Download PARALLEL LINES CUT BY A TRANSVERSAL

Similar Triangle Criteria
MCC9-12.G.SRT.2 Given two figures, use the definition of similarity
in terms of similarity transformations to decide if they are similar;
explain using similarity transformations the meaning of similarity for
triangles as the equality of all corresponding pairs of angles and the
proportionality of all corresponding pairs of sides.
MCC9-12.G.SRT.3 Use the properties of similarity transformations to
establish the AA criterion for two triangles to be similar
EQ: How do I know which method to use to prove two triangles
similar?
Warm Up
2
3
6
1
4
5
7 8
The m < 1 is 45 degrees, find the
remaining angles.
Vocabulary
 Angle: Angles are created by two distinct rays that share a common endpoint
(also known as a vertex). ∠ABC or ∠B denote angles with vertex B.
 Adjacent Angles: Angles in the same plane that have a common vertex
and a common side, but no common interior points.
 Alternate Exterior Angles : Alternate exterior angles are pairs
of angles formed when a third line (a transversal) crosses two other lines. These angles are
on opposite sides of the transversal and are outside the other two lines. When the two
other lines are parallel, the alternate exterior angles are equal.
 Alternate Interior Angles: Alternate interior angles are pairs of
angles formed when a third line (a transversal) crosses two other lines. These angles are
on opposite sides of the transversal and are in between the other two lines. When the two
other lines are parallel, the alternate interior angles are equal.
Vocabulary
 Complementary Angles: Two angles whose
sum is 90 degrees.
 Linear Pair: Adjacent, supplementary angles. Excluding their common
side, a linear pair forms a straight line.
 Supplementary Angles: Two angles whose sum is 180
degrees.
 Transversal: A line that crosses two or more lines.
 Vertical Angles: Two nonadjacent angles formed by intersecting lines or
segments. Also called opposite angles.
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 1 and < 2 are called SUPLEMENTARY ANGLES
They are a linear pair. ALL linear pairs are supplementary
(their measures add up to 180̊ ).
Name other supplementary pairs:
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 1 and < 3 are called VERTICAL ANGLES
They are congruent m<1 = m<3
Name other vertical pairs:
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 1 and < 5 are called CORRESPONDING ANGLES
They are congruent
m<1 = m<5
Corresponding angles occupy the same position on the top and
bottom parallel lines. Name other corresponding pairs:
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 4 and < 6 are called ALTERNATE INTERIOR ANGLES
They are congruent m<4 = m<6
Alternate Interior on the inside of the two parallel lines and on
opposite sides of the transversal. Name other alternate interior
angles.
Parallel lines cut by a transversal
2
3
6
1
4
5
7 8
< 4 and < 6 are called ALTERNATE EXTERIOR ANGLES
They are congruent m<2 = m<8
Alternate Interior on the inside of the two parallel lines and on
opposite sides of the transversal. Name other alternate exterior
angles.
TRY IT OUT
2
3
6
1
4
5
7 8
The m < 6 is 125 degrees, Find the rest
of the angles.
TRY IT OUT
2x + 20
What do you know about the angles?
Write the equation.
Solve for x.
x + 10
SUPPLEMENTARY
2x + 20 + x + 10 = 180
3x + 30 = 180
3x = 150
x = 30
TRY IT OUT
3x - 120
2x - 60
What do you know about the angles? ALTERNATE INTERIOR
Write the equation.
Solve for x.
Subtract 2x from both sides
Add 120 to both sides
3x - 120 = 2x - 60
x
=
60
Definition of Similar Triangles
 Two triangles are called similar if
they both have the same three angle
measurements.
 The two triangles shown are similar.
 Similar triangles have the same shape
but POSSIBLY different sizes.
 You can think of similar triangles as
one triangle being a magnification of
the other.
13
Similar Triangle Notation
 The two triangles shown are
C
similar because they have the same
three angle measures.
 The symbol for similarity is
Here we write:
ABC DEF
 The order of the letters is
important: corresponding letters
should name congruent angles.
55
A
45
E
80
55
F
14
80
45
B
D
Proportions from Similar Triangles
 Suppose
ABC DEF.
F
 Then the sides of the triangles are
proportional, which means:
AB AC BC


DE DF EF
E
D
C
 Notice that each ratio consists of
corresponding segments.
A
15
B
Example
ABC DEF , if the sides of
the triangles are as marked in the figure, find
the missing sides.
 First, we write:
AB AC BC


DE DF EF
 Then fill in the values:
AB 12 BC


7
8
6
 Then:
 Given that
16
12 21
12
A
AB  7  
 10.5 BC  6   9
8
2
8
F
8
D
6
7
C
12
E
9
10.5
B
 Let’s stress the order of the letters again.
When we write ABC DEF note
that the first letters are A and D, and
The second letters are B and E, and
The third letters are C and F, and
We can also write:
A  D.
B  E.
17
C  F.
but

BCA
DFE
orACB
not
CAB
FDE
BAC
EDF

DFE
C
55
A
45
E
80
55
F
80
45
B
D
Proving Triangles are Similar using
Angle Angle Similarity (AA) postulate
 One way to prove that two triangles
D
P
are similar, is to show that two pairs of
40
110
angles have the same measure.
 In the figure, ACB PTD.
30
 This is because the unmarked angles
are forced to have the same measure
C
T
because the three angles of any
triangle always add up to 180
30
 This is called the Third Angle
Theorem: If two angles of one
triangle are congruent to two angles
of another triangle, then the third
angles are also congruent.
110
40
A
18
B
When angles are congruent
 When trying to show that two triangles are similar, there are some






19
standard ways of establishing that a pair of angles (one from each
triangle) have the same measure:
They may be given to be congruent.
They may be vertical angles.
They may be the same angle (sometimes two triangles share an angle).
They may be a special pair of angles (like alternate interior angles)
related to parallel lines.
They may be in the same triangle opposite congruent sides.
There are numerous other ways of establishing a congruent pair of
angles.
Example 1
 In the figure,
 Show that
AB││ DE.
ABC EDC.
A
 First, note that ABC  EDC because
these are alternate interior angles.
 Also, BAC  DEC because these are
alternate interior angles too.
 This is enough to show the triangles are
similar, but notice the remaining pair of
angles are vertical.
20
B
C
D
E
Example 2
Explain why the triangles
are similar and write a
similarity statement.
By the Triangle Sum Theorem, mC = 47°, so C  F. B  E by
the Right Angle Congruence Theorem.
Therefore, ∆ABC ~ ∆DEF by AA ~.
Example 3
D
DAC and EBC
 are right angles, DA  12, and EB  9,
Find AB.
 First note that
BC  6. since
12
DAC EBC
 In the figure,
 and since the triangles share angle C.
E
9
DAC  EBC
 Let x denote AB. Then:
A x
22
DA AC
12 x  6
So, x  6  or
6   8. So, x  2.
EB BC
9
6
B
6
C
In addition to AA, triangles can also be proven similar using
the following criteria:
Example 4
Verify that the triangles are similar.
∆PQR and ∆STU
Therefore ∆PQR ~ ∆STU by SSS ~.
Example 5
Verify that the triangles are similar.
∆DEF and ∆HJK
D  H by the Definition of Congruent Angles.
Therefore ∆DEF ~ ∆HJK by SAS ~.
Example 6
Verify that ∆TXU ~ ∆VXW.
TXU  VXW by the Vertical
Angles Theorem.
Therefore ∆TXU ~ ∆VXW by SAS ~.
Lesson Quiz
1. Explain why the triangles are similar and
write a similarity statement.
2. Explain why the triangles are similar, then
find BE and CD.