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Similar Polygons Completely Explained w/ 23+ Step-by-Step Examples!

// Last Updated: January 21, 2020 - Watch Video //

In today’s geometry lesson, you’re going to learn about similar polygons.

Jenn (B.S., M.Ed.) of Calcworkshop® introducing similar polygons

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

We’re going to take a step-by-step approach to setup, identify, and use our detective skills once again to find missing side lengths and other unknown measures.

So let’s get started!

What Are Similar Polygons?

To define similar polygons we need to start with the concept of congruent polygons.

As you may recall, congruent polygons have the exact same size and are a perfect match because all corresponding parts are congruent (equal). Whereas, similar polygons have the same shape, but not the same size (i.e., one is bigger than the other).

This means that if two polygons are similar, then their corresponding angles are congruent but their their corresponding sides are proportional as displayed in the figure below.

similar and congruent figures

Similar and Congruent Figures

Remember, a ratio is a fraction comparing two quantities, and a proportion is when we set two ratios equal to each other. And we can use cross multiplication to solve a proportion. Checkout the video below for a review of ratios and proportions.

Scale Factor

So how do we create a proportion?

We need a scale factor!

If two polygons are similar, then the ratio of the lengths of any two corresponding sides is called the scale factor. This means that the ratio of all parts of a polygon is the same as the ratio of the sides .

For example, using the figure above, the simplified ratio of the lengths of the corresponding sides of the similar trapezoids is the scale factor.

And as ck-12 accurately states, if two polygons are similar then not only are their side lengths proportional, but their perimeters, areas, diagonals, medians, midsegments, and altitudes are proportional too.

And why are scale factors important?

Because if we have a scale factor then we can find all missing side lengths as well!

How To Find Scale Factor?

To find the scale factor, we simply create a ratio of the lengths of two corresponding sides of two polygons. If the ratio is the same for all corresponding sides, then this is called the scale factor and the polygons are similar.

scale factor example

Scale Factor Example

The above example indicates that the scale factor for the two quadrilaterals is 3/2 and proves that the two polygons are indeed similar.

In the video below we are going to review how to solve proportions, determine if two polygons are similar by creating scale factors, and learn how to solve for unknown measures.

Similar Polygons – Lesson & Examples (Video)

  • Introduction
  • 00:00:33 – Overview of the topic including properties of proportions
  • 00:06:09 – Solve each proportion (Examples #1-3)
  • 00:13:11 – Write the ratio as a fraction in simplest form (Examples #4-7)
  • Exclusive Content for Member’s Only
  • 00:16:32 – Determine whether the proportion is true or false (Examples #8-13)
  • 00:25:38 – Using the diagram and given proportion find the unknown length (Examples #14)
  • 00:28:16 – Using the diagram and given proportion find the unknown length (Examples #15-16)
  • 00:34:07 – Overview of scale factor
  • 00:36:50 – Determine if the polygons are similar. If yes, find the scale factor (Examples #17-22)
  • 00:50:57 – Find the indicated measures for the given problems (Examples #23-24)
  • Practice Problems with Step-by-Step Solutions
  • Chapter Tests with Video Solutions

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7.3: Similar Polygons and Scale Factors

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Similar Polygons

Similar polygons are two polygons with the same shape, but not the same size. Similar polygons have corresponding angles that are congruent , and corresponding sides that are proportional.

f-d_4c684343dd23b5763ad8777b6f12cca816961b1cc01793afd0e979ba+IMAGE_TINY+IMAGE_TINY.png

These polygons are not similar:

f-d_efa22797ae949a277ee0c20e5551515326b07d822a67798060ea0ad4+IMAGE_TINY+IMAGE_TINY.png

Scale Factors

Think about similar polygons as enlarging or shrinking the same shape. The symbol \sim\) is used to represent similarity. Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, all equilateral triangles are similar and all squares are similar . If two polygons are similar, we know the lengths of corresponding sides are proportional. In similar polygons, the ratio of one side of a polygon to the corresponding side of the other is called the scale factor . The ratio of all parts of a polygon (including the perimeters, diagonals, medians, midsegments, altitudes) is the same as the ratio of the sides.

What if you were told that two pentagons were similar and you were given the lengths of each pentagon's sides. How could you determine the scale factor of pentagon #1 to pentagon #2?

Example \(\PageIndex{1}\)

\(ABCD\) and \(UVWX\) are below. Are these two rectangles similar?

f-d_ceb4023630b2ede3a3117308b3d71b3b7255b45acfca8a57879fd1ca+IMAGE_TINY+IMAGE_TINY.png

All the corresponding angles are congruent because the shapes are rectangles.

Let’s see if the sides are proportional. \(\dfrac{8}{12}=\dfrac{2}{3}\) and \(\dfrac{18}{24}=\dfrac{3}{4}\). \(\dfrac{2}{3}\neq \dfrac{3}{4}\), so the sides are not in the same proportion , and the rectangles are not similar.

Example \(\PageIndex{2}\)

\(\Delta ABC\sim \Delta MNP\). The perimeter of \(\Delta ABC\) is 150, \(AB=32\) and \(MN=48\). Find the perimeter of \(\Delta MNP\).

From the similarity statement, \(AB\) and \(MN\) are corresponding sides. The scale factor is \(\dfrac{32}{48}=\dfrac{2}{3}\) or \(\dfrac{3}{2}\). \Delta ABC\) is the smaller triangle, so the perimeter of \(\Delta MNP\) is \(\dfrac{3}{2}(150)=225\).

Example \(\PageIndex{3}\)

Suppose \(\Delta ABC\sim \Delta JKL\). Based on the similarity statement, which angles are congruent and which sides are proportional?

Just like in a congruence statement, the congruent angles line up within the similarity statement. So, \(\angle A\cong \angle J\), \(\angle B\cong \angle K\), and \angle C\cong \angle L\). Write the sides in a proportion: \(\dfrac{AB}{JK}=\dfrac{BC}{KL}=\dfrac{AC}{JL}\). Note that the proportion could be written in different ways. For example, \(\dfrac{AB}{BC}=\dfrac{JK}{KL}\) is also true.

Example \(\PageIndex{4}\)

\(MNPQ \sim RSTU\). What are the values of \(x\), \(y\) and \(z\)?

f-d_617a3f2f7d46746718561152566c9bc889b0cd66cc82b3680471947b+IMAGE_TINY+IMAGE_TINY.png

In the similarity statement, \(\angle M\cong \angle R\), so \(z=115^{\circ}\). For \(x\) and \(y\), set up proportions.

\(\dfrac{18}{30}=\dfrac{x}{25} \qquad \dfrac{18}{30}=\dfrac{15}{y}\)

\(450=30x \qquad 18y=450\)

\(x=15\qquad y=25\)

Example \(\PageIndex{5}\)

\(ABCD\sim AMNP\). Find the scale factor and the length of \(BC\).

f-d_eaf044b3d91c642f259308a77a955262a109b3afa4c4b9c0f935853c+IMAGE_TINY+IMAGE_TINY.png

Line up the corresponding sides, \(AB\) and \(AM=CD\), so the scale factor is \(\dfrac{30}{45}=\dfrac{2}{3}\)or \(\dfrac{3}{2}\). Because \(BC\) is in the bigger rectangle, we will multiply 40 by \(\dfrac{3}{2}\) because \(\dfrac{3}{2}\) is greater than 1. \(BC=\dfrac{3}{2}(40)=60\).

For questions 1-8, determine whether the following statements are true or false.

  • All equilateral triangles are similar.
  • All isosceles triangles are similar.
  • All rectangles are similar.
  • All rhombuses are similar.
  • All squares are similar.
  • All congruent polygons are similar.
  • All similar polygons are congruent.
  • All regular pentagons are similar.
  • \(\Delta BIG \sim \Delta HAT\). List the congruent angles and proportions for the sides.
  • If \(BI=9\) and \(HA=15\), find the scale factor.
  • If \(BG=21\), find \(HT\).
  • If \(AT=45\), find \(IG\).
  • Find the perimeter of \(\Delta BIG\) and \(\Delta HAT\). What is the ratio of the perimeters?
  • An NBA basketball court is a rectangle that is 94 feet by 50 feet. A high school basketball court is a rectangle that is 84 feet by 50 feet. Are the two rectangles similar?
  • HD TVs have sides in a ratio of 16:9. Non-HD TVs have sides in a ratio of 4:3. Are these two ratios equivalent?

Use the picture to the right to answer questions 16-20.

f-d_ff937542a1cec934530e672ce08978600f9c2c81aff28344bde681f6+IMAGE_TINY+IMAGE_TINY.png

  • Find \(m\angle E\) and \(m\angle Q\).
  • \(ABCDE\sim QLMNP\), find the scale factor.

Determine if the following triangles and quadrilaterals are similar. If they are, write the similarity statement.

f-d_3b2cf162ad39c5a57481b8556b516ecb7d49daf622441f4b679468da+IMAGE_TINY+IMAGE_TINY.png

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Unit 12: Similarity

About this unit.

Learn what it means for two figures to be similar, and how to determine whether two figures are similar or not. Use this concept to prove geometric theorems and solve some problems with polygons.

Definitions of similarity

  • Similar shapes & transformations (Opens a modal)
  • Similarity & transformations 4 questions Practice

Introduction to triangle similarity

  • Intro to triangle similarity (Opens a modal)
  • Triangle similarity postulates/criteria (Opens a modal)
  • Determining similar triangles (Opens a modal)
  • Proving slope is constant using similarity (Opens a modal)
  • Triangle similarity review (Opens a modal)
  • Determine similar triangles: Angles 4 questions Practice
  • Determine similar triangles: SSS 4 questions Practice

Solving similar triangles

  • Solving similar triangles (Opens a modal)
  • Solving similar triangles: same side plays different roles (Opens a modal)
  • Solve similar triangles (basic) 4 questions Practice
  • Solve similar triangles (advanced) 4 questions Practice

Angle bisector theorem

  • Intro to angle bisector theorem (Opens a modal)
  • Using the angle bisector theorem (Opens a modal)
  • Solve triangles: angle bisector theorem 4 questions Practice

Solving problems with similar and congruent triangles

  • Using similar & congruent triangles (Opens a modal)
  • Challenging similarity problem (Opens a modal)
  • Use similar triangles 4 questions Practice

Solving modeling problems with similar and congruent triangles

  • Geometry word problem: the golden ratio (Opens a modal)
  • Geometry word problem: Earth & Moon radii (Opens a modal)
  • Geometry word problem: a perfect pool shot (Opens a modal)

geometry assignment state if the polygons are similar

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Table of Contents

Last modified on August 3rd, 2023

#ezw_tco-2 .ez-toc-title{ font-size: 120%; ; ; } #ezw_tco-2 .ez-toc-widget-container ul.ez-toc-list li.active{ background-color: #ededed; } chapter outline

Similar polygons.

In similar polygons, there are two aspects: ‘similarity’ and ‘polygon’. As we have already learned about polygons, here, let us understand what ‘similarity’ means.

What is Similarity in Polygons

In mathematics, a pair of polygons are said to be similar when their:

  • Corresponding angles are congruent
  • Corresponding sides are proportional

The symbol ∼ is commonly used to represent similarity.

Let’s take an example of two squares given below,

geometry assignment state if the polygons are similar

The four interior angles of one square are identical to the other. Also, their sides are found to be proportional. The smaller square could scale up to become the larger square. The two squares are thus similar.

E.g. 1. Prove whether the given pairs of rhombus are similar.

geometry assignment state if the polygons are similar

In rhombus ABCD, ∠ABC = ∠ADC… (1) ∠BAD = ∠BCD… (2) and, AB = BC = CD = DA = 4 cm In rhombus EFGH, ∠EFG = ∠EHG… (3) ∠FEH = ∠FGH… (4) and, EF = FG = GH = HE = 2 cm

Taking together (1), (2), (3), (4), we can say that corresponding angles of rhombus ABCD and EFGH are congruent, and also the corresponding sides are proportional. Hence, rhombus ABCD and EFGH are similar (Proved).

Scale Factor of Similar Polygons

In similar polygons, the scale factor is the ratio of one side of a polygon to the corresponding side of the other polygon.

How to Find the Scale Factor

When the corresponding sides of the polygons are similar, the number of times the smaller sides of one polygon becomes the larger sides of the other is their scale factor.

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Similar Polygons

Table of contents.

03 November 2020                

Read time: 5 minutes

Introduction

Sizes and shapes are the backbones of geometry. One of the most encountered shapes in geometry is polygons. The Greek word ‘Polygon’ consists of Poly meaning ‘many’ and gon meaning ‘angle.’

polygons are most encountered shapes in geometry.

Polygons are two-dimensional shapes composed of straight lines. They are said to have a ‘closed shape’ as all the lines are connected. In this article, we will discuss the concept of similarity in Polygons.

Similar Polygons - PDF

If you ever want to read it again as many times as you want, here is a downloadable PDF to explore more.

First, let us get clear with what ‘similar’ means. Two things are called similar when they both have a lot of the same properties but still may not be identical. The same can be said about polygons.

Congruent polygons

As you might have studied, Congruent shapes are the shapes that are an exact match. Congruent polygons have the same size, and they are a perfect match as all corresponding parts are congruent or equal.

Similar polygons definition

On the other hand, In Similar polygons , the corresponding angles are congruent, but the corresponding sides are proportional. So, similar polygons have the same shape, whereas their sizes are different. There would be certain uniform ratios in similar polygons.

Properties of similar polygons

There are two crucial properties of similar polygons:

  • The corresponding angles are equal/congruent. (Both interior and exterior angles are the same)
  • The ratio of the corresponding sides is the same for all sides. Hence, the perimeters are different.

two similar polygons(triangles), whose corresponding angles are equal

The above image shows two similar polygons(triangles), ABC, and A’B’C’. We can see that corresponding angles are equal.

\[<A=<A', <B=<B',<c=<C'\]

The corresponding sides have the same ratios.

\[\frac{AB}{A'B'}=\frac{BC}{B'C'}=\frac{CA}{C'A'}\]

Similar Quadrilaterals

Quadrilaterals are polygons that have four sides. The sum of the interior angles of a quadrilateral is 360 degrees. Two quadrilaterals are similar quadrilaterals when the three corresponding angles are the same( the fourth angles automatically become the same as the interior angle sum is 360 degrees), and two adjacent sides have equal ratios.

Are all squares similar?

Let us discuss the similarity of squares. According to the similarity of quadrilaterals, the corresponding angles of similar quadrilaterals should be equal. We know that all angles are 90 degrees in the square, so all the corresponding angles of any two squares will be the same.

Similarity of squares, all sides of a square are equal

All sides of a square are equal. If let’s say, square1 has a side length equal to ‘a’ and square2 has a side length equal to ‘b’, then all the corresponding sides' ratios will be the same and equivalent to a/b.

Hence, all squares are similar squares.

Are all rhombuses similar?

In a Rhombus, all the sides are equal. So, just like squares, rhombuses satisfy the condition of the ratio of corresponding sides being equal.

In Rhombus all the sides are equal

In a Rhombus, the opposite sides are parallel, and hence the opposite angles are equal. But the value of those angles can be anything. So, it can very much happen that two rhombuses have different angles. Hence, all rhombuses are not similar.

Similar Rectangles

Two rectangles are similar when the corresponding adjacent sides have the same ratio. We do not need to check the angles as all angles in a rectangle are 90 degrees.

Two rectangles are similar when the corresponding adjacent sides have same ratio.

In the above image, the ratios of the adjacent side are . Hence, these are similar rectangles.

Are all rectangles similar?

No, all rectangles are not similar rectangles. The ratio of the corresponding adjacent sides may be different. For example, let’s take a 1 by 2 rectangle and take another rectangle with dimensions 1 by 4. Here the ratios will not be equal.

\[\frac{1}{1}\ne\frac{4}{2}\]

Congruent rectangles

Two rectangles are called congruent rectangles if the corresponding adjacent sides are equal. It means they should have the same size. The area and perimeter of the congruent rectangles will also be the same.

Similarity and congruency are some important concepts of geometry. A solid understanding of these topics helps in building a good foundation in geometry. This article discussed the concepts of similarity in polygons looking at some specific cases of similar quadrilaterals like similar squares, similar rectangles, and similar rhombuses.

Frequently Asked Questions (FAQs)

What are similar polygons.

Two polygons are similar when the corresponding angles are equal/congruent, and the corresponding sides are in the same proportion.

If two rectangles have the same perimeter, are they congruent?

No, rectangles are not always congruent when they have the same perimeter. The ratio of lengths of corresponding sides may be different even when the perimeter is the same. For example, a rectangle of 5 by 4 and another rectangle of 6 by 3 has the same perimeter(equal to 18), but the corresponding sides' ratios are different \(\frac{5}{6}\ne\frac{4}{3}\).

Are all regular hexagons similar?

A regular hexagon is one with all equal sides, and since it is made of 6 equilateral triangles, all regular hexagons would be similar with equal angles but different sides measurements.

COMMENTS

  1. PDF Similar Polygons Date Period

    Kuta Software - Infinite Geometry Name_____ Similar Polygons Date_____ Period____ State if the polygons are similar. 1) 14 10 14 10 21 15 21 15 similar 2) 24 18 24 18 36 24 36 24 not similar 3) 5 7 5 7 40 ° 15 21 15 21 130 ° not similar 4) 40 20 40 20 100 ° 48 24 48 24 100 ° similar 5) 9.1 8 9.1 14 16.7 10 16.7 21 not similar 6) 12.4 20 12.4 28

  2. Similar Polygons (Explained w/ 23+ Step-by-Step Examples!)

    00:28:16 - Using the diagram and given proportion find the unknown length (Examples #15-16) 00:34:07 - Overview of scale factor. 00:36:50 - Determine if the polygons are similar. If yes, find the scale factor (Examples #17-22) 00:50:57 - Find the indicated measures for the given problems (Examples #23-24) Practice Problems with Step-by ...

  3. PDF Similar Triangles Date Period

    State if the triangles in each pair are similar. If so, state how you know they are similar and complete the similarity statement. 7) similar; SSS similarity; ∆QRS8) not similar. Find the missing length. The triangles in each pair are similar. Free trial available at KutaSoftware.com.

  4. 7.3: Similar Polygons and Scale Factors

    Scale Factors. Think about similar polygons as enlarging or shrinking the same shape. The symbol \sim\) is used to represent similarity. Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, all equilateral triangles are similar and all squares are similar. If two polygons are similar, we know the ...

  5. PDF Similarity Review Guide

    List the pairs of congruent angles and the extended proportion that relates the corresponding sides for the similar polygons. 9. ∆ ~∆ 10. ~ Determine whether the polygons are similar. If so, write a similarity statement and give the scale factor. If not explain. 11. J L K M N O G F D E O P N M A B D C . .

  6. PDF Chapter 10 Similar Polygons

    Theorem. If two polygons are similar, the ratio of their perimeters equals the ratio of the lengths of any two corresponding sides. To prove this theorem, all we have to do is use the last property of proportion we studied. As always, the best way to address any new information is by drawing and labeling a picture. A.

  7. PDF Similar Polygons

    a side length of 3 cm are similar. e) Any two parallelograms. f) A right triangle with legs of 3 ft and 4 ft and a right triangle with legs of 6 cm and 8 cm are similar. g) Any two isosceles triangles are similar. 3. Find the length of each missing side in the two similar polygons below. 4. The ratio of the perimeters of two hexagons is 5:4.

  8. PDF 7-Using Similar Polygons

    The polygons in each pair are similar. 15) 8 x − 2 42 63 49 49 16) x − 2 27 18 12 36 36 24 17) 30 6x − 6 42 35 63 49 18) 16 2x + 4 35 40 35 45 19) 3x + 11 A 42 B scale factor from A to B = 5 : 6 20) 30 A 3x B scale factor from A to B = 5 : 6 21) 14 A 8x − 7 B scale factor from A to B = 2 : 7 22) 48 A 8x B scale factor from A to B = 6 : 7-2-

  9. Answered: Geometry Assignment State if the…

    Solution for Geometry Assignment State if the polygons are similar. 1) 8.9 8.9 14 10 16.4 The 16.4 *ja eack pair ar 21

  10. Similarity

    Unit test. Test your understanding of Similarity with these NaN questions. Learn what it means for two figures to be similar, and how to determine whether two figures are similar or not. Use this concept to prove geometric theorems and solve some problems with polygons.

  11. PDF 7.2 Similar Polygons

    7.2 Similar Polygons 365 Goal Identify similar polygons. Key Words • similar polygons • scale factor 7.2 Similar Polygons TPRQ STSTU. a. List all pairs of congruent angles. b. Write the ratios of the corresponding sides in a statement of proportionality. c. Check that the ratios of corresponding sides are equal. Solution a. aP ca S, aR ca T ...

  12. Infinite Geometry

    Infinite Geometry covers all typical Geometry material, beginning with a review of important Algebra 1 concepts and going through transformations. There are over 85 topics in all, from multi-step equations to constructions. Suitable for any class with geometry content. Designed for all levels of learners, from remedial to advanced. Topics.

  13. PDF Chapter 8 Similarity

    8.1 Similar Polygons Understand the relationship between similar polygons. • I can use similarity statements. • I can find corresponding lengths in similar polygons. • I can find perimeters and areas of similar polygons. • I can decide whether polygons are similar. 8.2 Proving Triangle Similarity by AA Understand and use the Angle-Angle

  14. PDF Infinite Geometry

    Geometry: Chapter 7 - Similarity 7.2 Similar Polygons Name_____ ID: 1 Date_____ Period____ ©o h2K0C1M5o PKruvtYay jSkoFfvtvwZakrJeJ eLDLfCb.X _ PAilVlf FrciRglhDtbsW GryeSsweFryvferd_.-1-State if the polygons are similar. 1) 2.8 4 2.8 6 5.6 8 5.6 12 2) 18 12 18 12 24 16 24 16 3) 5 5 5 5 60° 9 15 9 15 120° 4) 4 6 4 6 6 12 6 12 5) 15 18 15

  15. PDF Using Similar Polygons Worksheet

    Author. KONICA MINOLTA bizhub PRO 951. Created Date. 11/19/2015 2:12:05 PM.

  16. How to Determine whether Polygons are Similar

    Step 1: Check that the angles of the two polygons are congruent. This means that the angles in the first shape should match the angles in the second shape. They should be exactly the same number ...

  17. Similar Polygons

    In mathematics, a pair of polygons are said to be similar when their: Corresponding angles are congruent. Corresponding sides are proportional. The symbol ∼ is commonly used to represent similarity. Let's take an example of two squares given below, Similar Polygons. The four interior angles of one square are identical to the other.

  18. THEOREMS-SIMILAR POLYGONS Flashcards

    Polygons. The smaller of two similar rectangles has dimensions 4 and 6. Find the dimensions of the larger rectangle if the ratio of the perimeters is 2 to 3. 6 by 9. The perimeters of two similar polygons are 20 and 28. One side of the smaller polygon is 4. Find the corresponding side of the larger polygon. 5 3/5.

  19. Similar Polygons, Rectangles, Quadrilaterals and More!

    Quadrilaterals are polygons that have four sides. The sum of the interior angles of a quadrilateral is 360 degrees. Two quadrilaterals are similar quadrilaterals when the three corresponding angles are the same ( the fourth angles automatically become the same as the interior angle sum is 360 degrees), and two adjacent sides have equal ratios.

  20. PDF Similar Polygons

    Math 9 HW Section 7.3 Similar Polygons 1. Given that the following polygons are similar, find the lengths of PT and DE. b) Find the lengths of BC and PT: ... Microsoft Word - Ch 7 Assignments Author: Danny Young Created Date: 3/12/2013 10:25:29 PM ...

  21. Geometry, assignment 7. Area Comparison of Polygons

    rectangles. Two similar polygons have areas of 50 and 100 sq. in. Find the ratio of the length of a pair of corresponding sides. √2/2. One side of a triangle is 15 inches, and the area of the triangle is 90 sq. inches. Find the area of a similar triangle in which the corresponding side is 9 inches. 150 sq. in.

  22. Geometry B Unit 3: Similarity Lesson 7: Similarity Unit Test

    The Empire State Building in New York City is 1,454 feet tall. A model of the building is 24 inches tall. ... math geometry theorems, definitions, postulates, axioms. 37 terms. daddymatthew222. Preview. ... The polygons are similar, but not necessarily drawn to scale. Find the value of x. Polygon 1: x - 3 , 8 , ...

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    Step Two - Angle Bisector Theorem. Next, it's time to apply the angle bisector theorem, which deals with relative lengths of polygons, to find the length of a missing side of a right triangle ...