Lesson 11.1 (581)

You know that figures that have the same shape and size are congruent figures.

Figures that have the same shape but not necessarily the same size are similar figures. To say that two figures have the same shape but not necessarily the same size is not, however, a precise definition of similarity.

Polygons are 'similar' if they are exactly the same shape, but can be different sizes.

Similar polygons have the same shape, but can be different sizes.

Two polygons are similar if two things are true:

1. The corresponding sides of each are in the same proportion 2. The corresponding interior angles are congruent.

Link:

Silimar Polygons Given two similar polygons ABCD and JKLM, we can write ABCD ~ JKLM which is read as "polygon ABCD is similar to polygon JKLM". The wavy line symbol means 'similar to'.

JRLeon Geometry Chapter 11.1

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Lesson 11.1 (581)

Figures that have the same shape and size

are congruent figures.

Figures that have the same shape but not necessarily the same size are similar figures. Polygons are 'similar' if they are exactly the same shape, but different sizes.

Two polygons are similar if:

1. The corresponding sides of each are in

the same proportion

2. The corresponding interior angles are congruent.

Link:

Silimar Polygons Given two similar polygons ABCD and JKLM, we can write ABCD ~ JKLM which is read as "polygon ABCD is similar to polygon JKLM". The wavy line symbol means 'similar to'.

Figuras que tienen la misma forma y tamaño

son figuras congruentes. Figuras que tienen la misma forma pero no necesariamente el mismo tamaño son figuras

similares.

Los polígonos son 'similares' si son

exactamente la misma forma , pero diferentes tamaños.

Dos polígonos son similares si:

1. Los lados correspondientes de cada uno

están en la misma proporción

2. Los ángulos interiores correspondientes son congruentes .

Dados dos polígonos semejantes ABCD y JKLM, podemos escribir ABCD ~ JKLM que se lee como "ABCD polígono es similar a JKLM polígono ". El símbolo de línea ondulada significa 'similar a' JRLeon Geometry Chapter 11.1

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Lesson 11.1

The statement CORN ~ PEAS says that quadrilateral CORN is similar to quadrilateral PEAS. Just as in statements of congruence, the order of the letters tells you which segments and which angles in the two polygons correspond.

Notice that the ratio of the lengths of any two segments in one polygon is equal to the ratio of the corresponding two segments in the similar polygon.

Observe que la relación de las longitudes de dos segmentos en un polígono es igual a la relación de los dos segmentos correspondientes en el polígono similar.

JRLeon Geometry Chapter 11.1

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Lesson 11.1

Do you need both conditions— congruent angles and proportional sides—to guarantee that the two polygons are similar?

If only the corresponding angles of two polygons are congruent, are the polygons similar?

JRLeon Geometry Chapter 11.1

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Lesson 11.1

Do you need both conditions— congruent angles and proportional sides—to guarantee that the two polygons are similar?

If corresponding sides of two polygons are proportional, are the polygons necessarily similar?

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Lesson 11.1

You can use the definition of similar polygons to find missing measures in similar polygons.

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Lesson 11.1

A transformation in which a polygon is enlarged or reduced by a given factor around a given center point.

Una transformación en el que un polígono se amplía o reduce por un factor dado alrededor de un punto central dado.

Dilation - of a polygon

JRLeon Geometry Chapter 11.1

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Lesson 11.2 (589)

In Lesson 11.1, you concluded that you must know about both the angles and the sides of two quadrilaterals in order to make a valid conclusion about their similarity.

However, triangles are unique. Recall from Chapter 4 that you found four shortcuts for triangle congruence: SSS, SAS, ASA, and SAA.

Are there shortcuts for triangle similarity as well? ( SSS, SAS, ASA, SAA or AAA)?

Let’s first look for shortcuts using only angles.

The figures below illustrate that you cannot conclude that two triangles are similar given that only one set of corresponding angles are congruent.

How about two sets of congruent angles?

JRLeon Geometry Chapter 11.1

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Lesson 11.2

The AA Similarity Postulate

The AA (angle angle) similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar

You know from the Triangle Sum Conjecture that:



mA +



mB +



mC = 180 °, and



mD +



mE +



mF =180 °. By the transitive property,



You also know that



mA =



mA +



mB +

mD, and

 

mC =



mB =



mE.

mD +



mE +



mF.

You can substitute for

 

mA +



mB +



mC =



mD and

mA +

 

mE in the longer equation to get:

mB +



mF.

Subtracting equal terms from both sides, you are left with



mC =



mF.

As you may have guessed, there is no need to investigate the AAA, ASA, or SAA Similarity Conjectures. Thanks to the Triangle Sum Conjecture, or more specifically the Third Angle Conjecture, the AA Similarity Conjecture is all you need.

JRLeon Geometry Chapter 11.1

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Lesson 11.2

Now let’s look for shortcuts for similarity that use only sides.

The figures below illustrate that you cannot conclude that two triangles are similar given that two sets of corresponding sides are proportional.

How about all three sets of corresponding sides?

JRLeon Geometry Chapter 11.1

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Lesson 11.2

SSS Similarity

Side-side-side similarity. When two triangles have corresponding sides with identical ratios as shown, the triangles are similar.

Three sides in proportion (SSS)

So SSS, AAA, ASA, and SAA are shortcuts for triangle similarity. That leaves SAS and SSA as possible shortcuts to consider.

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Lesson 11.2

SAS Similarity

Side-angle-side similarity. When two triangles have corresponding sides with identical ratios and the included angles are congruent as shown, the triangles are similar.

Two sides and included angle (SAS) JRLeon Geometry Chapter 11.1

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Lesson 11.2

One question remains: Is SSA a shortcut for similarity?

Recall from Chapter 4 that SSA did not work for congruence because you could create two different triangles.

Those two different triangles were neither congruent nor similar. So, no, SSA is not a shortcut for similarity.

JRLeon Geometry Chapter 11.1

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Lessons 11.1 – 11.2

Class work: 11.1- Pg. 585 – Problems 2 through 16 EVEN 11.2- Pg. 591 – Problems 2 through 16 EVEN Homework: 11.1- Pg. 585 – Problems 1 through 15 ODD 11.2- Pg. 591 – Problems 1 through 15 ODD

JRLeon Geometry Chapter 9.1-9.2

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