SSS Congruence Rule Definition Construction Steps and Solved Examples
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.
[Image will be Uploaded Soon]
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
[Image will be Uploaded Soon]
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)
[Image will be Uploaded Soon]
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.
[Image will be Uploaded Soon]
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.
[Image will be Uploaded Soon]
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
[Image will be Uploaded Soon]
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)
[Image will be Uploaded Soon]
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.
[Image will be Uploaded Soon]
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.
[Image will be Uploaded Soon]
Thus, the obtained triangle shown above is the required triangle ABC with the given measurements.
FAQs on How to Construct SSS Congruent Triangles in Geometry
1. What is SSS congruence in triangles?
SSS congruence states that two triangles are congruent if all three corresponding sides are equal in length. This means:
- If side A = side A′
- Side B = side B′
- Side C = side C′
2. How do you construct a triangle using SSS congruence?
To construct a triangle using SSS congruence, draw three sides with the given lengths using a ruler and compass.
- Step 1: Draw a base side of given length.
- Step 2: From one endpoint, draw an arc with radius equal to the second side.
- Step 3: From the other endpoint, draw an arc with radius equal to the third side.
- Step 4: Join the intersection point of the arcs to both endpoints.
3. Why does SSS prove triangle congruence?
SSS proves triangle congruence because fixing all three side lengths determines a unique triangle. When three sides are fixed, the angles are automatically determined, leaving no variation in shape. Therefore, triangles with equal three corresponding sides must be congruent.
4. What tools are needed to construct SSS congruent triangles?
To construct SSS congruent triangles, you need basic geometric tools.
- Ruler – to measure and draw line segments
- Compass – to draw arcs of given radius
- Pencil
5. Can you give an example of constructing a triangle with sides 5 cm, 6 cm, and 7 cm?
Yes, a triangle with sides 5 cm, 6 cm, and 7 cm can be constructed using the SSS method.
- Draw a base of 7 cm.
- From one endpoint, draw an arc of radius 5 cm.
- From the other endpoint, draw an arc of radius 6 cm.
- Mark their intersection and join it to both ends of the base.
6. What is the difference between SSS and SAS congruence?
The difference is that SSS uses three sides, while SAS (Side–Angle–Side) uses two sides and the included angle to prove congruence.
- SSS: All three sides are equal.
- SAS: Two sides and the angle between them are equal.
7. Is SSS enough to prove two triangles are similar?
SSS proves similarity only when the three corresponding sides are proportional, not necessarily equal. For SSS similarity:
- a₁/a₂ = b₁/b₂ = c₁/c₂
8. What conditions must be satisfied to construct a triangle using SSS?
To construct a triangle using SSS, the given sides must satisfy the Triangle Inequality Theorem. This means:
- Sum of any two sides > third side
9. What are common mistakes when constructing SSS triangles?
Common mistakes in constructing SSS congruent triangles include:
- Incorrect compass width measurement
- Not checking the triangle inequality condition
- Misplacing the arc intersection point
10. How do you prove two triangles are congruent using SSS in a geometry proof?
To prove two triangles are congruent using SSS, show that all three pairs of corresponding sides are equal. In a proof:
- State the three given equal sides.
- Match corresponding sides correctly.
- Conclude: Triangles are congruent by SSS Congruence Rule.
