1. A triangle is a closed figure formed by three intersecting lines, with three sides, three angles, and three vertices. 2. There are five criteria to determine if two triangles are congruent: side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), side-side-side (SSS), and right-angle-hypotenuse-side (RHS). 3. Key properties of triangles include: angles opposite equal sides are equal; sides opposite equal angles are equal; the sum of any two sides is greater than the third side.

1. INTRODUCTION TO TRIANGLES
Prepared By:
Sharda Chauhan
TGT Mathematics
Class 9

2. We know that a closed figure formed by three intersecting
lines is called a triangle(‘Tri’ means ‘three’).A triangle has
three sides, three angles and three vertices. For e.g.-in
Triangle ABC, denoted as ∆ABC AB,BC,CA are the three
sides, ∠A,∠B,∠C are three angles and A,B,C are three
vertices.
A
B C
INTRODUCTION

3. OBJECTIVES
• INEQUALITIES IN A TRIANGLE.
2
• STATE THE CRITERIA FOR THE CONGRUENCE OF TWO
TRIANGLES.
3
• SOME PROPERTIES OF A TRIANGLE.
4
•DEFINE THE CONGRUENCE OF TRIANGLE.
1

4. DEFINING THE CONGRUENCE OF TRIANGLE:-
Let us take ∆ABC and ∆XYZ such that
corresponding angles are equal and
corresponding sides areequal :-
A
B C
X
Y Z
CORRESPONDING PARTS
∠A=∠X
∠B=∠Y
∠C=∠Z
AB=XY
BC=YZ
AC=XZ

5. Now we see that sides of ∆ABC coincides with sides of
∆XYZ.
B C
A X
Y Z
SO WE GET THAT
TWO TRIANGLES ARE CONGRUENT, IF ALL THE
SIDES AND ALL THE ANGLES OF ONE TRIANGLE
ARE EQUAL TO THE CORRESPONDING SIDES AND
ANGLES OF THE OTHER TRIANGLE.
Here, ∆ABC ≅ ∆XYZ

6. This also means that:-
A corresponds to X
B corresponds to Y
C corresponds to Z
Forany twocongruent triangles the corresponding
parts areequal and are termed as:-
CPCT – Corresponding Parts of Congruent Triangles

7. CRITERIAS FOR CONGRUENCE OF TWO TRIANGLES
SAS(side-angle-side) congruence
• Twotrianglesarecongruent if twosidesand the included angleof one triangleare
equal to the twosidesand the included angleof othertriangle.
ASA(angle-side-angle) congruence
• Twotrianglesarecongruent if twoanglesand the included sideof one triangleare
equal to twoanglesand the included sideof othertriangle.
AAS(angle-angle-side) congruence
• Twotrianglesarecongruent if any two pairs of angleand one pairof corresponding
sidesareequal.
SSS(side-side-side) congruence
• If three sidesof one triangleareequal to the three sidesof another triangle, then the
two trianglesarecongruent.
RHS(right angle-hypotenuse-side) congruence
• If in two right-angled triangles the hypotenuseand onesideof one triangleareequal to
the hypotenuseand onesideof theother triangle, then the two trianglesarecongruent.

8. A
B C
P
Q R
S(1) AC = PQ
A(2) ∠C = ∠R
S(3) BC = QR
Now If,
Then ∆ABC ≅ ∆PQR (by SAS congruence)

23. A
B C
D
E F
Now If, A(1) ∠BAC = ∠EDF
S(2) AC = DF
A(3) ∠ACB = ∠DFE
Then ∆ABC ≅ ∆DEF (by ASA congruence)

24. A
B C P
Q
R
Now If, A(1) ∠BAC = ∠QPR
A(2) ∠CBA = ∠RQP
S(3) BC = QR
Then ∆ABC ≅ ∆PQR (by AAS congruence)

25. Now If, S(1) AB = PQ
S(2) BC = QR
S(3) CA = RP
A
B C
P
Q R
Then ∆ABC ≅ ∆PQR (by SSS congruence)

26. Now If, R(1) ∠ABC = ∠DEF = 90°
H(2) AC = DF
S(3) BC = EF
A
B C
D
E F
Then ∆ABC ≅ ∆DEF (by RHS congruence)

27. PROPERTIES OF TRIANGLE
A
B C
A Triangle in which two sides are equal in length is called
ISOSCELES TRIANGLE. So, ∆ABC isa isosceles trianglewith
AB = BC.

28. Angles opposite to equal sides of an isosceles triangle are equal.
B C
A
Here, ∠ABC = ∠ ACB

29. The sides opposite to equal angles of a triangle are equal.
C
B
A
Here, AB = AC

30. Theorem on inequalities in a triangle
If two sides of a triangleare unequal, theangleopposite to the longer side is
larger( orgreater)
10
9
8
Here, bycomparing wewill get that-
Angleopposite to the longer side(10) is greater(i.e. 90°)
INEQUALITIES IN A TRIANGLE

31. In any triangle, the side opposite to the longer angle is longer.
10
9
8
Here, bycomparing we will get that-
Side(i.e. 10) opposite to longerangle (90°) is longer.

32. The sum of any two side of a triangle is greater than the third
side.
10
9
8
Here bycomparing weget-
9+8>10
8+10>9
10+9>8
So, sum of any two sides isgreater than the third side.

33. SUMMARY
1.Twofiguresarecongruent, if theyareof the same shape and size.
2.If twosides and the included angleof one triangle is equal to the twosides and
the included angle then the two trianglesarecongruent(by SAS).
3.If two angles and the included side of one triangle are equal to the two angles
and the included side of othertriangle then the two trianglesarecongruent( by
ASA).
4.If two angles and the one side of one triangle is equal to the two angles and the
corresponding side of othertriangle then the two trianglesarecongruent(byAAS).
5.If three sides of a triangle is equal to the three sides of other triangle then the
twotrianglesarecongruent(by SSS).
6.If in tworight-angled triangle, hypotenuseoneside of the triangleareequal to
the hypotenuseand one side of the other triangle then the two triangle are
congruent.(byRHS)
7.Angles opposite to equal sides of a triangle are equal.
8.Sides opposite to equal angles of a triangle are equal.
9.Eachangleof equilateral triangleare 60°
10.Ina triangle, anglesopposite tothe longerside is larger
11.Ina triangle, sideopposite to the largerangle is longer.
12.Sum of any twosides of triangle is greaterthan the third side.

34. THANKYOU!
ANY QUESTIONS?