1) The document discusses properties and theorems related to triangles, including similarity, congruence, medians, altitudes, angle bisectors, and the Pythagorean theorem. 2) It provides criteria for establishing similarity between triangles, including having equal corresponding angles or proportional corresponding sides. 3) Theorems are presented on parallel lines cutting across two sides of a triangle in the same ratio, and on perpendiculars drawn from right triangle vertices cutting the hypotenuse.

1. Presenter Name
Presentation Date

2. •Thales
•Pythagoras
•Triangle
•Triangle Properties
• Similarity And Congruence
• Key words
• Similar Figure
• Similarity Of Triangle
•BPT
•Converse Of BPT
•Criteria For Similarity Of Triangle

3. *Area Of Similar Triangle
*Pythagoras Theorem
*Converse Of Pythagoras Theorem

11. *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

12. 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.

13. • 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.

14. • 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.

15. 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).

17. :- 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.

18. To Proof:- AD / DB = AE / EC
Given:- In ΔABC , 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.

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

22. 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.

24. 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

25. 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.

26. 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 || EF
5) By converse of basic
proportionality theorem
6) ∠DPQ = ∠E and ∠DQP = ∠F 6) Corresponding angles
7) ΔDPQ ~ ΔDEF
7) By AAA similarity
…………………..(2)
8) ΔABC ~ ΔDEF 8) From (1) and (2)

27. 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.

28. 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)

29. 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)

31. ½ (BC × AM)/½( QR × PN)

33. :- 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.