Discovering the Magic of ASA and AAS Congruence in Triangles

Step-by-step Guide: ASA and AAS Congruence

1. ASA (Angle-Side-Angle) Congruence Postulate:
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. To use this postulate:

• Ensure you know the magnitudes of two angles and the length of the included side for both triangles.
• Compare these angles and the side.
• If both conditions are met, then the triangles are congruent.

2. AAS (Angle-Angle-Side) Congruence Postulate:
If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent. To use this postulate:

• Make sure you know the magnitudes of two angles and the length of a side (that’s not between the two angles) for both triangles.
• Compare these angles and the side.
• If all match, then the triangles are congruent.

Examples

Example 1:
Are triangles with angles \(40^{\circ}\), \(90^{\circ}\) and a side of \(5 \text{ cm}\) (between the two angles) and another triangle with angles \(40^{\circ}\), \(90^{\circ}\) and a side of \(5 \text{ cm}\) congruent based on the ASA postulate?

Solution:
Both triangles have:
Angles of \(40^{\circ}\) and \(90^{\circ}\)
An included side of \(5 \text{ cm}\)
Since both the angles and the included side match, the triangles are congruent by the ASA postulate.

Example 2:
Are triangles with angles \(30^{\circ}\) and \(50^{\circ}\), and a non-included side of \(7 \text{ cm}\), and another triangle with angles \(30^{\circ}\) and \(50^{\circ}\), with a non-included side of \(7 \text{ cm}\) congruent based on the AAS postulate?

Solution:
Both triangles have:
Angles of \(30^{\circ}\) and \(50^{\circ}\)
A non-included side of \(7 \text{ cm}\)
Since the angles and the non-included side match, the triangles are congruent by the AAS postulate.

Practice Questions:

1. Are triangles with angles \(60^{\circ}\) and \(80^{\circ}\), and a non-included side of \(10 \text{ cm}\), and another triangle with angles \(60^{\circ}\) and \(80^{\circ}\), with a non-included side of \(11 \text{ cm}\) congruent by the AAS postulate?
2. Are triangles with angles \(70^{\circ}\) and \(110^{\circ}\), and an included side of \(12 \text{ cm}\), and another triangle with angles \(70^{\circ}\) and \(110^{\circ}\), with an included side of \(12 \text{ cm}\) congruent by the ASA postulate?

Answers:

1. No
2. Yes