When we think of two objects being similar, we mean that they have many commonalities but are not identical. Mathematically, if two objects are similar, they will have the same shape but not necessarily the same size.
Two figures are congruent if they are the same size and the same shape.
Two figures are similar if they have the same shape, but are not necessarily the same size.
Similarity in Regular Polygons
A regular polygon has sides that are all the same length.
Common examples of regular polygons include squares, equilateral triangles, and circles, but can have any number of sides as long as each side is the same length.
Two of the same type of regular polygon will always be similar to each other, since by definition their sides will be proportional and their angle measures will be the same (each angle is equal).
A square will always be identifiable as a square, regardless of its size. Two different sized squares are similar to one another because they have the same angle measures (4 right angles) and proportional side lengths (all sides are equal).
Similarity in Rectangles
By definition, rectangles have four right angles and two pairs of parallel sides. Two rectangles will be similar if the proportions of their corresponding sides are equal, since we know they will have equal angle measures.
To determine if the two rectangles are similar, they must have proportional sides. We can create two proportions and set them equal to each other to see if this is true.
Since the proportions are equal, and the angles are equal, the two rectangles are similar.
Since we know that the proportions of the side lengths of two similar shapes must be equal, if we know two shapes are similar, we can solve for a missing side.
The two rectangles are similar. Find the length of the missing side, x.
We can set up proportions to solve for x.
By definition, two triangles are similar if their corresponding angles are the same and their corresponding sides are proportional. Sometimes, we may not be given every side measurement or angle measurement of each triangle, but we may still be able to determine if the two triangles are similar.
Angle-Angle (AA) Theorem
The Angle-Angle Theorem states that two triangles are similar if they have two pairs of corresponding angles.
This is also called the Angle-Angle-Angle (AAA) Theorem, because if we have found two angles in a triangle, we also will know the third angle, since the angles in a triangle must add up to 180°.
The triangles may be facing different directions, as above, so it may help to draw the triangles facing the same direction to match up the corresponding angles
Side Side Side (SSS) Theorem
The Side-Side-Side Theorem states that two triangles are similar if they have three corresponding pairs of proportional sides.
Even if we do not have the angle measures, if the triangles have proportional sides, it means they are the same shape, which means the angle measures will be equal.
The triangles may not be facing the same directions, so it may be helpful to redraw the triangles before determining if their sides are proportional.
Side Angle Side (SAS) Theorem
The Side-Angle-Side Theorem states that two triangles are similar if they have a pair of corresponding proportional side, followed by a pair of congruent angles, followed by another pair of corresponding proportional sides.
This theorem only works if the given angle is between the two given sides. There is no SSA Theorem.
Note: Another common Theorem is the Angle-Side-Angle (ASA) Theorem, but this can be proven by the Angle-Angle (AA) Theorem.
Similarity Example Problem
Are the two triangles below similar? (Figures not drawn to scale).
We can see that both triangles have a congruent angle, and we are given side lengths on either side of the angle. Therefore, we can use SAS to see if the triangles are similar.
To test the proportionality of the sides, set the two proportions equal to each other.
Since the proportions are not equal, these triangles are not similar.
Answers to Practice Problems