Congruent Polygons

Definition

Two polygons are congruent if they have the same size and shape. This means that all corresponding sides and angles are equal.

Criteria for Congruent Polygons

In order to determine if two polygons are congruent, we can use the following criteria:

• Side-Side-Side (SSS) Congruence: If the three sides of one polygon are congruent to the three sides of another polygon, then the two polygons are congruent.
• Angle-Angle-Angle (AAA) Congruence: If the three angles of one polygon are congruent to the three angles of another polygon, then the two polygons are congruent.
• Side-Angle-Side (SAS) Congruence: If two sides and the included angle of one polygon are congruent to two sides and the included angle of another polygon, then the two polygons are congruent.
• Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one polygon are congruent to two angles and the included side of another polygon, then the two polygons are congruent.
• Hypotenuse-Leg (HL) Congruence: If the hypotenuse and a leg of one right-angled triangle are congruent to the hypotenuse and a leg of another right-angled triangle, then the two triangles are congruent.

Examples and Practice Problems

Let's work through some examples and practice problems to reinforce our understanding of congruent polygons:

1. Determine if the following pairs of triangles are congruent:
1. Triangle ABC with side lengths 3, 4, 5 and Triangle DEF with side lengths 5, 12, 13
2. Triangle XYZ with angles 30°, 60°, 90° and Triangle PQR with angles 30°, 60°, 90°
2. Using the given information, determine if the following pairs of polygons are congruent:
1. Polygon ABCD with side lengths AB = 8, BC = 6, CD = 10, and Polygon WXYZ with side lengths WX = 8, XY = 6, YZ = 10
2. Polygon LMNO with angles ∠L = 60°, ∠M = 90°, ∠N = 30° and Polygon PQRS with angles ∠P = 60°, ∠Q = 90°, ∠R = 30°

Conclusion

Understanding congruent polygons and the criteria for determining their congruence is essential in geometry. Practice is key to mastering this topic, so be sure to work through plenty of examples and problems to reinforce your knowledge.