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6. Polygons
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United States of AmericaVirginia
Grade 7

6.01 Parts of similar figures

Lesson
Practice

Parts of similar figures

Remember that congruent figures have the exact same size and shape. In other words, two figures are congruent if all corresponding sides and all corresponding angles are congruent.

Corresponding sides and angles are a pair of matching angles or sides that are in the same spot in two different shapes.

Recall, we use markings on figures to show congruency.

The curved markings on the angles show us which corresponding angles are congruent.

The hash or hatch marks on the sides show which sides are congruent.

Different numbers of markings represent different measures.

Two triangles, ABC and DEF, are shown with congruent markings. Triangle ABC has sides AB, AC, and BC marked with double, single, and triple tick marks, respectively, while triangle DEF has sides DE, DF, and EF marked with the same corresponding tick marks. The angles at vertices A, B, and C have one, two, and three arcs respectively, matching the angles at vertices D, E, and F in triangle DEF.

We can use notation to write that different parts of a figure are congruent. The symbol \cong means 'is congruent to'.

For the angles:

\angle C \cong \angle F, \angle B \cong \angle E , and \angle A \cong \angle D.

For the sides:

\overline{AC} \cong \overline{DF}, \overline{AB} \cong \overline{DE}, and \overline{BC} \cong \overline{EF}.

For the triangles:

\triangle ABC \cong \triangle DEF

Similar Figures

Similar figures have corresponding sides that are proportional and corresponding interior angles that are congruent. The symbol \sim is used to represent similarity.

Parallelograms CATS and BIRD are similar because all of their corresponding angles are congruent and their corresponding sides are proportional. Specifically, each side of BIRD is 2 times longer than the corresponding side of CATS.

We can write the similarity statement: CATS \sim BIRD.

Two parallelograms, CAST and BIRD, are shown. Parallelogram CAST has sides of 3 units and 2 units, with angles of 110 degrees and 70 degrees alternating at its vertices. Parallelogram BIRD has sides of 6 units and 4 units, with the same alternating angles of 110 degrees and 70 degree at its vertices.

We can identify corresponding angles directly from the similarity statement. Corresponding interior angles are congruent in similar figures.

\angle C \cong \angle B , \angle A \cong \angle I, \angle T \cong \angle R, and \angle S \cong \angle D

The sides connecting corresponding angles will be corresponding sides.

\overline{CA} corresponds to \overline{BI}, \overline{AT} corresponds to \overline{IR}, \overline{TS} corresponds to \overline{RD}, and \overline{CS} corresponds to \overline{BD}.

Similarity does not depend on the position or orientation of the figures. Figures can be turned differently and still be similar. We need to be extra careful with identifying corresponding parts when this is the case.

Here we have two similar triangles. Because their corresponding angles are congruent and their corresponding sides are proportional (by a factor of 3).

Two right triangles, PQR and DEF, are shown. Triangle PQR has sides PQ = 2, PR = 4, and QR = 3, with angles 40 degree at Q, 50 degree at R, and a right angle at P. Triangle DEF has sides DE = 6, DF = 12, and EF = unknown, with angles 40 degree at D, 50 degrees at F, and a right angle at E.

Identify congruent angles to find corresponding angles:

\angle Q \cong\angle D, \angle R \cong\angle F, and \angle P \cong\angle E.

The sides connecting the corresponding angles will be corresponding sides.

Side \overline{PQ} corresponds to side \overline{ED}. Side \overline{QR} corresponds to side \overline{DF}. Side \overline{PR} corresponds to side \overline{EF}.

We can write a similarity statement for the above triangles:

\triangle PQR \sim \triangle EDF

This statement is read as "Triangle PQR is similar to triangle EDF."

Congruent polygons are similar polygons for which the ratio of the corresponding sides is 1:1. However, similar polygons are not necessarily congruent.

Examples

Example 1

The two figures below are similar with a similarity statement NOPM \sim TQRS.

Two quadrilaterals are shown. The first quadrilateral, MNOP, is a trapezoid with parallel sides NO and MP. The second quadrilateral, RSTQ, is an irregular quadrilateral with no parallel sides.
a

Identify all corresponding angles.

Worked Solution
Create a strategy

Use the given similarity statement to match up corresponding angles.

Apply the idea

\angle N corresponds to \angle T

\angle O corresponds to \angle Q

\angle P corresponds to \angle R

\angle M corresponds to \angle S

b

Identify all corresponding sides.

Worked Solution
Create a strategy

Use the similarity statement to match up corresponding sides.

Apply the idea

Side \overline{NO} corresponds to side \overline{TQ}

Side \overline{OP} corresponds to side \overline{QR}

Side \overline{PM} corresponds to side \overline{RS}

Side \overline{NM} corresponds to side \overline{TS}

c

Write a different similarity statement for these two figures.

Worked Solution
Create a strategy

Name both figures using a different, but still corresponding, order of letters. Write a similarity statement using the symbol \sim for similar figures.

Apply the idea

PMNO \sim RSTQ

Reflect and check

There are multiple ways we could have written our similarity statement, ensuring corresponding parts. Another correct way could have been MNOP \sim STQR.

Example 2

The two triangles below are similar.

Two right triangles, ABC and XYZ, are shown. Triangle ABC has sides AC = 24, BC = 32, and AB = 40, with angles 35 degrees at A, 55 degrees at B, and a right angle at C. Triangle XYZ has sides YZ = 20, XZ = 15, and XY = 25, with angles 55 degrees at Y, 35 degrees at X, and a right angle at Z.
a

Identify all corresponding angles.

Worked Solution
Create a strategy

Corresponding angles are congruent. This means they have the same measure.

Apply the idea

\angle A corresponds to \angle X

\angle B corresponds to \angle Y

\angle C corresponds to \angle Z

b

Identify all corresponding sides.

Worked Solution
Create a strategy

Use the corresponding angles found in part (a) to help identify corresponding sides.

Apply the idea

Side \overline{AB} corresponds to side \overline{XY}

Side \overline{AC} corresponds to side \overline{XZ}

Side \overline{BC} corresponds to side \overline{YZ}

c

Write a similarity statement for these two figures.

Worked Solution
Create a strategy

Name both figures using a corresponding order of letters. Write a similarity statement using the symbol \sim for similar figures.

Apply the idea

\triangle ABC \sim \triangle XYZ

Reflect and check

There are multiple ways we could have written our similarity statement, ensuring corresponding parts. Another correct way could have been \triangle BCA \sim \triangle YZX.

Idea summary

Figures are similar if:

  • All corresponding sides are proportional

  • All corresponding angles are equal

The symbol \sim is used to represent similarity. Similarity statements can be used to determine corresponding parts of similar figures. \triangle ABC \sim \triangle DEF is read as "Triangle ABC is similar to triangle DEF."

Similarity does not depend on the position or orientation of the figures.

Outcomes

7.MG.2

The student will solve problems and justify relationships of similarity using proportional reasoning.

7.MG.2a

Identify corresponding congruent angles of similar quadrilaterals and triangles, through the use of geometric markings.

7.MG.2b

Identify corresponding sides of similar quadrilaterals and triangles.

7.MG.2c

Given two similar quadrilaterals or triangles, write similarity statements using symbols.

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