Introduction to Similar Figures:

Similar figures have the same shape, but might not be the same size

  • When two shapes are similar, their corresponding sides are proportional (see ratios and proportions) and their corresponding angles are congruent.
  • An older sister and a younger sister might be considered similar

Congruent figures have the same shape and size.

  • When two shapes are congruent, their corresponding sides and angles are congruent.
  • Identical twins might be considered congruent

Here are two symbols that you need to know:

~ means similar
≅ means congruent

Look Out: be careful when reading which sides correspond to each other; the shapes may be rotated.

Similar Figures Practice:

Example 1

Quadrilaterals ABCD and EFGH are congruent (ABCD ≅ EFGH). Find the measure of each missing angle and side.

ABCD and EFGH quadrilaterals


Example 2

Triangles ABC and DEF are similar (ABC ~ DEF). What is the length of side DF?

Similar Triangles ABC and DEF


Example 3

The sun casts a shadow on two trees in a field. The smaller of the two is 10.5 ft high and has a shadow 11.25 ft long. The shadow of the taller tree is 17.5 ft long. How tall is that tree?

tree

This is actually how the heights of many tall structures are estimated!


Example 4

Pentagons OPQRS and TUVWX are both regular. Find the length of the apothem of TUVWX.

Pentagons

The apothem is the distance from the center of a regular polygon to the midpoint of one side.


Exercise 1

JKLM ~= NOPQ What are m ∠ K and m of line MJ?

Trapezoids


Exercise 2

Triangle RST ~ Triangle UVW. Find the length of sides ST and VU. Round to the nearest hundredths place.

Triangles RST and UVW


Exercise 3

The sun casts a 23 foot shadow on a flagpole. You are 5 feet 3 inches and cast an 11 foot shadow. Approximately how tall is the flagpole? Round to the nearest hundredths place.

flagpole


Exercise 4

One square has side lengths of 5 cm, and a diagonal ≅ 7.07 cm. Another square has side lengths of 8 cm. What is the length of its diagonal?


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