Similarity and Congruence
Warm-Up
1) Solve for x.
2) Solve for x:
𝑥 2 − 3𝑥 − 21 = 0
x4 9

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Congruent Figures
Two figures are congruent if they have the same shape and size. The
tessellation at right is produced by replicating the same figure over and
over. How do show mathematically that two figures are congruent?
Definition of Congruent Polygons
Two polygons are congruent polygons if and only if their corresponding
sides are congruent. Thus triangles that are the same size and shape are
congruent.
The symbol for “is congruent to” is ≅
Corresponding angles and corresponding sides are in the same position in polygons
with an equal number of sides.
Model Problem
Given polygon ABCD is congruent to polygon EFGH
The corresponding sides are congruent:
The corresponding angles are congruent:
____________
_____________
_____________
____________
____________ _____________
_____________
____________
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Exercise
Given triangle ABC is congruent to triangle DEF
List three pairs of congruent angles:
List three pairs of congruent sides:
__________________________
___________________________
__________________________
___________________________
__________________________
___________________________
Using Corresponding Parts of Congruent Triangles
Model Problem
Given: ∆ABC  ∆DBC
a) Find the value of x.
b) Find the measure of angle DBC.
Exercise
Given: ∆ABC  ∆DEF
a) Find the value of x.
b) Find the measure of angle E.
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Similar Polygons
When we order different-sized prints from a photography
studio, we expect the original image to be enlarged or
reduced without distortion. For example, if we divide the
width by 4 to make a smaller print, we must also divide
the length by 4 also.
A photo and its enlargement are an example of similar
polygons. Two figures that are similar have the same
shape but not necessarily the same size.
Definition of Similar Polygons
Two polygons are similar polygons if and only if their corresponding angles are
congruent and their corresponding side lengths are proportional. The side lengths are
proportional (or in proportion) if all pairs of corresponding sides are in the same ratio.
For example, the photos above are in proportion because both the length and width are
divided by 4. We can say the sides of the largest photo and the smallest photo are in a
4:1 ratio.
The symbol for “is similar to” is ~
More Examples
Similar Figures
Congruent Figures
Neither Similar Nor
Congruent
Critical Thinking
1) What gives congruent and similar figures their identical shapes?
2) Look at the above pictures of similar and congruent triangles. What part of the
triangle makes congruent triangles the same size but similar figures different
sizes?
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Using Corresponding Parts of Similar Polygons
Just like with congruent polygons, we can match up corresponding sides and angles.
Remember that corresponding angles are congruent and corresponding sides are in
proportion.
Given: ∆QRT ~ ∆NPM
Corresponding angles are CONGRUENT:
__________________________________
__________________________________
__________________________________
Corresponding sides are IN PROPORTION. We write this as equal ratios (fractions):
The similarity ratio is the ratio of the lengths of the corresponding sides of two similar
polygons.
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1
The similarity ratio of ∆ABC to ∆DEF is 6, or 2 .
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The similarity ratio of ∆DEF to ∆ABC is 3, or 2.
When writing the similarity statement, be sure to write
the congruent sides and angles in corresponding order.
Correct Way
Incorrect Way
Triangle VUT ~ triangle LKJ
Triangle VUT ~ triangle JLK
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Model Problem A
Determine whether the given polygons are similar. If so, write
the similarity ratio and a similarity statement.
1) Determine if corresponding sides are in proportion:
2) Determine if corresponding angles are congruent:
Model Problem B
Determine whether the given polygons are similar. If so, write
the similarity ratio and a similarity statement.
1) Determine if corresponding sides are in proportion:
2) Determine if corresponding angles are congruent:
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Exercise
Determine if ∆JLM ~ ∆NPS. If so, write the similarity ratio
and a similarity statement.
Lesson Quiz
1) Two figures are congruent if and only if their corresponding sides are
_________________ and their corresponding angles are _____________.
2) Two figures are similar if and only if their corresponding sides are
_________________ and their corresponding angles are _____________.
3) Draw two right triangles that are congruent, two right triangles that are
similar but not congruent, and two right triangles that are neither similar
nor congruent. Label sides and angles with numerical values.
4) Determine if the following figures are similar. Show mathematically why or
why not.
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Homework
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Similar Polygon Problems
Warm-up
1) Determine if the following polygons are similar.
2) Multiply and express your answer as a trinomial: (x – 2)(x + 6)
Solving Proportions
Recall that in similar figures, the corresponding sides are in proportion.
In a proportion, the cross-products are equal. That is, when
𝑎
𝑏
=
𝑐
𝑑
we have
ad = bc
Model Problems
A) A tree 24 feet tall casts a shadow 16 feet long at the same time a man 6 feet tall
casts a shadow x feet long. What is the length of the man’s shadow?
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B) In the accompanying diagram,
is similar to
,
, and
, if AB = 3, BC = 12, DE = x + 2, and EF = 18, find the value of x.
C)
Exercises
1) Find the length of the model to the nearest tenth of a centimeter.
2) A triangle has sides of length 3, 5, and 7. In a similar triangle, the shortest side has a
length of x – 3, and the longest side has a length of x + 5. Find the value of x.
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3) Two triangles are similar. If the lengths of the sides of the larger triangle are 12,
18, and 24 and the length of the shortest side of a similar triangle is 4, what is the
perimeter of the smaller triangle?
More Similar Triangle Problems
Model Problem A: Using Quadratics
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Model Problem B: Twisted Triangles
Exercise
1)
2)
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Homework
6)
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