Presenter Name

Presentation Date

•Thales

•Pythagoras

•Triangle

•Triangle Properties

•Similarity And Congruence

•Key words

•Similar Figure

•Similarity Of Triangle

•BPT

•Converse Of BPT

•Criteria For Similarity Of Triangle

*Area Of Similar Triangle

*Pythagoras Theorem

*Converse Of Pythagoras Theorem

*The sum of the measure of the 3 angles of a triangle is 180 degrees.

*The sum of the lengths of any 2 sides of a triangle is greater than the length of the third side.

*In a triangle the line joining a vertex to the mid point of the opposite side is called a median. The three medians of a triangle are concurrent at a point called the "Centroid".

*The perpendicular from a vertex to the opposite side is called the "altitude".The three altitudes of a triangle are concurrent at a point called the "Orthocentre".

*The bisectors of the three angles of a triangle meet at a point called the "Incentre".

*The perpendicular bisectors of the three sides of a triangle are concurrent at a point called the "Circumcentre".

*If one side of a triangle is produced, the exterior angle so formed is equal to the sum of the interior opposite angles

1) Similarity:-

I. Two triangles are said to be similar if every angle of one

triangle has the same measure as the corresponding

angle in the other triangle.

II. The corresponding sides of similar triangles have lengths

that are in the same proportion, and this property is also

sufficient to establish similarity.

2) Congruence :-

I. Two triangles that are congruent have exactly the same

size and shape all pairs of corresponding interior angles

are equal in measure, and all pairs of corresponding

sides have the same length.

• Median of a triangle:- A median of a triangle is a line

joining a vertex to the mid point of the opposite side.

• Equiangular Triangles:- If corresponding angles of two

triangles are equal, then they are known as equiangular

triangles.

• Scale factor or Representative fraction:- The same ratio

of the corresponding sides of two polygons is known as

the scale factor or the representative fraction for the

polygons.

• Angle bisector of a triangle:- The angle bisector of a

triangle is a line segment that bisects one of the vertex

angle of a triangle.

• Altitude of a triangle:- The altitude of a triangle is a line

that extends from one vertex of a triangle and

perpendicular to the opposite side.

• Angle of elevation of the Sun:- The angle of elevation of

the Sun is the angle between the direction of the geometric

center of the sun’s apparent disc and the horizontal level.

Two polygons of the same number of sides

are similar, if

i) Their corresponding angles are equal,

and

ii) Their corresponding sides are in the

same ratios (or proportion).

:- If a line is drawn parallel to one side of

a triangle to intersect the other two sides

in distinct points, the other two sides are

divided in the same ratio.

To Proof:- AD / DB = AE / EC

Given:- In ∆ABC , DE || BC and intersects

AB in D and AC in E.

Construction :- Join BC,CD and draw

EF ┴ BA and DG ┴ CA.

Proof:-

Area (BDE) = (1/2) (BD) (EF)Area (ADE) = (1/2) (DA) (EF)Therefore Area (BDE) / Area (ADE) = BD / DA … (1)Area (CDE) = (1/2) (CE) (DG)

Area (ADE) = (1/2) (EA) (DG)Area (CDE) / Area (ADE) = CE / EA … (2)

But Area (CDE) = Area (BDE) since they are on the same base between the same parallels.

So (2) can be written as:Area (ADE) / Area (BDE) = EA / EC … (3)From (1) and (3)AD / DB = EA / ECTherefore DE divides AC and BC in the same ratio.

PROVED

i) If in two triangles, corresponding angles are

equal, then their corresponding sides are in the

same ratio (or proportion) and hence the two

triangles are similar.

Given: Triangles ABC and DEF such that A = D; B

= E; C = F

To Prove: Δ ABC ~ Δ DEF

Construction: We mark point P on the line DE and

Q on the line DF such that AB = DP and AC = DQ,

we join PQ.

Consequently, PQ || EF

DP/DE = DQ/DF (Corollary to basic proportionality

theorem)

i.e., AB/DE = BC/EF (construction) ---------- (1)

Similarly AB/DE = AC/DF --------------(2)

From (1) and (2) we get,

Since corresponding angles are equal, we conclude

that

Δ ABC ~ Δ DEF

ii) If in two triangles, sides of one triangle are proportional to

(i.e., in the same ratio of) the sides of the other triangle, then

their corresponding angles are equal and hence the two triangles

are similar.

Statements Reasons

1) AB = DP ; ∠A = ∠D and AC = DQ 1) Given and by construction

2) ΔABC ≅ ΔDPQ

2) By SAS

postulate……………………..

(1)

3)AB/DE = AC/DF 3) Given

4)DP/DE = DQ/DF 4) By substitution

5) PQ || EF5) By converse of basic

proportionality theorem

6) ∠DPQ = ∠E and ∠DQP = ∠F 6) Corresponding angles

7) ΔDPQ ~ ΔDEF7) By AAA similarity

…………………..(2)

8) ΔABC ~ ΔDEF 8) From (1) and (2)

iii) If one angle of a triangle is equal to one angle of

the other triangle and the sides including these

angles are proportional, then the two triangles are

similar.

Statements Reasons

1) AB/DE = AC/DF 1) Given …………………………(1)

2) DP/DE = DQ/DF 2) As AB = DP and AC = DQ.

3) PQ || EF 3) By converse of basic proportionality theorem

4) ∠DPQ = ∠E and ∠DQP = ∠F 4) Corresponding angles

5) ΔDPQ ~ ΔDEF 5) By AA similarity………………….. (2)

6) DP/DE = PQ/EF 6) By definition of similar triangles ………….(3)

7) AB/DF = PQ/EF 7) As DP = AB …………………………(4)

8) PQ/EF = BC/EF 8) { From (1) (3) and (4)} ………(5)

9) PQ = BC 9) From (5)

10) ΔABC ≅ ΔDPQ 10) By S-S-S postulate…………..(6)

11) ΔABC ~ ΔDEF 11) From (2) and (6)

½ (BC × AM)/½( QR × PN)

:- If a perpendicular is drawn from

the vertex of the right angle of a right

triangle to the hypotenuse then

triangles on both sides of the

perpendicular are similar to the whole

triangle and to each other.