The document discusses properties of triangles. It defines a triangle as a closed figure formed by three intersecting lines with three sides, three angles, and three vertices. Two triangles are congruent if corresponding sides and angles are equal. The five criteria for congruence are stated as SAS, ASA, AAS, SSS, and RHS. Properties of triangles discussed include corresponding parts of congruent triangles, angles opposite equal sides of an isosceles triangle being equal, and sides opposite equal angles being equal. Inequalities in a triangle are that the angle opposite the longer side is larger, and the side opposite the longer angle is longer. The sum of any two sides of a triangle is greater than the third side.

1. CLASS – XTH
TOPIC - TRINAGLES
Name:- Sandeep Jaiswal
11813483
Integrated B.Sc. B.Ed.

2. A
B C
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
TriangleABC, denoted as ∆ABC AB,BC,CAare the three sides, ∠A,∠B,∠C are
three angles and A,B,C are three vertices.

3. 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

4. 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

5. THISALSO 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

6. 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.

7. 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)

8. 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)

9. 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)

10. 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)

11. 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)

12. 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.

13. ANGLES OPPOSITE TO EQUAL SIDES OF AN ISOSCELES
TRIANGLE ARE EQUAL.
B C
A
Here, ∠ABC = ∠ ACB

14. THE SIDES OPPOSITE TO EQUAL ANGLES OF A
TRIANGLE ARE EQUAL.
C
B
A
Here, AB = AC

15. 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°)

16. 10
9
8
Here, bycomparing we will get that-
Side(i.e. 10) opposite to longerangle (90°) is longer.
In any triangle, the side opposite to
the longer angle is longer.

17. 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.
The sum of any two side of a triangle is
greater than the third side.