SIMILARITY

AND CONGRUENCE

Insisivi Eka SMutiara Aura KSri Ayu Pujiati

SIMILARITY

CONGRUENCE

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SIM

ILA

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SIMILAR FIGURES

SIMILAR FIGURES

SIMILAR FIGURES

SIMILAR FIGURES

SIMILAR FIGURES

SIMILAR FIGURES

SIMILAR FIGURES

SIMILAR FIGURES

SIMILAR FIGURES

SIMILAR

FIGURESSIMILAR FIGURES

SIMILAR

FIGURES

SIMILAR FIGURES

SIMILAR FIGURES

SIMILAR FIGURES

SIMILAR FIGURES

Similar figures are two figures that are the

same shape and whose sides are proportional

SIMILAR FIGURES

~ This is the symbol that means “similar.”

These figures are the same

shape but different

sizes.

SIMILAR FIGURES

Example :A 25 cm x 15 cm rectangle and a 10 cm x 6 cm rectangle are given. Are

the rectangles similar? 15 cm 6cm

10 cm 25 cm

ANSWER

ANSWER \( )/≧∇≦

The two rectangles have equal corresponding angles each of which is a

right angle.Ratio of the length = 25 cm : 10 cm = 5 : 2Ratio of the width = 15 cm : 6 cm = 5 : 2

Thus, Two rectangles are similar . Because the corresponding angles are equal and the

corresponding sides are proportional.

SIMILAR

TRIANGLES

SIMILAR TRIANGLES

Similar triangles are two triangles that have the

same shape but not necessarily the same size.

The Corresponding sides are in proportion

Corresponding pairs of sides are in proportion

TWO TRIANGLE ARE SIMILAR IF :

SIMILAR TRIANGLE

Angle A ~ Angle DAngle B ~ Angle EAngle C ~ Angle F

AB = BC = AC DE EF DF

Proving Similarity (AAA) - Angle, Angle, Angle

If three angles of one triangle are congruent, respectively, to three angles of a second triangle, then the triangles

are similar.AAA AA

(` ´)-σ ▽ Example I :

In △ABC and △DEF, AB = 9 cm, BC = 6 cm , CA = 12 cm, DE = 15 cm, EF = 10 cm, FD = 20 cm. Explain why the two triangles are considered similar. Name all the pairs of equal angles !

ANSWER

C

12 6 A B F

20 10

D 15 E

ᾈňšὠὲ ŕ (• •) "̮� AB : DE = 9 cm : 15 cm= 3 : 5

BC : EF = 6 cm : 10 cm= 3 : 5

CA : FD = 12 cm : 20 cm

= 3 : 5Thus, △ABC and △FED are similar since all the corresponding sides are proportional

• The Pairs of equal angles are : A = D , B = E , C = F

In △ABC :AB = 9 cmBC = 6 cmCA = 12 cmIn △ DEF :DE = 15 cmEF = 10 cmFD = 20 cm

•

CONGRUENCE

CONGRUENT FIGURES

CONGRUENT TRIANGLES

CONGRUENCE

CONGRUENCE

CONGRUENCE

CONGRUENCE CONGRUENCE

CONGRUENT FIGURES

Two figures are congruent if they have same size

and same shape.

This is the symbol that means “congruence.”These figures are the same shape and same sizes.

≅

The Properties of Two Congruent Figures

Has same shape and same size

All corresponding pairs of angles are congruentCorresponding pairs of sides are congruent.

EXAMPLE:

D C

A B

H G

E

‘ F

Since parallelogram ABCD and EFGH are congruent :

EH = AB, thus AB = 7 cmAD = GH , thus AD = 12 cm

CONGRUENT TRIANGLES

When we talk about congruent triangles,we mean everything about them Is

congruent. All 3 pairs of corresponding angles are

equal….

And all 3 pairs of corresponding sides are equal

• To prove that two triangles are congruent it is only necessary to show that some corresponding parts are congruent.

• For example, suppose that in and in that

• Then intuition tells us that the remaining sides must be congruent, and…

• The triangles themselves must be congruent.

Proving Triangles Congruent

AB DEand AC DFand A D

A

B C

D

E F

The properties of congruent triangle

If we can show all 3 pairs of corr.sides are congruent, the triangles

have to be congruent.

Show 2 pairs of sides and the included angles are congruent

and the triangles have to be

congruent

Includedangle

Non-includedangles

THIS MEANS WE ARE GIVEN ALL THREE ANGLES OF A TRIANGLE, BUT NO SIDES.

AAA PROPERTY (ANGLE,ANGLE, ANGLE)

IN TWO TRIANGLES, IF ONE PAIR OF ANGLES ARE CONGRUENT, ANOTHER PAIR OF ANGLES ARE CONGRUENT, AND THE PAIR OF SIDES IN BETWEEN THE PAIRS OF CONGRUENT ANGLES ARE CONGRUENT, THEN THE TRIANGLES ARE CONGRUENT.FOR EXAMPLE, IN THE FIGURE, IF THE CORRESPONDING PARTS ARE CONGRUENT AS MARKED, THEN

WE CITE “ANGLE-SIDE-ANGLE (ASA)” AS THE REASON THE TRIANGLES ARE CONGRUENT.

ASA PROPERTY (ANGLE,SIDE, ANGLE)

A B

C

D E

F

AAS PROPERTY (ANGLE,ANGLE, SIDE)

In two triangles, if one pair of angles are congruent, another pair of

angles are congruent, and a pair of sides not between the two angles

are congruent, then the triangles are congruent.

For example, in the figure, if the corresponding parts are congruent

as marked, then

We cite “angle-angle-side (AAS)” as the reason the triangles are

congruent.

A B

C

D E

F

THANKS FOR YOUR

ATTENTION

THE END