Constructing Triangles Through SSS Congruence

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Constructing triangles through SSS congruence requires an understanding of congruent triangles and the SSS congruence rule. SSS stands for Side-Side-Side. Congruent triangles are triangles having the same shape and same size. Both corresponding sides and angles of the triangles should be equal through this application.

Table of Contents

Congruent triangles
Rules of congruency
SSS Criterion
Construction of 3 sided triangle
Things to Remember
Sample Questions

Key Terms: Congruent triangles, SSS Congruence rule, Construct, Three sides, Angles, Triangle

Congruent Triangles

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A triangle is a polygon having three line segments joined together and forming three angles. If two triangles are congruent, it means that all the sides and their corresponding angles are congruent. Thus, the sides have the equal length or the angles have the same measure.

congruent triangle

Congruent Triangles

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Area of a Triangle

Congruence of Triangles

Pascal’s Triangle

Rules of Congruence

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There are four rules of congruency of triangles:

SSS Criterion: Side-Side-Side; Three sides

SAS Criterion: Side-Angle-Side; Two sides and the angle between them.

ASA Criterion: Angle-Side-Angle; Two angles and the side between them.

RHS Criterion: Right Angle-Hypotenuse; Side hypotenuse and a leg in the case of a right-angled triangle.

Rules of Congruent Triangles

Rules of Congruent Triangles

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Triangles

Triangle Important Question

Similarities of Triangles

SSS Criterion

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Theorem: Two triangles are congruent if the three sides of one triangle are equal to the corresponding three sides of the other triangle.

SSS Congruency Theorem

SSS Congruency Theorem

We construct SSS congruent triangles when measures of all three sides are given. The materials required are a ruler, a pencil and a compass

Read more: Pythagoras Theorem

Construction of Three-sided Triangle

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Important properties of triangles to keep in mind before constructing Congruent Triangles:

The exterior angle of a triangle is equal in measure to the sum of interior opposite angles.
The total measure of the three angles of a triangle is 180°.
Sum of the lengths of any two sides of a triangle is greater than the length of the third side.
In any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

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Trigonometry

Applications of Trigonometry

Heights and Distances

Construction

Example: Construct a triangle ABC, given that AB=6 cm, BC=7 cm and AC=8 cm.

The steps for constructing triangles are:

Step 1: Draw a line segment BC length 7 cm.

Step 2: Point A lies above point b at a distance of 6 cm. Take the compass and place it on point B, as its centre. Draw the arc of 6 cm above point B. Point A lies somewhere on it.

Step 3: Now with point C as the centre, make the arc at the radius of 8 cm. This arc should intersect the previous arc.

step 3

Step 4: The intersection point between both the arcs is Point A. Mark it as A.

step 4

Step 5: Join AB and AC. This is the required triangle ABC.

step 5

Now construct another triangle PQR such that PQ = 6 cm, BC= 7 cm and AC= 8 cm in the same way shown above. Take the cut out of this triangle and place it on Triangle ABC. We observe that both the triangles coincide. They were constructed when all the sides were given.

Here, the three sides of a triangle are equal to the three sides of another triangle. Therefore, both the triangles are congruent.

Note: For constructing SSS criterion triangles, the sum of the length of any two sides must be greater than the length of the third side.

Read more:

Circle important questions

Tangent to a circle

Construction formula

Things to Remember

Congruent triangles are the triangles in which all sides and their corresponding angles are congruent.
There are four criteria to prove whether the triangles are congruent.
SSS Criterion states that two triangles are congruent if three sides of one triangle are equal to the three sides of another triangle.
Triangles can be constructed through a ruler and a compass if they follow any congruent criteria.

Read more:

Arithmetic Progressions

Quadratic Equations

Nature of roots of quadratic equations

Sample Questions

Ques: Are the following two triangles congruent? Prove with the help of a criterion. (2 marks)

Ans: In the

∆ABC and ∆PQR,

AC = PR

AB = RQ

BC = PQ

Hence following the SSS criterion, both the triangles are congruent.

two triangles congruent

Ques: Construct two equilateral triangles sides, 5 cm and 6 cm. Are these two congruent? Why or why not? (2 marks)

Ans: The triangles,

∆ABC and ∆PQR, are not congruent because they do not fulfil the SSS Criterion. The equilateral triangle has equal sides, but to fulfil this criterion that the corresponding sides should be equal.

equilateral triangles

Ques: In the adjoining figure, apply the SSS congruence condition and state the result in the symbolic form. (2 marks)

Ans: In

∆LNM and ∆OQP

SSS congruence

LN = OQ = 3 cm

NM = PQ = 5cm

LM = PO = 8.5cm

Therefore,

∆LNM ≅ ∆OQP, by Side Side Side (SSS) congruence condition

Ques: Draw ? XYZ in which XY = 5.5 cm, YZ = 4.5 cm and XZ = 5.5 cm. What type of triangle is this? (2 marks)

Ans:

isosceles triangle

This is an isosceles triangle where XY= XZ = 5.5 cm.

Ques: Construct PQR in which PQ = 2.5 cm, QR = 6 cm and PR = 6.5 cm. Measure ∠Q. (2 marks)

Ans: ∠Q = 90°, hence

∆PQR is a right-angled triangle.

right-angled triangle.

Ques: By Side Side Side congruence prove that 'Diagonal of the rhombus bisects each other at right angles’. (5 marks)

Ans: Diagonal BD and AC of the rhombus ABCD intersect each other at E.

rhombus

It is required to prove that CD ⊥ AB and DE = EB and CE = EA.

Proof: ABCD is a rhombus.

Therefore, ABCD is a parallelogram.

Therefore, DE = EB and CE = EA.

In AED and CED; AD = CD, [Since, sides of a rhombus are equal] Side ED is common

AE = EC, [Since diagonal of a parallelogram bisects each other]

Therefore, AED ≅ CED, [by SSS congruence condition]

But, ∠AED + ∠CED = 2 right angle

Therefore, 2∠AED = 2 right angle

or, ∠AED = 1 right. angle

Therefore, CD ⊥ AB

i.e., CD ⊥ AB (Proved)

[Since diagonals of a square are perpendicular to each other]

Ques: If the opposite sides of a quadrilateral are equal, prove that the quadrilateral will be a parallelogram. (5 marks)

Ans: PQRS is a parallelogram quadrilateral, whose sides PQ = SR and PS = QR. It is required to prove that LMNO is a parallelogram.

parallelogram

Construction: Diagonal PR is drawn.

Proof: In PSR and PQR,

PQ = SR and PS = QR, [By hypothesis] PR is the common side.

Therefore, PSR ≅ PQR, [by Side Side Side congruence condition]

Therefore, ∠PRS = ∠QPR, [Corresponding angles of congruent triangles] Since PR cuts SR and PQ and both alternate angles are equal.

Therefore, SR || PQ

Again, ∠PRQ = ∠SPR [Corresponding angles of congruent triangles] But PR cuts PS and QR, and the alternate angles are equal.

Therefore, PS || QR

Therefore, In quadrilateral PQRS,

SR || PQ and

PS || QR.

Therefore, PQRS is a parallelogram. [Proved]

Read More:

Congruence of Triangles	Congruence Of Triangles	Triangles
Important Terms, Scalene, Acute, and Obtuse Triangle	Class 10 Mathematics Chapter 6 Triangles	Isosceles Triangle Theorems
Isosceles triangle	Triangles Important Questions	Similarity of Triangles
Properties of Triangle	Triangle Theorems	Scalene Triangle
NCERT Solutions for Class 7 Mathematics Chapter 5	Equilateral Triangle	Quadrilateral Formulas

Constructing Triangles Through SSS Congruence

Congruent Triangles

Rules of Congruence

SSS Criterion

Construction of Three-sided Triangle

Construction

Things to Remember

Sample Questions

CBSE X Related Questions

1.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

2.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

3.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

4.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

5.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

6.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

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Constructing Triangles Through SSS Congruence

Congruent Triangles

Rules of Congruence

SSS Criterion

Construction of Three-sided Triangle

Construction

Things to Remember

Sample Questions

CBSE X Related Questions

1.An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

2.PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

3.Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

4.A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

5.Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

6.The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

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1.
An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

2.
PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

3.
Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \tan \theta + \cot \theta$

4.
A bag contains 25 balls. Some of them are yellow and others are green. One ball is drawn at random. If probability of getting a green ball is $3/5$, then find the number of yellow balls.

5.
Two water taps together can fill a tank in $8\frac{8}{9}$ hours. The tap of larger diameter takes 4 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

6.
The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.