SAS Congruence Rule
Two triangles are congruent if they have exactly the same shape and size — all corresponding sides and angles are equal. But you do not need to check all six measurements. Certain combinations of three measurements are enough.
The SAS (Side-Angle-Side) Congruence Rule states that if two sides and the included angle (the angle between those two sides) of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent.
SAS is one of the four main congruence criteria: SSS, SAS, ASA, and RHS.
What is SAS Congruence Rule - Grade 7 Maths (Congruence of Triangles)?
SAS Congruence Rule:
- If in two triangles, two sides and the included angle of one are equal to the corresponding two sides and included angle of the other, then the triangles are congruent.
Important: The angle MUST be the included angle — the angle formed between the two given sides. If the angle is not between the two sides, SAS does not apply.
SAS Congruence Rule Formula
If AB = PQ, ∠B = ∠Q, and BC = QR
Then △ABC ≅ △PQR (by SAS)
Notation: The order of letters matters. △ABC ≅ △PQR means A↔P, B↔Q, C↔R.
Types and Properties
When to use SAS:
- You know two sides and the angle between them.
- The angle must be the included angle.
When NOT to use SAS:
- If the known angle is NOT between the two known sides — this is the SSA case, which does NOT guarantee congruence.
Solved Examples
Example 1: Proving Triangles Congruent
Problem: In triangles ABC and DEF: AB = DE = 5 cm, ∠A = ∠D = 60°, AC = DF = 7 cm. Are they congruent?
Solution:
- AB = DE = 5 cm (side)
- ∠A = ∠D = 60° (included angle between AB & AC, and DE & DF)
- AC = DF = 7 cm (side)
- By SAS: △ABC ≅ △DEF
Answer: Yes, △ABC ≅ △DEF by SAS.
Example 2: Finding Unknown Side
Problem: △ABC ≅ △PQR by SAS. AB = 6 cm, BC = 8 cm, ∠B = 50°. Find PQ, QR, and ∠Q.
Solution:
- Corresponding parts of congruent triangles are equal (CPCT).
- PQ = AB = 6 cm, QR = BC = 8 cm, ∠Q = ∠B = 50°
Answer: PQ = 6 cm, QR = 8 cm, ∠Q = 50°.
Example 3: Not SAS — Angle Not Included
Problem: AB = PQ = 5, BC = QR = 7, ∠C = ∠R = 40°. Is this SAS?
Solution:
- ∠C is opposite to AB, not between AB and BC.
- The angle is NOT included between the two given sides.
- This is SSA, not SAS.
Answer: No, this is NOT SAS. We cannot conclude congruence.
Example 4: Word Problem — Garden
Problem: Two triangular garden plots have: Plot A — two sides 10 m and 8 m with an included angle of 45°. Plot B — two sides 10 m and 8 m with an included angle of 45°. Are the plots the same shape and size?
Solution:
- Both have two equal sides and the same included angle.
- By SAS, the triangles are congruent.
Answer: Yes, the plots are congruent (same shape and size).
Real-World Applications
Real-world uses:
- Construction: Two triangular supports are congruent if they share two sides and the included angle — ensuring equal strength.
- Engineering: Verifying identical parts in manufacturing.
- Surveying: Proving two plots of land are identical in shape.
Key Points to Remember
- SAS stands for Side-Angle-Side.
- The angle MUST be the included angle (between the two sides).
- If two sides and the included angle of one triangle equal those of another, the triangles are congruent.
- SSA (non-included angle) does NOT prove congruence.
- Congruent triangles have all corresponding parts equal (CPCT).
- The order of vertices in the congruence statement matters.
Practice Problems
- AB = DE = 4 cm, BC = EF = 6 cm, ∠B = ∠E = 70°. Are triangles ABC and DEF congruent? By which rule?
- Two sides of a triangle are 5 cm and 9 cm with included angle 55°. Another triangle has sides 5 cm and 9 cm with included angle 55°. Prove congruence.
- Can SAS be applied if the angle is not between the two given sides?
- △PQR ≅ △XYZ by SAS. PQ = 7 cm, ∠Q = 80°, QR = 10 cm. Find XY, ∠Y, and YZ.
- Identify the criterion: AB = PQ, ∠A = ∠P, AC = PR.
Frequently Asked Questions
Q1. What is the SAS congruence rule?
If two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another, the triangles are congruent.
Q2. What does 'included angle' mean?
The angle formed between the two given sides. For example, if the sides are AB and BC, the included angle is ∠B.
Q3. Is SSA the same as SAS?
No. SSA means the angle is not between the two sides. SSA does NOT guarantee congruence. SAS requires the angle to be between the two given sides.
Q4. What does CPCT mean?
Corresponding Parts of Congruent Triangles. Once you prove two triangles congruent, all their corresponding sides and angles are equal.
