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22 Sept 2015 9:50 - 11:20
Geometry Agenda
Homework: 7.1 Proportions Review
Dilations Homework quiz
Similar Polygons (Similar Triangles)
Homework
YOUR NAME, Due: Tuesday, September 22.
Proportions Practice
p. 384 textbook
Qu. 18, 19, 20, 21, 22, 23, 24, 25, 28 and 29.
Write out the question each time and show all
the steps you take to solve the problem.
Check and correct (Document camera)
Dilations Homework quiz
Answer on the piece of paper.
The properties established with our dilation
will be found in all dilations. Every dilation
will create similar shapes represented with
the symbol
so for the picture, we would say
△ABC (is similar to) △A’B’C’
A’
A
C
C’
B
B’
Similar shapes have two major properties
that we found last time
1) Similar shapes have congruent
corresponding angles.
2) Similar shapes have proportional sides.
We generated similar shapes by
performing dilations on them.
7-2 Similar Polygons
The order of the vertices in a similarity
statement is important.
It identifies the corresponding angles and
the corresponding sides.
Example.
1) Determine whether the pair of triangles
is similar. Justify your answer.
F
B
6
4.5√3
4.5
12
E
30°
C
6√3
9
A
30°
D
Example.
1) If you can do a rotation, translation and
then a dilation to map one onto the other
F
B
6
4.5√3
4.5
12
E
30°
C
6√3
9
A
then the shapes are similar.
30°
D
Example.
1) Determine whether the pair of triangles
is similar. Justify your answer.
F
B
6
4.5√3
4.5
12
E
30°
C
6√3
9
A
30°
D
F
B
6
4.5√3
4.5
12
E
30°
C
9
A
30°
D
6√3
All right angles are congruent, so ∠C (is congruent to)∠F
Since m∠A = m∠D, ∠A (is congruent to)∠D
F
B
6
4.5√3
4.5
12
E
30°
C
9
A
30°
D
6√3
Third Angle Theorem
If two angles of one triangle are congruent to two angles of a
second triangle, then the third angles must also be
congruent, since the sum of the measures of the internal
angles of a triangle are always 180°. (p 211)
F
B
6
4.5√3
4.5
12
E
30°
C
9
30°
A
D
6√3
All right angles are congruent, so ∠C (is congruent to)∠F
Since m∠A = m∠D, ∠A (is congruent to)∠D
By the Third Angle Theorem, ∠B (is congruent to)∠E.
Thus, all corresponding angles are congruent.
Now...
F
B
6
4.5√3
4.5
12
E
30°
C
9
30°
A
6√3
Now determine whether the corresponding sides are
proportional.
Sides opposite 90° angle
Sides opposite 30°angle
AB = 12 or 1.3
BC = 6
or 1.3
DE 9
FE 4.5
D
F
B
6
4.5√3
4.5
12
E
30°
C
9
30°
A
6√3
Now determine whether the corresponding sides are
proportional.
Sides opposite 60° angle
AC =
DF
D
F
B
6
4.5√3
4.5
12
E
30°
C
9
A
30°
D
6√3
Now determine whether the corresponding sides are
proportional.
Sides opposite 60° angle
AC = 6√3 or 1.3
(We’ve found the scale factor for the
DF 4.5√3
dilation).
F
B
6
4.5√3
4.5
12
E
30°
C
9
A
30°
D
6√3
The ratios of the measures of corresponding sides are equal,
and the corresponding angles are congruent, so we have
shown that △ABC (is similar to) △DEF.
Note: When we compare the lengths of corresponding sides
of similar figures, we can find a numerical ratio - the scale
factor for the two figures.
Real World Example
Model Car
6.5” long
Car
13’ long
Scale factor of the model compared to the car?
Real World Example
We need both measurements to have the same
units. We know 1 foot = 12 inches
1 = 12”
Use this conversion factor
1’
Car
13’ x 12” = 156”
1’
Real World Example
Length of model = 6.5 inches
length of car
156 inches
=1
24
The scale factor is 1/24.
The model is 1/24 the length of the real car.
(Or the car is 24 times the length of the model).
Similar Triangles
Postulate: Angle-Angle (AA) Similarity
If the two angles of one triangle are congruent
to two angles of another triangle, then the
triangles are similar.
Example ∠P (is congruent to)∠T and ∠Q (is congruent
to)∠S, so △PQR (is similar to) △TSU.
Write using the correct symbols and draw the diagram that
illustrates this situation.
Theorems
Side-Side-Side (SSS) Similarity: If the
measures of the corresponding sides of two
triangles are proportional, then the triangles are
similar.
Example PQ = QR = RP
ST TU US ,so △PQR (is similar to) △STU.
Write using the correct symbols and draw the diagram that
illustrates this situation.
Side-Angle-Side (SAS) Similarity: If the
measures of two sides of a triangle are
proportional to the measures of two
corresponding sides of another triangle and the
included angles are congruent,then the triangles
are similar.
Example PQ = QR and ∠Q (is congruent to)∠T,
ST TU
so △PQR (is similar to) △STU.
Write using the correct symbols and draw the diagram that
illustrates this situation.
Similarity of triangles is:
● Reflexive △ABC (is similar to) △ABC.
● Symmetric If △ABC (is similar to) △DEF, then △DEF
(is similar to) △ABC.
● Transitive If △ABC (is similar to) △DEF and △DEF (is
similar to) △GHI, then △ABC (is similar to) △GHI.
G-SRT Are They Similar
Task.
In the picture below, line segments AD and BC
intersect at X. Line segments AB and CD are
drawn, forming two triangles AXB and CXD.
In each part (a)-(d) below, some additional
assumptions about the picture are given.
G-SRT Are They Similar
In each part (a)-(d) below, some additional
assumptions about the picture are given. In each
problem, determine whether the given
assumptions are enough to prove that the two
triangles are similar; and if so, what the correct
correspondence of vertices is. If the two triangles
must be similar, prove this result by describing a
sequence of similarity transformations that maps
one triangle on to the other. If not, explain why not.
G-SRT Are They Similar
(a)The lengths AX and XD satisfy the equation
2AX = 3XD.
Let’s draw the diagram and mark on exactly what
we know now.
G-SRT Are They Similar
(b)The lengths AX, BX, CX and DX satisfy the
equation
AX = DX
BX CX
Draw the new diagram and mark on exactly what
we know this time (note, what we were told in part
a) does not apply anymore).
Homework
Complete part b) of Are they similar worksheet.
Textbook 7-3 Similar Triangles
Check Understanding (p. 400) 1, 3, 5.
Exercises (p.401) 7, 10, 14, 21 (proof - does
not have to be a 2 column proof) and 26.