Congruence and Similarity
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More Lessons for High School Regents Exam
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High School Math based on the topics required for the Regents Exam conducted by NYSED. Learn to identify whether figures are congruent, similar, or neither.
Geometry definition for Congruent:
Two shapes are congruent if they have the same shape and the same size.
Geometry definition for Similar:
Two shapes are similar if they have the same shape, but not necessarily the same size.
The following diagrams gives the conditions for similar triangles. Scroll down the page for more examples and solutions.
Geometry Worksheets
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Properties of Congruent Figures:
Corresponding sides are equal in length.
Corresponding angles are equal in measure.
They have the same perimeter and the same area (for 2D shapes).
How to Prove Congruence (for Triangles):
For triangles, you don’t always need to check all three sides and three angles. There are specific congruence theorems:
SSS (Side-Side-Side): If all three sides of one triangle are congruent to the three corresponding sides of another triangle.
SAS (Side-Angle-Side): If two sides and the included angle (the angle between those two sides) of one triangle are congruent to the corresponding two sides and included angle of another triangle.
ASA (Angle-Side-Angle): If two angles and the included side (the side between those two angles) of one triangle are congruent to the corresponding two angles and included side of another triangle.
AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle.
HL (Hypotenuse-Leg): If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle.
Properties of Similar Figures:
Corresponding angles are equal in measure.
Corresponding sides are in proportion (meaning the ratio of corresponding side lengths is constant). This constant ratio is called the scale factor.
The ratio of their perimeters is equal to the scale factor.
The ratio of their areas is equal to the square of the scale factor.
How to Prove Similarity (for Triangles):
AA (Angle-Angle): If two angles of one triangle are congruent to two angles of another triangle.
SSS (Side-Side-Side) Similarity: If the corresponding sides of two triangles are proportional.
SAS (Side-Angle-Side) Similarity: If two sides of one triangle are proportional to two corresponding sides of another triangle, and the included angles are congruent.
Similar, Congruent, or Neither
Example:
Determine if the following solid shapes are similar, congruent, or neither. If similar, find the scale factor from the left figure to the right.
Investigate Congruent Triangles
Investigate Similar Triangles
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