure for a tower. Use SAS to explain why ∆

XYZ

 ∆

VWZ

.

It is given that

XZ



VZ

and that the Vertical  s Theorem. 

XZY

Therefore ∆

XYZ

 ∆

VWZ

by SAS.

YZ

  

VZW

.

WZ

. By

Example 5

Does the diagram give enough information to use the SAS Congruence Postulate? Explain your reasoning.

a.

b.

SOLUTION a.

From the diagram you know that

AB



CB

and

DB



DB

.

The angle included between

AB

and

DB

is 

ABD

.

The angle included between

CB

and

DB

is 

CBD

.

Because the included angles are congruent, you can use the SAS Congruence Postulate to conclude that ∆

ABD

 ∆

CBD

.

Example 5

b.

You know that

GF



GH

and

GE



GE

. However, the congruent angles are not included between the congruent sides, so you cannot use the SAS Congruence Postulate.

Your Turn

Use SAS to explain why ∆

ABC

∆

DBC

.

 It is given that

BA



BD

and Reflexive Property of  ,

BC

 

ABC BC

  DBC. By the . So ∆

ABC

 ∆

DBC

by SAS.

Example 6A: Verifying Triangle Congruence

Show that the triangles are congruent for the given value of the variable.

∆

MNO

 ∆

PQR

, when

x

= 5.

PQ = x + 2 = 5 + 2 = 7 QR = x = 5

PQ

∆

MNO

 

MN

,

QR

∆

PQR



NO

by SSS.

,

PR



MO

PR = 3x – 9 = 3 (5) – 9 = 6

Example 6B: Verifying Triangle Congruence

Show that the triangles are congruent for the given value of the variable.

∆

STU

 ∆

VWX

, when

y

= 4.

ST

 ∆

STU VW

,

TU

 ∆

VWX



WX

, and by SAS.



T

ST = 2y + 3 = 2 (4) + 3 = 11 TU = y + 3 = 4 + 3 = 7  

W

m  T = 20y + 12 = 20 (4) +12 = 92° .

Your Turn

Show that ∆

ADB

 ∆

CDB

,

t

= 4.

DA = 3t + 1 = 3 (4) + 1 = 13 DC = 4t – 3 = 4 (4) m  D = 2t

2

– 3 = 13 

ADB

= 2 (16) = 32°  

CDB Def. of



.

DB



DB Reflexive Prop. of



.

∆

ADB

 ∆

CDB

by SAS.

#1 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent.

R S

ΔRST

 T

ΔYZX by SSS

#2 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent.

P R

ΔPQS



Q S

ΔPRS by SAS

#3 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent.

R S T

Not congruent.

Not enough Information to Tell

#4 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent.

P S U Q R

ΔPQR



T

ΔSTU by SSS

#5 Determine if whether the triangles are congruent. If they are, write a congruency statement explaining why they are congruent.

M P R Q N

Not congruent.

Not enough Information to Tell

Proof

• • A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.

An important part of writing a proof is giving justifications to show that every step is valid.

Two-Column Proof

• • •

Two-Column Proof

geometry in which an argument is presented with two columns,

statements

conjectures and theorems are true. Also referred to as a

formal proof

.

– A proof format used in and

reasons

, to prove Two- Column Proof is a formal proof because it has a specific format. Two-Column Proof ▫ Left column – statements in a logical progression.

▫ Right column – Reason for each statement (definition, postulate, theorem or property).

Two-Column Proof

Given: Prove: Proof:

Statements Reasons

How to Write A Proof

1.

2.

3.

4.

5.

6.

List the given information first Use information from the diagram Give a reason for every statement Use given information, definitions, postulates, and theorems as reasons List statements in order. If a statement relies on another statement, list it later than the statement it relies on End the proof with the statement you are trying to prove

Geometric Proof

• Since geometry also uses variables, numbers, and operations, many of the algebraic properties of equality are true in geometry. For example:

Property

Reflexive

Segments

AB = AB

Angles

m ∠ 1 = m ∠ 1 Symmetric Transitive If AB = CD, then CD = AB.

If AB = CD and CD = EF, then AB = EF.

If m ∠ 1 = m ∠ 2, then m ∠ 2 = m ∠ 1.

If m ∠ 1 = m ∠ 2 and m ∠ 2 = m ∠ 3, then m ∠ 1 = m ∠ 3.

• These properties can be used to write geometric proofs.

Remember!

Numbers are equal (=) and figures are congruent (  ).

Example 7

Write a two-column proof that shows ∆

JKL

 ∆

NML

.

JL



NL L

is the midpoint of

KM

.

∆

JKL

 ∆

NML

SOLUTION The proof can be set up in two columns. The proof begins with the given information and ends with the statement you are trying to prove.

Example 7

Statements 1.

JL



NL

2.

3.

L

is the midpoint of

KM

.



JKL

 

NML

Reasons 1.

Given 2.

Given These are the given statements.

3. Vertical Angles Theorem This information is from the diagram.

Example 7

Statements 4.

KL



ML

5.

∆

JKL

 ∆

NML

Reasons 4.

Definition of midpoint Statement 4 follows from Statement 2.

5. SAS Congruence Postulate Statement 5 follows from the congruences of Statements 1, 3, and 4.

Example 8

You are making a model of the window shown in the figure. You know that

DR

proof to show that ∆

DRA

 ∆ 

AG DRG

.

and

RA



RG

. Write a

D A R

SOLUTION 1. Make a diagram and label it with the given information.

G

Example 8

2. Write the given information and the statement you need to prove.

DR



AG

,

RA



RG

∆

DRA

 ∆

DRG

3.

Write a two-column proof. List the given statements first.

Example 8

Statements 1.

RA



RG

2.

3.

4.

DR



AG



DRA

and 

DRG

are right angles.



DRA

 

DRG

5.

DR



DR

6.

∆

DRA

 ∆

DRG

Reasons 1. Given 2. Given 3.

 lines form right angles.

4. Right angles are congruent.

5. Reflexive Property of Congruence 6. SAS Congruence Postulate

Your Turn: 1.

Fill in the missing statements and reasons.

CB



CE

, ∆

BCA



AC

∆

ECD



DC

Statements Reasons 1.

CB



CE

2.

1.

2.

Given ANSWER ANSWER Given

AC



DC

3.



BCA

 

ECD

3.

4.

∆

BCA

 ∆

ECD

4.

ANSWER ANSWER Vertical Angles Theorem SAS Congruence Postulate

Example 9:

Given: QR



UT, RS



Prove: ΔQRS



ΔUTS TS, QS = 10, US = 10

Example 9:

Given: QR



UT, RS



Prove: ΔQRS



ΔUTS TS, QS = 10, US = 10

Q 10 U 10 R S Statements Reasons________ 1. QR



UT, RS



TS, 1. Given QS=10, US=10 T 2. QS = US 2. Substitution 3. QS



US 4. ΔQRS



3. Def of



segs. ΔUTS 4. SSS Postulate

Your turn: Proof by SSS Given: Prove:

RP



RT RX

bisects

PRX



PT TRX

1) Statement

RP



RT

(s) 2) 3) 4) 5) 6)

RX

bisects

PT X

is midpoint of

PT PX



XT

(s)

RX



PRX RX

 (s)

TRX

R 48 P X T Reason 1) 2) 3) 4) 5) Given Given Def. line bisector Def. midpoint Reflexive Postulate 6) SSS Postulate

Example 10: Proving Triangles Congruent

Given:

BC

║

AD

,

BC



AD

Prove: ∆

ABD

 ∆

CDB

Statements 1.

BC || AD

2.



CBD

 

ABD

3.

BC



AD

4.

BD



BD

5.

∆

ABD

 ∆

CDB

Reasons 1. Given 2. Alt. Int.  s Thm.

3. Given 4. Reflex. Prop. of  5. SAS

Your Turn

Given:

QP

bisects 

RQS

.

QR



QS

Prove: ∆

RQP

 ∆

SQP

Statements 1.

QR



QS

2.

QP

bisects 

RQS

3.



RQP

 

SQP

4.

QP



QP

5.

∆

RQP

 ∆

SQP

Reasons 1. Given 2. Given 3. Def. of bisector 4. Reflex. Prop. of  5. SAS

Summary – SSS & SAS

Hints to Proofs!

• If two triangles

share a side

The reason is the , in the statement column state that the side is congruent to itself.

Reflexive Property

.

• If two triangles

share a vertex

congruent. The reason is the , in the statement column state that the angles are

Vertical Angles

Theorem.

Hints to Proofs!

• If a

vocabulary word

need its

definition

is in the given, you will as a reason somewhere in the proof, otherwise you wouldn’t need to know that piece of information.

• If two lines are parallel , look for congruent angles pairs, such as Angles.

Alternate Interior Angles, Corresponding Angles, or Alternate Exterior

Assignment

• Pg. 245 -249 #1 – 25 odd, 29 – 37 odd