SSS Congruence

Key Concepts

• Use the SSS congruence Postulate

Introduction

Side-Side-Side (SSS) Congruence Postulate: 

If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. 

Prove Triangles Congruent by SSS 

Example 1: Use the SSS congruence postulate to show that ∆ LON ≅ ∆ NML. 

Solution: 

From the given figure,

∴ By the definition of SSS congruency, ∆ LON ≅ ∆ NML.  

Example 2: Use the SSS congruence postulate to prove that ∆ KLM ≅ ∆ NLM. 

Solution: 

Given: KL ≅ NL, and KM ≅ NM 

TO Prove: ∆ KLM ≅ ∆ NLM 

Proof:  

Therefore, by the SSS congruence postulate,  

∆ KLM ≅ ∆ NLM. 

Example 3: Prove that ΔABC ≅ ΔFGH, using the below diagram. 

Solution: 

From the given diagram, in ∆ ABC, AB = 5 and in ∆ FGH, FG = 5. 

Therefore, AB ≅ FG 

Similarly, in ∆ ABC, AC = 3 and in ∆ FGH, FH = 3. 

Therefore, AC ≅ FH 

Now, find the lengths of BC and GH using the distance formula. 

Since, BC = GH = √34 

Then BC ≅ GH 

Therefore, ΔABC ≅ ΔFGH (By SSS congruence postulate) 

Hence proved. 

Example 4: Prove that the given two triangles are congruent.  

Solution: 

Given triangles are △ABC and △PQR. 

From the given triangles, we have 

Therefore, the given two triangles are congruent. 

5: From the given image, the first bench with diagonal support is stable. The second bench without support is unstable. Justify your answer. 

Solution:  

From the given image, the first bench with diagonal support has triangles with fixed side lengths. So, the triangles do not change the shape. Hence, by SSS congruent postulate, the bench is stable. 

While the bench without support is not stable because the table has many possible quadrilaterals. 

Example 6: Use the below image and find the stable gate. Explain your reason. 

Solution: From the given figure, the first gate with diagonal support has two triangles with fixed side lengths. So, the triangles do not change the shape. Hence, by SSS congruent postulate, gate 1 is stable.  

While the second gate without support is not stable because gate 2 has many possible quadrilaterals. 

Exercise

1. In a given quadrilateral LMNP, LM = LP and MN = NP. Prove that LN ⊥ MP and
   MO = OP.

2. If the opposite sides of a quadrilateral are equal, prove that the quadrilateral is a parallelogram.

3. By side-side-side congruence, prove that ‘diagonal of the rhombus bisects each other at right angles.’
4. Write the congruence statement for the given picture.

5. Prove that the given triangles are congruent.

6. Find the lengths of the line segments for the given figure and show that the two triangles are congruent.

7. Use the given coordinates to determine if ΔABC ≅ ΔDEF.
   • A(-2,-2),B(4,-2),C(4,6),D(5,7),E(5,1),F(13,1)

8. Find whether the below figure is stable or not.

9. In the given diagram, if ¯PK≅¯PLand ¯JK≅¯JL. Show that ΔJPK≅ΔJPL.

10. Prove that ΔGHJ≅ΔJKG, if ¯GH≅¯JKand ¯HJ≅¯KG.

What have we learned

• Understand the concept of congruency by SSS congruent postulate.
• Prove the theorem on side-side-side congruent postulate.
• Find the lengths of sides of the two triangles in the coordinate plane.
• Solve the real-world problems using SSS congruent postulate.
• Solve the problems using SSS congruent postulate.

Summary

Side-Side-Side (SSS) Congruence Postulate:
If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

If side ¯AB≅¯RS,
Side ¯BC≅¯ST, and
Side ¯CA≅¯TR,
then ΔABC≅ΔRST.

Practice problems 

Q1 ABC is a triangle in which AB = AC and is any point on the side AB. Through D, a line is drawn parallel to BC and intersecting AC at E. Prove that PD = PE. 

Q.2: If the angles formed by the perpendiculars from any point within the range are congruent. Establish its location on the angle’s bisector. 

FAQs 

1. What are Congruent Triangles? 

Ans) If the three sides and the three angles of both triangles are equal in any orientation, then two triangles are said to be congruent. 

2. What is SSS congruency of triangles? 

Ans) According to the SSS rule, two triangles are said to be congruent if all three sides of one triangle are equal to the corresponding three sides of the second triangle. 

3. What is triangular congruency with ASA? 

Ans) According to the ASA rule, two triangles are said to be congruent if any two angles and sides included between the angles of one triangle are equal to the corresponding two angles and sides included between the angles of the second triangle. 

4. RHS congruency: what is it? 

Ans) The two right triangles are said to be congruent by the RHS rule if the hypotenuse and a side of one right-angled triangle are equal to the hypotenuse and a side of the second right-angled triangle.