Learn about congruent figures, especially triangles, and the rules to determine if two triangles are congruent.
Example 1: What is Congruence?
Explain the concept of congruence in geometry.
Step 1: **Congruence** means that two figures have the exact same size and shape.
Step 2: If two figures are congruent, one can be placed exactly on top of the other, matching perfectly.
Step 3: The symbol for congruence is ≅.
Example 2: Congruence of Line Segments and Angles
Explain when line segments and angles are congruent.
Step 1: **Congruent Line Segments:** Two line segments are congruent if they have the same length.
Example: If segment AB is 5 cm long and segment CD is 5 cm long, then segment AB ≅ segment CD.
Step 2: **Congruent Angles:** Two angles are congruent if they have the same measure.
Example: If angle PQR measures 45 degrees and angle XYZ measures 45 degrees, then angle PQR ≅ angle XYZ.
Example 3: Congruence of Triangles
Explain when two triangles are congruent.
Step 1: Two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent.
Step 2: This means if triangle ABC ≅ triangle PQR, then:
Side AB ≅ Side PQ
Side BC ≅ Side QR
Side CA ≅ Side RP
Angle A ≅ Angle P
Angle B ≅ Angle Q
Angle C ≅ Angle R
Step 3: However, we do not need to check all six pairs of corresponding parts. Specific criteria can be used to prove triangle congruence.
Example 4: Congruence Criteria (SSS, SAS, ASA, AAS, RHS)
List and briefly explain the main criteria used to prove triangle congruence.
Criterion 1: SSS (Side-Side-Side) If three sides of one triangle are congruent to the corresponding three sides of another triangle, then the triangles are congruent.
Criterion 2: SAS (Side-Angle-Side) If two sides and the included angle (the angle between the two sides) of one triangle are congruent to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.
Criterion 3: ASA (Angle-Side-Angle) If two angles and the included side (the side between the two angles) of one triangle are congruent to the corresponding two angles and the included side of another triangle, then the triangles are congruent.
Criterion 4: AAS (Angle-Angle-Side) If two angles and a non-included side of one triangle are congruent to the corresponding two angles and the non-included side of another triangle, then the triangles are congruent.
Criterion 5: RHS (Right Angle-Hypotenuse-Side) If the hypotenuse and one side of a right-angled triangle are congruent to the corresponding hypotenuse and one side of another right-angled triangle, then the triangles are congruent.
Practice Mode
Enter a problem about triangle congruence.
Note: This basic solver can help identify the congruence criterion for simple pairs of triangles if you describe the congruent parts (e.g., "Are triangles ABC and PQR congruent if AB=PQ, BC=QR, AC=PR?", "criterion for SAS").
Related Concepts
Explore these related mathematical concepts to deepen your understanding of congruence.
Triangle
A polygon with three sides and three angles.
Corresponding Parts
Matching sides and angles in two or more figures.
Side
A line segment forming a boundary of a polygon.
Angle
Formed by two rays sharing a common endpoint.
Vertex
A point where two or more sides or edges meet.
Hypotenuse
The longest side of a right-angled triangle, opposite the right angle.
Right Angle
An angle that measures exactly 90 degrees.
Included Angle/Side
The angle between two sides, or the side between two angles.