ASA and AAS Congruence

Learn to prove triangles congruent using the ASA and AAS theorems.

Intermediate25 minLesson
Printable worksheet

Definition

The ASA (Angle-Side-Angle) and AAS (Angle-Angle-Side) theorems are two ways to prove that triangles are congruent.
ASA Congruence Theorem: If two angles and the included side (the side between those angles) of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
If ∠A≅∠D,AB‾≅DE‾,∠B≅∠E\text{If } \angle A \cong \angle D, \quad \overline{AB} \cong \overline{DE}, \quad \angle B \cong \angle E
Then △ABC≅△DEF\text{Then } \triangle ABC \cong \triangle DEF
AAS Congruence Theorem: If two angles and a non-included side (a side not between those angles) of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
If ∠A≅∠D,∠B≅∠E,BC‾≅EF‾\text{If } \angle A \cong \angle D, \quad \angle B \cong \angle E, \quad \overline{BC} \cong \overline{EF}
Then △ABC≅△DEF\text{Then } \triangle ABC \cong \triangle DEF

Try it now

In the ASA congruence theorem, what does the 'S' stand for?

Worked Examples

In triangles ABCABC and DEFDEF: ∠A=50°\angle A = 50°, ∠B=70°\angle B = 70°, AB=8AB = 8 cm. In triangle DEFDEF: ∠D=50°\angle D = 50°, ∠E=70°\angle E = 70°, DE=8DE = 8 cm. Are the triangles congruent?
1

Identify what we know

∠A=∠D=50°\angle A = \angle D = 50°, ∠B=∠E=70°\angle B = \angle E = 70°, AB=DE=8AB = DE = 8 cm → Two angles and one side match
2

Check if the side is included

Side ABAB is between angles AA and BB. Side DEDE is between angles DD and EE. → Yes, the side is included
3

Apply ASA theorem

Two angles and the included side are congruent → △ABC≅△DEF\triangle ABC \cong \triangle DEF by ASA

Common Mistakes

Confusing ASA with AAS

Why it's wrong: Students mix up which theorem to use based on where the known side is located.

Correct: ASA: the side is BETWEEN the two angles (included). AAS: the side is NOT between the two angles (non-included).

Trying to use AAA (three angles) to prove congruence

Why it's wrong: Three pairs of equal angles only prove similarity, not congruence. Triangles can have the same angles but different sizes.

Correct: You need at least one pair of congruent sides. AAA proves similarity, but ASA or AAS (with a side) proves congruence.

Not matching corresponding parts correctly

Why it's wrong: The order of vertices matters when stating congruence.

Correct: When writing △ABC≅△DEF\triangle ABC \cong \triangle DEF, make sure AA corresponds to DD, BB to EE, and CC to FF.

Interactive Visual

Congruent triangles

8ABC8DEF△ABC ≅ △DEF

ASATwo pairs of angles and the side between them are equal.

  • AB = DE = 8
  • ∠A = ∠D = 50°
  • ∠B = ∠E = 70°

Tap a criterion to see which matching parts it needs. Parts with the same marks are equal.

Congruent triangles

6PQR6STU△PQR ≅ △STU

AASTwo pairs of angles and a side that is not between them are equal.

  • QR = TU = 6
  • ∠P = ∠S = 45°
  • ∠Q = ∠T = 85°

Tap a criterion to see which matching parts it needs. Parts with the same marks are equal.

Interactive Sandbox

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Practice Problems

15 problems
Problem 1 of 15
Easy
In the ASA congruence theorem, what does the 'S' stand for?

Why It Matters

ASA and AAS are essential tools in geometry because:
  • Architecture and Engineering: When designing structures, engineers must ensure triangular components are identical for stability
  • Navigation: Surveyors use triangle congruence to measure distances that cannot be measured directly
  • Art and Design: Creating symmetric patterns requires congruent triangles
  • Proof Writing: These theorems are fundamental for constructing logical arguments in geometry
Unlike SSS or SAS, ASA and AAS allow you to prove congruence when you know two angles and only one side. This is powerful because knowing two angles of a triangle automatically tells you the third angle (since angles sum to 180°180°)!

Real World Applications

Bridge Construction

Engineers use triangle congruence to ensure identical support structures in bridges.

Example:

A bridge has two triangular supports. If both have angles of 60°60° and 75°75° with an included beam of 5 meters, ASA guarantees they are identical.

1Try It Yourself

Two bridge supports have angles of 55°55° and 65°65° with an included beam of 4 meters each.

Are the supports guaranteed to be congruent?

Step 1: Choose the right answer

Which congruence criterion proves it?

Surveying Land

Surveyors measure angles from two points to determine distances across rivers or valleys.

Example:

A surveyor measures a baseline of 100 meters and angles of 40°40° and 72°72° from each end toward a distant point. Using AAS, they can calculate the exact triangle formed.

2Try It Yourself

Two surveyors create triangles. Both measure angles of 50°50° and 80°80°, and the side opposite the 50°50° angle is 25 meters.

Are their triangles congruent?

Step 1: Choose the right answer

Which congruence criterion proves it?

Key Takeaways

  • 1ASA (Angle-Side-Angle): Two angles and the included side prove congruence
  • 2AAS (Angle-Angle-Side): Two angles and a non-included side prove congruence
  • 3The key difference is the position of the known side relative to the angles
  • 4Remember: AAA only proves similarity, not congruence (you need a side!)
  • 5Always match corresponding vertices when writing congruence statements

Frequently Asked Questions

The angles in any triangle always sum to 180°180°. So if you know two angles, you can find the third: ∠C=180°−∠A−∠B\angle C = 180° - \angle A - \angle B.
The angles in any triangle always sum to 180°180°. So if you know two angles, you can find the third: ∠C=180°−∠A−∠B\angle C = 180° - \angle A - \angle B.
Similar triangles have equal angles but different sizes. A small triangle and a large triangle can have the same angles but not be congruent. You need a side measurement to fix the size.
Check where the known side is located. If it is between the two known angles, use ASA. If it is not between them (it is a non-included side), use AAS.

Glossary

Congruent triangles
Triangles that have exactly the same shape and size; all corresponding sides and angles are equal
Included side
The side that is between two given angles
Non-included side
A side that is not between the two given angles
Corresponding parts
Parts of congruent figures that match; written in the same order in a congruence statement
Today's challengeOne problem, three tries

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Introduction to Congruence

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Introduction to Similarity

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Congruent Triangles

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AA Similarity

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