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DAY 3 - SIMILARITY AND CONGRUENCE
Title
Goal
Congruence and Similarity
Apply the concepts of similarity and congruence to solve
problems
Standard
Addressed
Current Standards: 3MG2.2; 3.MG.2.3; 4MG3.3; 4MG3.7;
5MG1.1; 5MG2.1; 6MG 2.2
Materials for
Teacher
Materials for
Students
Description
Reflection
Looking Ahead
Common Core: 3.G.CA-1; 4.G.1; 5.G.CA-3; 5.G.CA-1; 7.G.5
Overhead, handouts, colored pencils, Geoboards, rubber
bands, ruler, compass, paper clips and Anglegs.
Lesson handouts, colored pencils, Geoboards, rubber bands,
ruler, compass, and Anglegs.
In this activity we discuss the ideas of similarity and
congruence. These two ideas have extensive real-world
applications and provide useful strategies to solve geometry
and algebra problems.
Similarity and congruence of triangles are important ideas in
geometric thinking. On one side they are useful ideas to solve
problems and on the other side they generalize to other
polygons. The ideas of similar and congruent polygons arise
naturally from triangle similarity and congruence.
Link to text
Thales (ca 624BC – 547BC) was also the first one who started thinking of the idea of similarity
and congruence of triangles. He used the idea of congruence to measure the distance of a ship to
the shore, and the idea of similarity to measure the height of the Pyramid of Giza in Egypt.
MEASURING THE DISTANCE OF A SHIP TO THE SHORE
Thales used the idea of ASA (Angle-Side-Angle) to measure the distance of a ship to the shore.
He would go on top of a lighthouse with a device that looked like a compass. Then he would
position one leg of the compass on the top of the lighthouse and the other leg in alignment with
the ship. Next, he would identify a point on the shore (by rotating the compass without affecting
the angle) that could be measured. That was the distance that they needed. This useful idea
helped people in war because it allowed them to know how far a ship was from the shore.
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Now, what are the mathematical pieces that Thales used? As we mentioned earlier he used the
idea of ASA (Angle-Side-Angle). The first angle is the angle determined by the compass, the
side is the height of the lighthouse and the compass, and the second angle is the right angle
formed by the lighthouse and the shore.
Thales had discovered that for two triangles to be congruent (meaning to be equal in all pieces),
he only needed to check three pieces: two angles and the side between those angles.
CONGRUENCE OF TRIANGLES
Two triangles are said to be congruent if they are identical copies of each other. This means that
their three sides AND their three angles match when we overlap them. Learning to recognize
congruent shapes is an important component of geometric thinking. Students have trouble
recognizing congruent triangles because they are not typically in the same position. Fortunately
we do not have to check the three angles AND the three sides of the two triangles to declare
them congruent. We only need to check three pieces. Once we know that these corresponding
three pieces match, the other pieces of the triangle also match. This is very useful when solving
problems. Below we give the conditions for Congruence of Triangles.
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I. SSS, Side-Side-Side
Two triangles are congruent if their three (corresponding) sides are congruent. We illustrate this
with Anglegs. Make two triangles, each with a blue strip, a red strip and a yellow strip. Do the
triangles look the same? How could we check that they are the same?
What colored sides match?
What colored angles match?
Notice that in using SSS, Side-Side-Side we never checked any of the angles. They
automatically match. This SSS criterion is very useful when we have the measures of segments
and know nothing about angles.
Problems
1. Are these two triangles congruent? Color the corresponding parts with the same color.
E
2
D
C
4
5
5
4
F
B
2
A
4
2. Are these two triangles congruent? Explain. Color the corresponding parts with the same
color.
E
F
1
C
1
A
5
5
3
3
D
B
3. Are triangles FAD and FCD congruent triangles? Color the corresponding parts with the same
color. Explain. Illustrate this with Anglegs. An important idea to notice here is that of common
side.
F
C
1
1
A
3
3
D
4. Are these triangles congruent? Explain. Illustrate with Anglegs.
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6
6
6
5
5. Do the following triangles look congruent? If so, what would the corresponding parts be?
Color the corresponding parts with the same color? What could we do to check they are
congruent?
6. Explain why in any rectangle RSTU (see figure), the diagonals RT and SU have the same
length (in geometry this means that segments RT and SU are congruent). Next, use this fact to
explain why triangles RST and TUR are congruent. Find two other triangles congruent to
triangle RST.
R
b
U
a
a
S
b
T
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II. A-S-A, Angle-Side-Angle
This is the idea that Thales used to measure the distance from the ship to the shore. In this case,
we check two angles and the side between those angles. In the example below, we know the
angles CAB and DBA, and the length of segment AB (in this case, 8). Extend segments AC and
BD using a ruler. Call the point of intersection E. Notice that this point of intersection is
uniquely determined (since two lines intersect at one point). This point determines the lengths of
side AE and BE, and one single triangle is determined.
A
C
8
B
D
It is important to notice that we automatically know the measure of the third angle of the triangle.
Sometimes that information is irrelevant, but in other cases, it becomes a key piece in solving a
problem.
Problems
1. Are these two triangles congruent?
4
71°
88°
88°
4
71°
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2. Are these two triangles congruent?
39°
12
122°
39°
122°
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3. Are the following triangles congruent?
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48°
48°
75°
57°
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3. In rectangle RSTU, we constructed the diagonals RT and SU. We labeled the intersection of
the diagonals, X. Are triangles RXU and TXS congruent? What about triangles UXT and SXR?
U
R
64°
64°
8
X
64°
S
64°
15
T
8
3. S-A-S, Side-Angle-Side
This is the counterpart for A-S-A. In SAS, Side-Angle-Side, we check the angle between two
known sides. Notice that in this case, the length of the third side is automatically determined for
there is only one segment that satisfies these conditions. When completing this triangle (with
colored pencil) you will notice that there is only one option for completing it.
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α
2
Problems
1. Consider rectangle RSTU. Are triangles RST and TUR congruent?
U
24
R
64°
10
10
64°
S
T
24
2. Quadrilateral PQRS is a rhombus (ie, all sides are congruent). Are triangles SQR and QSP
congruent?
R
S
45
°
°
45
Q
P
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3. Quadrilateral PQRS is a kite (see figure). Explain notation. Are triangles PRS and PRQ
congruent?
S
119°
P
R
119°
Q
Similarity
Thales (ca 624BC – 547BC) visited Egypt and asked how high the pyramid of Giza was.
Nobody but Thales was able to figure this out. He positioned himself during the day until his
shadow was equal to his height. Then he knew that the shadow of the pyramid was equal to its
height.
More specifically Thales was comparing two similar triangles. On one side he has the pyramid,
its shadow and the angle between those measurements, and on the other side he is standing with
his shadow and the angle between them. He observed that the sides were proportional. This is
the main idea of similarity.
Pyramid
Thales
shadow
shadow
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Two triangles are similar if they are exact copies of each other but with sides that are not the
same length. This idea is well understood when we think of miniatures, models, or re-sizing on a
Xerox machine. You may think of similar figures as dilated or shrunk.
1.64” by 1.46”
0.82” by 0.73”
When two triangles are similar their angles are the same, and their corresponding sides are in the
same proportion. In the following figure side AB corresponds to side A’B’, side BC corresponds
to side B’C’ and side CA corresponds to side C’A’. Notice that the corresponding sides are in
the same proportion. The sides of triangle ABC are half of the corresponding sides of triangle
A’B’C’.
A'
A
4
2
B
8
4
5
C
B'
10
C'
As with congruence, we only have to check three parts to determine if two triangles are similar.
The pieces that we have to check are the same as with congruence:
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SSS- Side-Side-Side Similarity
Two triangles are similar if their corresponding sides are in the same proportion. Triangles ACB
and PRQ are isosceles. Fill out the following ratios to conclude that the triangles are similar.
C
9
9
Q
1.5
3
P
3
R
A
4.5
B
Notice that we automatically know that the corresponding angles must be congruent. Color the
corresponding angles with the same color.
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SAS, Side-Angle-Side Similarity
In this case, we check that two sides are in the same proportion, and that the angle between those
sides is the same. In the following figure we can observe two triangles, ABC and A’BC’.
Measures are given as follow:
AB = 6
BC = 9
A’B = 4
BC’ = 6
We will use this information to show that the triangles are similar. First, notice that they have a
common angle. Which one is it? Next, let’s identify the corresponding sides. Write them.
Next, let’s compute the ratio of the corresponding sides. Can we conclude that the triangles are
similar?
C
C'
A
A'
B
Notice that we automatically know that the third side is in the same proportion as the other two
sides AND that the corresponding angles are congruent. Color the corresponding angles with the
same color.
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AA, Angle-Angle Similarity
As the name indicates, in this case we only have to check two angles. For instance, the isosceles
triangles ABC and DEF are similar. Why?
C
A
E
108°
36°
F
D
B
Notice that automatically we know that the corresponding sides of the triangles are in the same
proportion.
Problems
1. Show that if two triangles are similar their heights are in the same proportion as their sides.
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2. In the following figure we are given right triangle ACB with right angle ACB. Show that
triangles ACB, CDB and ADC are similar.
C
A
D
B
3. If CB = 3, CA = 4 and AB = 5 in the problem above, what is the measure of CD?
4. In the problem above, what is the area of each triangle?
5. In rectangle ABCD, E is the midpoint of side AB, and F is the intersection of diagonal AC and
segment DE. If AB = 4 and BC = 2, determine the area of triangles AEF, CDF, AFD, and
quadrilateral FEBC.
C
D
F
A
E
B