What are Congruent Triangles?

A polygon is generally made of three line segments that form three angles known as a Triangle.

Two triangles are known to be congruent triangles if their sides have equal length and the angles in the triangle have the same measure. Therefore, any two triangles can be superimposed side to side and also angle to angle.

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In the figure given below, Δ ABC and Δ PQR are known to be congruent triangles. This means that ,

Vertices: A and P, B and Q, and C and R are the same.

Sides: AB is equal to PQ, QR is equal to BC and AC is equal to PR;

Angles: ∠A equals ∠P, ∠B equals ∠Q, and ∠C equals ∠R.

Congruent triangles are known to be the triangles that have corresponding sides and angles are known to be equal. Congruence is basically denoted by the symbol ≅. They have the same area and have the same perimeter.

What are the Rules of Congruency?

There are four main rules of congruence for triangles:

SSS Criterion: Side-Side-Side -Two triangles are known to be congruent if all the sides of any given triangle are equal in measure to all the corresponding sides of the other triangle.

SAS Criterion: Side-Angle-Side-Two triangles are known to be congruent if two sides and the included angle of one of the triangles are equal to the two sides and the included angle of the other triangle.

ASA Criterion: Angle-Side- Angle - Two triangles are known to be congruent if two angles and the included side of one of the triangles are equal to two angles and the included side of another triangle.

RHS Criterion: Right angle- Hypotenuse-Side

In this article we are going to discuss the SSS congruence & constructing triangles with sss congruence.

SSS Congruence Rule: If three sides of 1 triangle are similar to the corresponding sides of another triangle, then the triangles are known to be congruent. Constructing triangles with sss congruence criteria is possible when all the three sides are known to us. The necessities of constructing triangles with sss congruence are basically a ruler and a compass. Side-Side-Side is one among the properties of similar triangles.

How to Construct a Triangle with the Given Three Sides?

By the SSS(Side,Side,Side) rule, construction of a triangle is easily possible with three given side measures. For the construction of a triangle, you need to first identify the longest measure among the three side measures. Now, draw the longest side measure because of the base of the triangle, then take other measurements using a ruler to mark the arcs by taking the endpoints of the bottom as vertices. Finally, now you need to join the intersection of arcs with the endpoints of the base to get the specified triangle

Now, you may run into a "trick" question where the given segments will NOT form a triangle. You need to keep in mind that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side of the triangle. If this relationship does not occur, then you will NOT be able to draw a triangle.

Therefore this is how to construct a triangle with the given three sides.

Constructing SSS Triangles

Let us consider a triangle namely ABC, having the measurement of sides equal:

Side AB = 7 cm, Side BC = 4 cm and Side CA = 6 cm. Now the steps for construction of triangle are:

Step 1: First mark a point namely A

Step 2: Now you need to measure a length of 7 cm using compass and a ruler

Step 3: With the help of a compass and then mark an arc placing pointer at a point namely A

Step 4: Mark a point named B on the arc

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Step 5: Now measure the length of six(6) cm

Step 6: Now again using compass mark an arc above the point B using the same point namely (A)

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Step 7: Measure a length equal to 4 cm

Step 8: Now using the compass placed at point namely B cut an arc such that it crosses the previous arc.

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Step 9: Now the name the point as C, which is the point where the two arcs cross each other

Step 10: At the end,you need to join the points A, B and C with the help of a ruler to get the required triangle.

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Thus, the obtained triangle given above is the required triangle ABC with the given measurements.

Questions to be Solved :

Question 1) List down the steps for constructing sss triangles.

Solution) Constructing SSS Triangles

Let us consider a triangle namely ABC, having the measurement of sides equal:

Side AB = 7 cm, Side BC = 4 cm and Side CA = 6 cm. Now the steps for constructing triangles sss are:

Step 1: First mark a point namely A

Step 2: Now you need to measure a length of 7 cm using compass and a ruler

Step 3: With the help of a compass and then mark an arc placing pointer at a point namely A

Step 4: Mark a point named B on the arc

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Step 5: Now measure the length of six (6) cm

Step 6: Now again using compass mark an arc above the point B using the same point namely (A)

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Step 7: Measure a length equal to 4 cm

Step 8: Now using the compass placed at point namely B cut an arc such that it crosses the previous arc.

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Step 9: Now the name the point as C, which is the point where the two arcs cross each other

Step 10: At the end,you need to join the points A, B and C with the help of a ruler to get the required triangle.

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Thus, the obtained triangle shown above is the required triangle ABC with the given measurements.

FAQ (Frequently Asked Questions)

Question 1. What is sss Triangle Construction Condition?

Answer. SSS - Side, Side, Side

Remember for constructing triangles sss, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this relationship doesn't occur, you'll NOT be ready to draw a triangle. This sss triangle construction is going to be to "copy a segment" 3 times.

Question 2. What is SSS Triangle Congruence?

Answer. The term SSS means "side, side, side" and it basically denotes that we've two triangles with all three sides equal. Let’s take for example: is congruent to: (See Solving sss triangle construction to seek out more) If three sides of 1 triangle are adequate to three sides of another triangle, the triangles are congruent.

Question 3. What is Triangle Construction?

Answer. This is how to construct a triangle with the given three sides (Constructing Triangles):- If the measurements of three sides, the measurements of two sides and therefore the angle between them, or the measurements of two angles and therefore the side between them are given, then a triangle is often constructed by using a ruler, protractor and compass.

Question 4. What is a SAS Triangle?

Answer. "SAS" is once we know two sides and therefore the angle between them. to unravel an SAS triangle. use The Law of Cosines to calculate the unknown side, then use The Law of Sines to seek out the smaller of the opposite two angles, then use the three angles increase 180° to seek out the last angle