We have learned different types of 2-dimensional figures like square, rectangle, triangle, circle, etc. and their various properties. In triangles class 9 notes we will be discussing the figure triangle in detail.
A figure formed by the intersection of three lines is said to be a triangle. A triangle has three vertices, three sides, and three angles.
The above figure shows ABC, here AB, BC, AC, are the sides of the triangle. A, B, C are the vertex and ∠ A, ∠B, ∠ C are the three angles.
We have studied different properties of triangles. Now, let us learn different types of triangles, congruence of triangles, and inequality relations of triangles.
Triangles class 9 notes will help you to solve problems related to triangles easily.
Two geometrical figures having exactly the same shape and size are said to be congruent figures.
Two triangles are congruent to each other if one of them is superimposed on another such that they both cover each other completely.
Two triangles are said to be congruent if the sides and angles of one triangle are exactly equal to the corresponding sides and angles of the other triangle.
From the figure, ABC is congruent to DEF and it is written as ABC ≅ DEF
In two congruent triangles, corresponding parts of corresponding angles generally written as ‘c.p.c.t’ are equal they are:
∠ A = ∠ D,
∠ B = ∠ E,
∠ C = ∠ F
AB = DE ,
BC = EF ,
AC = DF.
Every triangle is congruent to itself i.e ABC ≅ ABC
If ABC ≅ DEF then DEF ≅ ABC
If ABC ≅ DEF and DEF ≅ PQR then ABC ≅ PQR
Criteria for congruence of triangles are well defined and proved. Congruent parts of the congruent triangle is written as c.p.c.t. Different rules for congruence are as given below:
SAS( Side-Angle-Side) congruence rule
If two sides and the angle included between the two sides of one triangle are equal to the corresponding sides and the included angle of the other triangle then the two triangles are congruent to each other.
In the above figure AB = PQ ;
AC = PR;
And the angle between the sides are equal
I.e ∠ A = ∠ P
therefore ABC ≅ PQR …….by SAS criteria
also,∠ B = ∠ Q; ∠ C = ∠ R; BC = QR (by c.p.c.t)
ASA ( Angle-Side-Angle) congruence rule
If two angles and the side included by the two angles of one triangle are equal to the corresponding angles and side included by the angles of other triangle then the two triangles are said to be congruent.
In the above figure ∠ B = ∠ Q
∠ C = ∠ R
And the side between the angles are equal
BC = QR;
therefore ABC ≅ PQR …….by ASA criteria
also,∠ A = ∠ P; AB = PQ ; AC = PR (by c.p.c.t)
AAS (Angle-Angle-Side) congruence rule
If two angles and one non-included side of one triangle are equal to corresponding angles and a non-included side of other triangle then the two triangles are said to be congruent to each other.
In the above figure ∠ B = ∠ E
∠ C = ∠ F
AC = DF;
Then ABC ≅ DEF …….by AAS criteria
also,∠ A = ∠ D; AB = PQ ; BC = QR (by c.p.c.t)
SSS ( Side-Side-Side) congruence rule
If all the three sides of a triangle are equal to the corresponding sides of another triangle then the two triangles are said to be congruent to each other.
In the above figure AB = PQ ;
BC = QR; and
AC = PR,
therefore ABC ≅ PQR …….by SSS criteria
Also,∠ A = ∠ D; ∠ B = ∠ E; ∠ C = ∠ F (by c.p.c.t)
RHS( Right angle-Hypotenuse-Side) congruence rule
Two right-angled triangles are congruent if one side and the hypotenuse of the one triangle are equal to the corresponding side and the hypotenuse of the other.
In above figure, hypotenuse XZ = RT
And side YZ=ST,
∠ XYZ = ∠ RST ( angle are of 900)
therefore XYZ ≅ RST …….by RHS criteria
Also,∠ X = ∠ R; ∠ Z = ∠ T; XY = RS (by c.p.c.t)
If two sides of a triangle are unequal, the longer side has a greater angle opposite to it, here if in ABC, BC > AB. then ∠ CAB > ∠ ACB
The Triangle Inequality theorem states that
The sum of the lengths of any two sides of a triangle is greater than the length of the third side of a triangle.
Let us see some triangle questions for class 9
Example 1: In the below figure AD = BC and ∠DAB = ∠ CBA.Prove that AC = BD and ∠BAC = ∠ABD
Solution: In DAB and CBA
AD = BC….. (Given)
∠DAB = ∠ CBA…. (Given)
AB =AB….. (common side)
So by SAS criteria of congruence we get,
DAB ≅ CBA
So by corresponding parts of congruent triangle
AC = BD and ∠BAC = ∠ABD
Example 2: In the right triangle ABC, the right angle at C, M is the midpoint of hypotenuse AB. Join CM and produce to a point D such that DM = CM. Point D is joined to point B(see figure). Show that:AMC ≅ BMD
Solution: In AMC and BMD
AM = MB ( M is the midpoint of AB given)
∠DMB = ∠ CMA ( vertically opposite angles)
CM = MD ( given )
therefore , by SAS criteria , we get
AMC ≅ BMD hence proved
Solve more problem triangles lessons for class 9 and be an expert at solving triangles problems.
In the given figure, ΔABD and ΔACE are equilateral triangles that are drawn on the sides of a ΔABC. Prove that CD = BE.
In the given figure, side AB = AD, side AC = AE and ∠BAD = ∠EAC, then prove that BC = DE.
1. What are Congruent Triangles?
Congruent means exactly the same replica of one another. Congruent means equal in all respects or figures whose shapes and sizes are both the same.
Two triangles are congruent to each other if one is superimposed on the other triangle it exactly covers one another.
If all three sides and all the three angles of one triangle are equal to corresponding sides and angles of another triangle, then the two triangles are congruent to each other. Congruence of two triangles is represented by the symbol ≅.
2. How to find if the triangles are congruent or not?
Two triangles are said to be congruent if all its sides and angles are equal. But it is not necessary that every time we know all the three sides and all the three angles. Hence we can show two triangles are congruent by different congruence rules. The five congruence rules are:
SSS( side-side-side) congruence rule
SAS ( side-angle-side) congruence rule
ASA ( angle-side-angle) congruence rule
AAS ( angle-angle-side) congruence rule
RHS ( right angle- hypotenuse-side) congruence rule.
From these rules we can show that the two triangles are congruent or not.