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
HGSH
Lesson 11.1 (581)
Figures that have the same shape and size
are congruent figures.
Figuras que tienen la misma forma y tamaño
son figuras congruentes.
Figures that have the same shape but not
necessarily the same size are similar figures.
Figuras que tienen la misma forma pero no
necesariamente el mismo tamaño son figuras
similares.
Polygons are 'similar' if they are exactly the
same shape, but different sizes.
Two polygons are similar if:
Los polígonos son 'similares' si son
exactamente la misma forma , pero
diferentes tamaños.
Dos polígonos son similares si:
1. The corresponding sides of each are in
the same proportion
1. Los lados correspondientes de cada uno
están en la misma proporción
2. The corresponding interior angles are
congruent.
2. Los ángulos interiores correspondientes
son congruentes .
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
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'
Geometry Chapter 11.1
HGSH
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
HGSH
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
HGSH
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?
JRLeon
Geometry Chapter 11.1
HGSH
Lesson 11.1
You can use the definition of similar polygons to find missing measures in similar
polygons.
JRLeon
Geometry Chapter 11.1
HGSH
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
HGSH
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
HGSH
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,  mA + mB + mC = mD + mE + mF.
You also know that  mA = mD, and  mB = mE.
You can substitute for  mD and  mE in the longer equation to get:
 mA + mB + mC = mA + 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
HGSH
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
HGSH
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.
JRLeon
Geometry Chapter 11.1
HGSH
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
HGSH
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
HGSH
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
HGSH