Similar Polygons

Similar polygons are polygons that have the same shape but are not the same size; their corresponding angles are congruent and their corresponding sides are proportional.

∼ is the symbol for "is similar to"

Vocabulary:

  • similar polygons
  • corresponding angles
  • corresponding sides
  • scale factor
  • perimeter scale factor

  • Angle Q corresponds to Angle M
  • Angle R corresponds to Angle N
  • Angle S corresponds to Angle O
  • Angle P corresponds to Angle L
  • QR corresponds to MN
  • NO corresponds to RS
  • LO corresponds to PS
  • ML corresponds to QP

Example #1

Quadrilaterals PQRS and LMNO are similar. List all the pairs of congruent angles. Write the ratios of the corresponding sides in a statement of proportionality.

Let's start by listing all pairs of congruent angles.

∠P ≅ ∠L

∠Q ≅ ∠M

∠R ≅ ∠N

∠S ≅ ∠O 

Now write the ratios of the sides as a statement of proportionality.

Note on Ratios

When you write your own statement of proportionality with the ratios, remember to:

  1. Always list the same polygon first
  2. Ratios must be set up with corresponding sides

 

Example #2

Decide whether the figures are similar. If they are similar, write a similarity statement.

Solution: Let's look at this step by step.

First, we need to find out if angles are congruent

The corresponding angles of triangles HIJ and BAC are congruent based on the given information.

The next step, by the definition of similar polygons, is that all corresponding sides are proportional, or have the same ratios of side lengths. Let's check.

Start by identifying the corresponding sides. Make sure you list corresponding points for the line segments in the same order.

Answer: BA, AC, CB

Substitute the measures of the sides into the ratios.

Answer: 7/14, 3/6, 5/10

From this, all sides are proportional.

Ok, since all angles are congruent and all sides proportional, the triangles are similar. The similarity statement is:

△ HIJ ∼ △ ___

Answer: BAC

Summary - Finding Polygons Similar

If asked to find polygons similar, follow these steps.

  1. Determine whether or not corresponding angles are congruent.
    • If no, the polygons are not similar
    • If yes, go to 2
  2. Determine whether or not corresponding sides are proportional by making sure the ratios of corresponding sides are equal
    • If no, the polygons are not similar
    • If yes, the polygons are similar
  3. Fill in the similarity statement.

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