~3.1 Congruent Triangles~

•Basis used to determine congruence in triangles

•Triangles have six parts by which we determine whether two

triangles are congruent:

•The three sides of the triangle

•The three angles of the triangle

~The two triangles are congruent if the corresponding parts are

equal.

• The corresponding parts of two triangle are the parts

which “match” or the parts which hold the same

relative or ”corresponding” positions with respect to

the other parts of the triangle.

The sides and angles that match are called

corresponding sides and corresponding angles.

A

B C

X

Y Z

• For both triangles, segment AB and XY are the shortest

sides, segment BC and YZ are the “medium” sides and

segment AC and XZ are the longest sides.

• Corresponding sides: segment AB & XY, BC & YZ

and AZ corresponds to XZ.

• We can say that:

∆ABC = ∆XYZ ∆CAB = ∆ZXY

∆BCA = ∆YZX ∆BAC = ∆YZX

• Example 01

~ ∆YES and ∆ART are congruent.Identify the

corresponding parts.

Y

E S

A

R

T

~CORRESPONDING PARTS ARE:

segment YE = segment RT angle Y = angle R

segment YS = segment RA angle S = angle A

~Properties of equality and equivalence

relations~

1.Addition property - - - - if a = b,then a + c = b + c

or, if a = b & c = d, then

a + c = b + d

2. Subtraction property- - -if a = b, then a – c = b – c

or, if a = b & c = d, then

a – c = b – d

3.Multiplication property-if a = b, then ac = bc

if½a=½b, then a = b

4.division property - - - - if a = b, then a/c = b/c

if a = b, then ½a = ½b

(halves of equals are equal).

• For the next properties, we will use congruence as equivalence relations.

5.Reflexive property- - - -any number or figure is

congruent to itself.

Ex. Segment AB = segment AB.

6.Symmetric property- - - -members on both sides of the

congruence symbol are

interchangeable.

7.Transitive property- - - - figures that are congruent to

the same figure are also

congruent to each other.

Exercises:Say if what property is applied in the following.

1.If X = Y and N=M, then X-N=Y-M

2.AB = CD , then AB= CD

2 2

3. ST = ST

4.IT = TO, then TO = IT

5.GH=LM and LM= OP, then GH=OP

Answers:

1.Subtraction Property

2.Multiplication Property

3. Reflexive Property

4.Symmetric Property

5.Transitive Property

~Rules used to prove Congruent Triangles~

1. Using all six corresponding parts

Postulate 25 CPCTC postulate

-if all six pairs (angles and sides) of two triangles

are congruent , the two triangles are congruent.

CPCTC stands for corresponding parts of congruent

triangles are congruent.

M

N O

P

Q R

The corresponding parts of ∆MNO and ∆PQR are

congruent. There may be times, however , when not all

the measures of the sides and the angles are given.

MN=PQ

NO=QR

MO=PR

angle M = angle P

angle N = angle Q

angle O = angle R

2.Three sides of the triangles

Postulate 26 side-side-side (SSS) postulate

~ if three sides of one triangle are congruent to three

sides of another triangle, the two triangles are

congruent.

4 4

8 8

5 5

A

B C

D

E F

Thus, by the SSS Postulate

3.Two sides and an included angle

• Postulate 27 side-angle-side (SAS) postulate

~ if two sides and the included angle of one triangle are

congruent to two sides and the included angle of

another triangle, the two triangles are congruent.

• An included angle created by two sides of a triangle.

G

H I

K

L M

55

Example:

GH=KL

HI= LM

Thus , ∆GHI = ∆KLM by the SAS postulate.

4.Two angles and the included side

• Postulate 27 angle-side-angle (ASA) postulate

~if two angles and the included side of one triangle are

congruent to two angles and the included side of

another triangle, the two triangles are congruent.

An included side is a side that is between two angles.

for example, in the figure used of another triangle, the

two triangles are congruent.

O

N

Q

P

Thus, by ASA Potulate

5.Two angles and non-incuded side

Postulate 28 side-angle-angle (SAA) or angle-angle-side

(AAS) postulate

~if two angles and one non-included side of one triangle

are congruent to two angles and the corresponding

non-included side of another triangle, the two

triangles are congruent.

R

S T

U

W V

Thus, by the AAS or SAA Postulate

6.Third angle postulate

Postulate 29

~ if two angles of one triangle are congruent to two

angles of a second triangle , then third angles of the

triangles are congruent.

A

B C

X

Y Z

Since m A=m X and m B=m Y,

Then m C= m Z

Therefore by TA Postulate ∆ABC=∆XYZ

N.b.

There are also postulate that do not work in

proving that triangles are congruent.

1. Angle-Angle-Angle(AAA)

AAA can show that the triangles are of the same

shape( similar) but it does not necessarily mean

that the triangles are congruent.

2. Angle-Side-Side (ASS)

This does not show congruency.

Summary:6 Postulates to prove that two triangles are congruent.

a. CPCTC Postulate – Corresponding angles and side of two

triangles are congruent.

b. SSS Postulate - All corresponding sides of two triangles

are congruent.

c. SAS Postulate – two sides and an included angle of two

triangles are congruent.

d. ASA Postulate - two angles and an included side of two

triangles are congruent.

e. SAA/AAS Postulate - two angles and a non- included

side of two triangles are congruent.

f. TA Postulate - if two corresponding side of two triangles

are congruent, then the third angles of the triangles are

congruent.

Exercise:

A.Give the name of the Postulate that is ask in each

of the ff:

1. Segment AB= segment DE ; segment BC = segment EF;

angle B= angle E

2. Angle R= angle H; segment RS= segment HI ; angle I=

angle S

3. Segment JK= segment MN ; segment KL= segment

NO; segment JL= segment MO

B. Write the corresponding parts needed:X

Y

Z

O

1. ∆YXO=∆_____

2. YO= _____

3. angle X= _____

4. angle O= _____

5. YX= _______

6.∆YOZ=∆_____

Answers:

A.

1. SAS Postulate

2. ASA Postulate

3. SSS Postulate

B.

1. ∆ YZO

2. YO

3. angle Z

4. angle O

5. YZ

6. ∆ YOX

3.2 Proving triangle congruence

• How to make a proof ?

1.Make a diagram of the figure and mark the given

information about the sides and angles and othe

relevant parts.

2.Select which rule (in this case SSS, SAS, AAS/SAA)

You will be using in proving the statement .

3.In one column, write the statements and in another

column write the reasons for the statements.

T U V

W X

Given: angle TUW = angle VUX

angle T = angle V

U is the midpoint if segment TV

Prove: ∆TUW = ∆VUX

1.Angle TUW = VUX (angle)

2.Angle T = angle V (angle)

3.U is the midpoint of segment

TV.

4.Segment TU = UV

(included side)

5.∆TUW = ∆VUX

1.Given

2.Given

3.Given

4.Definition of a midpoint-it

evenly divides the segment into

two.

5.ASA postulate

STATEMENT REASONS

Exercises:A. Complete each proof.

1. given: segment SU bisects angle TSV

segment SU is perpendicular to segment TV

PROVE:∆SUV = ∆SUT

T

S U

V

a. segment SU bisects angle TSV

b. Angle TSU = angle VSU

c. Segment SU =segment SU

d. Segment SU is perpendicular to

segment TV

e. Angle SUT & angle SUV are

right triangles

f. Angle SUT=90°

Angle SUV=90°

g. Measure of angle SUT = measure

of angle SUV

h. Angle SUT = angle SUV

i. ∆SUV = ∆SUT

a.

b.

c.

d.

e.

f.

g.

h.

i.

STATEMENTS REASONS

a. segment SU bisects angle TSV

b. Angle TSU = angle VSU

c. Segment SU =segment SU

d. Segment SU is perpendicular to

segment TV

e. Angle SUT & angle SUV are

right triangles

f. Angle SUT=90°

Angle SUV=90°

g. Measure of angle SUT = measure

of angle SUV

h. Angle SUT = angle SUV

i. ∆SUV = ∆SUT

a. Given

b. Definition of Angle bisector

c. RPC

d. Given

e. Definition of perpendicular

angles

f. Definition of right angles

g. Substitution

h. Definition of Congruent

Angles

i. ASA Postulate

STATEMENTS REASONS