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