ASA Congruence Rule
The ASA (Angle-Side-Angle) Congruence Rule says that if two angles and the included side (the side between those angles) of one triangle are equal to the corresponding parts of another triangle, the two triangles are congruent.
This means that knowing two angles and the side between them is enough to determine the entire triangle — there is only one triangle possible with those measurements.
What is ASA Congruence Rule - Grade 7 Maths (Congruence of Triangles)?
ASA Congruence Rule:
- If in two triangles, two angles and the included side of one are equal to the corresponding two angles and included side of the other, the triangles are congruent.
AAS (Angle-Angle-Side):
- If two angles and a non-included side of one triangle equal the corresponding parts of another, the triangles are still congruent.
- This works because if two angles are known, the third is automatically known (angle sum = 180°).
ASA Congruence Rule Formula
If ∠A = ∠P, AB = PQ, ∠B = ∠Q
Then △ABC ≅ △PQR (by ASA)
Types and Properties
When to use ASA:
- Two angles and the side between them are known.
When to use AAS:
- Two angles and a side not between them are known. Since the third angle can be calculated, it reduces to ASA.
Solved Examples
Example 1: Proving Congruence by ASA
Problem: In △ABC and △DEF: ∠A = ∠D = 50°, AB = DE = 6 cm, ∠B = ∠E = 70°. Prove congruence.
Solution:
- ∠A = ∠D = 50° (angle)
- AB = DE = 6 cm (included side between ∠A and ∠B)
- ∠B = ∠E = 70° (angle)
- By ASA: △ABC ≅ △DEF
Answer: △ABC ≅ △DEF by ASA.
Example 2: Finding Unknown Parts
Problem: △ABC ≅ △XYZ by ASA. ∠A = 55°, AB = 8 cm, ∠B = 65°. Find ∠C, ∠X, XY, and ∠Y.
Solution:
- ∠C = 180° − 55° − 65° = 60°
- By CPCT: ∠X = ∠A = 55°, XY = AB = 8 cm, ∠Y = ∠B = 65°
Answer: ∠C = 60°, ∠X = 55°, XY = 8 cm, ∠Y = 65°.
Example 3: Using AAS
Problem: ∠A = ∠P = 40°, ∠B = ∠Q = 80°, BC = QR = 5 cm. Are the triangles congruent?
Solution:
- Two angles and a non-included side are equal.
- Third angles: ∠C = ∠R = 180° − 40° − 80° = 60°
- Now we have ∠B = ∠Q, BC = QR, ∠C = ∠R — this is ASA.
Answer: Yes, congruent by AAS (which reduces to ASA).
Example 4: Word Problem
Problem: Two triangular tiles have angles 45° and 90° with the side between these angles being 10 cm on both. Are the tiles identical?
Solution:
- Both have ∠1 = 45°, included side = 10 cm, ∠2 = 90°.
- By ASA, the triangles are congruent.
Answer: Yes, the tiles are identical (congruent).
Real-World Applications
Real-world uses:
- Surveying: Two angles and a baseline distance determine a unique triangle — used in triangulation.
- Construction: If two angles and the included side match, the structure is identical.
- Navigation: ASA helps determine distances by measuring angles and one known distance.
Key Points to Remember
- ASA stands for Angle-Side-Angle.
- The side MUST be between (included by) the two angles.
- AAS also works because the third angle is determined by the other two.
- Congruent triangles have all corresponding parts equal (CPCT).
- AAA does NOT prove congruence — it only proves similarity.
Practice Problems
- ∠P = ∠X = 60°, PQ = XY = 5 cm, ∠Q = ∠Y = 70°. Prove △PQR ≅ △XYZ.
- Two angles of a triangle are 55° and 75°. The side between them is 9 cm. Another triangle has the same. Are they congruent?
- Can AAA prove congruence? Why or why not?
- △ABC ≅ △MNO by ASA. ∠A = 45°, AB = 10 cm, ∠B = 60°. Find all parts of △MNO.
Frequently Asked Questions
Q1. What is ASA congruence?
If two angles and the included side of one triangle equal the corresponding parts of another, the triangles are congruent.
Q2. What is the difference between ASA and AAS?
In ASA, the known side is between the two angles. In AAS, the known side is not between them. Both prove congruence.
Q3. Does AAA prove congruence?
No. AAA (all angles equal) only proves the triangles are similar (same shape) but not necessarily the same size.
Q4. What does 'included side' mean?
The side that connects the vertices of the two given angles. If angles are ∠A and ∠B, the included side is AB.
