Triangles Class 9: Complete Guide with Theorems and Examples

Triangle is a polygon or shape that has three sides that meet at a point. A triangle has three angles and three points. There are different types of triangles and each one has different characteristics and meaning.

The first type of triangle we will discuss is the right triangle. A right triangle is a triangle with a right angle. A right angle is an angle that measures exactly 90 degrees. One of the most common right triangles is a 45-45-90 triangle. This triangle is named as such as one of the angles measures 45 degrees and the other two measure 45 degrees as well.

Another type of triangle is the obtuse triangle, meaning that one of the angles is greater than 90 degrees. An example of an obtuse triangle is one that is shaped like the letter L. An isosceles triangle is a triangle with at least two equal sides and equal angles. An equilateral triangle has three equal sides and three equal angles and is the only triangle where the angles and sides all measure the same.

Importance of Studying Triangles

Triangles are important to study because they can be used to study certain formulas that measure the areas and the lengths of the sides of a triangle. Triangles can also be used to measure the area of a quadrilateral. Take a look at a few of the many ways to study triangles.

The following methods to study triangles are efficacious for finding the basics of triangles and how to measure the different parts of the triangle.

The first way of studying triangles is to read about them. Reading about triangles can show you the definitions of triangles and how they work, which is the most important thing to know.

The second way is to use visual aids. These can include drawings or actual triangles made out of foam board or rubber matting.

The third way of studying triangles is to build a triangle and measure the different parts such as the height, base, and the angle which forms the side opposite of the base. This will be helpful to the student because they can visually see how the parts work and what part is opposite of each.

The fourth way is to build a triangle- It is important to build a triangle out of a rubber band so you can see the shape and measure the dimensions.

The last way is to Learn the rules- It is important to learn the general rule that when studying a triangle that the area of the triangle is equal to one half the product of the base and height, and when measuring a quadrilateral, the area is equal to one half the product of the base and the height.

Maths Class 9 Triangles

• 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 the different properties of triangles. Now, let us learn different types of triangles, the congruence of triangles, and the inequality relations of triangles.

• Triangles class 9 notes will help you to solve problems related to triangles easily.

Congruent Triangles

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  

and

AB = DE ,

BC = EF ,

AC = DF.

Note:

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 

Criteria for the congruence of triangles are well defined and proved. Congruent parts of the congruent triangle are written as c.p.c.t. Different rules for congruence are as given below:

SAS (Side-Angle-Side) Congruence Rule

Suppose 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. In that case, the two triangles are congruent to each other.

In the above figure  AB = PQ ; 

AC = PR; 

And the angles 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

Suppose two angles and the side included by the two angles of one triangle are equal to the corresponding angles and sides included by the angles of the other triangles. In that case, 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 another triangle then the two triangles are said to be congruent to each other.

In the above figure  ∠ B = ∠ E 

∠ C = ∠ F 

and

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 the 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)

Inequality Relations In a Triangle

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

Solved Examples

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.

Triangle Questions for Class 9

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.