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Congruency and Similarity - Coggle Diagram
Congruency and Similarity
Congruency
Two triangles are congruent if their corresponding sides are equal and their corresponding interior angles are equal
Denoted by this sign: "≡"
Tests for Congruent Triangles
SSS Property
If all three sides of one triangles are equal to each of their corresponding sides in the other triangle, then the two triangles are congruent
Example:
SAS Property
If two sides and the included angle of the first triangle are equal to the two corresponding sides and included angle of the second triangle, then the two triangles are congruent
Example:
AAS Property
If two angles and the non-included side of the first triangle are equal to the corresponding two angles and non-included side of the second triangle, then the two triangles are congruent
Example:
ASA Property
If two angles and the included side of the first triangle are equal to the corresponding two angles and included side of the second triangle, then the two triangles are congruent
Example:
RHS Property
If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, then the two right-angled triangles are congruent
Example:
Similarity
Two triangles are similar if all the corresponding angles are equal and all the corresponding sides are proportional
Tests for Similar Triangles
AAA Property
Condition: Two angles in one triangle are equal to two other angles in the other triangle
Example:
Side-Side-Side Property
Condition: All sides in one triangle are proportional to their corresponding side in the other triangle
Example:
Side-Angle-Side Property
Condition 1: Two sides of the first triangle are proportional to the corresponding sides in the second triangle Condition 2: The included angle in both triangles is equal
Example:
Midpoint Theorem
Condition: Requires a straight line that joins the midpoint of any two sides in the triangle
The straight line is parallel to the third side
The straight line is half the length of the third side
Intercept Theorem
Condition: Requires a straight line in the triangle that is parallel to any one side
The straight line will divide the other two sides in the same ratio
Important: Midpoint and Intercept Theorem cannot be used to prove similarity
Area and Volume Ratios
In two similar shapes, the length ratio squared gives the area ratio
This is helpful when obtaining the area ratio from the scale of a map
In two similar solids, the length ratio cubed gives the volume ratio