Congruence and similarity in geometry
Geometry and measures • Mensuration and calculation
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Key concepts
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Definition of congruence
Congruence occurs when two figures match exactly in shape and size under rigid motions (translation, rotation, reflection). Corresponding sides are equal in length and corresponding angles are equal in measure. Limiting factor: congruence requires exact equality of all corresponding dimensions; proportionality is not sufficient. Congruence criteria for triangles include SSS, SAS, ASA and RHS. Correct application of a criterion establishes full correspondence and allows direct transfer of length and angle measures between congruent triangles.
Congruence criteria for triangles
SSS (side-side-side) proves congruence when three pairs of corresponding sides are equal. SAS (side-angle-side) proves congruence when two pairs of corresponding sides and the included angle are equal. ASA (angle-side-angle) proves congruence when two pairs of angles and the included side are equal. RHS (right angle-hypotenuse-side) proves congruence for right-angled triangles when hypotenuse and one other side are equal. Cause → effect: matching specified elements causes all remaining corresponding elements to match, enabling calculation of unknown sides or angles by direct equality.
Definition of similarity
Similarity occurs when two figures have the same shape but not necessarily the same size. Corresponding angles are equal and corresponding sides are proportional. Limiting factor: similarity requires consistent proportional ratios for all corresponding side pairs and equality for all corresponding angles. Cause → effect: equal angles cause shape preservation; proportional sides cause consistent scaling, enabling conversion between lengths, areas and volumes using the scale factor.
Similarity conditions and scale factor
AA (angle-angle) proves similarity for triangles when two pairs of corresponding angles are equal. SSS and SAS prove similarity when all corresponding sides are in the same ratio or two sides and the included angle are proportional and equal respectively. The scale factor (k) equals corresponding length in image divided by corresponding length in original. Cause → effect: a scale factor of k multiplies every length by k. Therefore, lengths scale by k, areas scale by k^2, and volumes scale by k^3. Limiting factor: sign or orientation does not affect similarity; scale factor must be positive for standard similarity.
Lengths, areas and volumes in similar figures
For similar plane figures, corresponding lengths have ratio k. Corresponding areas have ratio k^2 because area depends on two linear dimensions. For similar solids, corresponding volumes have ratio k^3 because volume depends on three linear dimensions. Cause → effect: doubling linear dimensions (k = 2) causes area to increase by factor 4 and volume to increase by factor 8. Correct application requires consistent identification of corresponding dimensions and use of exact scale factor.
Applications and problem-solving strategy
Identify corresponding parts and determine whether congruence or similarity applies. Prove required condition (e.g., SSS or AA) before using equality or proportionality. Use the scale factor to convert lengths, square it for areas, and cube it for volumes. Cause → effect: proving similarity or congruence causes direct use of equalities or proportionalities, which simplifies calculation of unknown measures. Limiting factor: mismatched correspondence or incorrect identification of the included angle invalidates the chosen criterion.
Key notes
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