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Lesson: Similar Triangles
供稿人: Pappal
Congruent triangles
Congruent triangles
are triangles that have the same size and shape.
Congruent triangles
are triangles that have the same size and shape.
Congruent triangles
Similar triangles
are triangles that have the same size and shape.
Congruent triangles
Similar triangles
are triangles that have the same size and shape.
Congruent triangles
Similar triangles
have the same shape but different sizes.
Congruent
?
Similar
?
How do we know when triangles are similar?
How do we know when triangles are similar?
2
5
4
1
2.5
2
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
2
5
4
1
2.5
2
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
2
5
4
1
2.5
2
These are congruent
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
2
5
4
1
2.5
2
These are congruent
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
2
5
4
1
2.5
2
These are congruent
Their corresponding sides have to be proportional.
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
2
5
4
1
2.5
2
Their corresponding sides have to be proportional.
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
(Proportional means they have to divide to the
same thing.)
2
5
4
1
2.5
2
Their corresponding sides have to be proportional.
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
2
5
4
1
2.5
2
4 ÷ 2 = 2
Their corresponding sides have to be proportional.
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
2
5
4
1
2.5
2
4 ÷ 2 = 2
2 ÷ 1 = 2
Their corresponding sides have to be proportional.
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
2
5
4
1
2.5
2
4 ÷ 2 = 2
2 ÷ 1 = 2
5 ÷ 2.5 = 2
Their corresponding sides have to be proportional.
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
2
5
4
1
2.5
2
4 ÷ 2 = 2
2 ÷ 1 = 2
5 ÷ 2.5 = 2
Their corresponding sides have to be proportional.
How do we know when triangles are similar?
Their corresponding angles have to be congruent.
2
5
4
1
2.5
2
4 ÷ 2 = 2
2 ÷ 1 = 2
5 ÷ 2.5 = 2
All the same
For shapes to be similar
their corresponding angles must be
their corresponding sides must be
proportional
?
congruent
?
We can use similarity
We can use similarity
to find unknown sides
We can use similarity
to find unknown sides
2
5
x
10
We can use similarity
to find unknown sides
2
5
x
10
corresponding sides
These are
We can use similarity
to find unknown sides
2
5
x
10
corresponding sides
These are
We can use similarity
to find unknown sides
2
5
x
10
2
x
=
10
5
We can use similarity
to find unknown sides
2
5
x
10
Solve
2
x
=
10
5
We can use similarity
to find unknown sides
2
5
x
10
Solve
2
x
5x = 20
=
10
5
We can use similarity
to find unknown sides
2
5
x
10
Solve
2
x
5x = 20
x = 4
=
10
5
x
x =
5
8
10
There are three ways to prove trangles similar
There are three ways to prove trangles similar
AA
There are three ways to prove trangles similar
sAs
AA
There are three ways to prove trangles similar
sAs
sss
AA
For AA you have to have two sets of angles congruent
For AA you have to have two sets of angles congruent
These are congruent
For AA you have to have two sets of angles congruent
These are congruent
and these are too
For sAs one set of angles needs to be congruent
10
6
5
3
For sAs one set of angles needs to be congruent
These are congruent
10
6
5
3
For sAs one set of angles needs to be congruent
and the sides making up that angle must
10
6
5
3
proportional
For sAs one set of angles needs to be congruent
and the sides making up that angle must
10
6
5
3
10∕5 = 2
proportional
For sAs one set of angles needs to be congruent
and the sides making up that angle must
10
6
5
3
10∕5 = 2
6/3 = 2
proportional
For sAs one set of angles needs to be congruent
and the sides making up that angle must
10
6
5
3
10∕5 = 2
6/3 = 2
proportional
sAs
AA
Side Angle Side
?
Angle Angle
?
For sss all the side need to be proportional
For sss all the side need to be proportional
10
4
8
5
2
4
For sss all the side need to be proportional
10
4
8
5
2
4
10/5 = 2
For sss all the side need to be proportional
10
4
8
5
2
4
10/5 = 2
8/4 = 2
For sss all the side need to be proportional
10
4
8
5
2
4
10/5 = 2
8/4 = 2
4/2 = 2
9
3
sss
?
4
6
1
3
1
sAs
?
2
AA
?
12
3
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