S - Proving Similarity Lesson

Proving Similarity

Similar Polygons

image stating: "two shapes are similar if
#1 Their corresponding angles are congruent
AND
#2 Their corresponding sides are proportional" There are a couple of ways to show similarity. One way, which we have discussed already is the idea that shapes are similar if we can use our transformations (Translate, Reflect, Rotate and Dilate) to map one of the shapes onto the other. A second way of showing similarity is by comparing corresponding angles and sides.

Corresponding sides are sides that "match up" in the shapes. They are in the same place. Corresponding angles will also match up. Check out this video for an explanation of how to easily identify corresponding sides and angles. You will also learn about the importance of reading and writing similarity statements carefully.

Use the following activity to test your skills at identifying corresponding sides and angles.

In order to prove similarity of polygons, we have to check their angles and their sides.

Checking the angles is simple, either corresponding angles are congruent, or they are not. Since the sum of the angles in any given polygon is fixed, knowing all but one angle is sufficient.

But what does it mean for the sides to be proportional?

Check out this video to learn how to set up these ratios and check for proportionality.

Similar Triangles

When it comes to similarity, triangles are a special case. Because triangles are so simple, we don't need to check everything. Check out this video for an explanation of 3 "short cuts" to proving triangles similar.

In summary, there are three ways to prove that two triangles are similar. AA Similarity Theorem, SAS Similarity Theorem, and SSS Similarity Theorem.

AA Similarity Theorem

If any two angles of a triangle are congruent to any two angles of another triangle, then the two triangles are similar to each other.

In the diagram below, we are given that $LaTeX: \angle A\cong\angle D$ $\angle A\cong\angle D$ and $LaTeX: \angle B\cong\angle E$ $\angle B\cong\angle E$ . By the triangle sum theorem (Angles in a triangle sum to 180) we can conclude that $LaTeX: \angle C\cong\angle F$ $\angle C\cong\angle F$ . Therefore, $LaTeX: \bigtriangleup ABC\sim\bigtriangleup DEF$ $\bigtriangleup ABC\sim\bigtriangleup DEF$ .

image depicting two different triangles that have 3 equal angles

SSS Similarity Theorem

If all three sets of corresponding sides of two triangles are proportional, then the triangles are similar.

image depicting two similar triangles that are also proportional

To see if these two triangles are similar, we will set up a proportionality statement between the corresponding sides and see if they have the same scale factor.

$LaTeX: \frac{3}{6\:}=\frac{4}{8}=\frac{6}{12}$ $\frac{3}{6\:}=\frac{4}{8}=\frac{6}{12}$

Since they all reduce to 1/2, they are proportional.

This shows us that the sides are proportional and the scale factor is 1/2.

SAS Similarity Theorem

If two pairs of corresponding sides are proportional and the angles between them are congruent, then the triangles are similar.

In the above diagram, $LaTeX: \angle ACB\cong\angle ECD$ $\angle ACB\cong\angle ECD$ because the two angles are vertical angles. We know that vertical angles are congruent.

Since we know that two triangles have one pair of congruent angles, let's write a proportionality statement to see if the sides are proportional.

$LaTeX: \frac{15}{18}=\frac{20}{24}$ $\frac{15}{18}=\frac{20}{24}$

Since they both reduce to 5/6, they are proportional.

You could also check with cross products.

15*24=18*20

360=360

Once again, they are proportional!

From the proportionality statement we see that the 2 pairs of sides adjacent to the congruent angles are proportional.

This shows that equation image indicator $LaTeX: \bigtriangleup ABC\sim\bigtriangleup EDC$ $\bigtriangleup ABC\sim\bigtriangleup EDC$ by SAS similarity postulate and that the scale factor is 5/6.

IMAGES CREATED BY GAVS