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Warm up
Congruence vs. Similarity
Polygon
Polygon
Congruence Similarity
Triangle
Triangle
Congruence Similarity
Shortcuts
Shortcuts
1. Need to prove
that corresponding
angles are
congruent
1. Need to prove
that corresponding
angles are
congruent
SSS
SAS
ASA
AAS
HL
2. Need to prove
that corresponding
sides are congruent
2. Need to prove
that corresponding
sides are
proportional
AA~
SAS~
SSS~
Let’s practice our new knowledge of similarity shortcuts.
•Determine whether the triangles are similar.
•If so, explain why then write a similarity statement and name the
postulate or theorem you used.
•If not, explain.
Yes, they are similar because
AB || CD (given) which creates
congruent alternate interior angles:
<A and <D are congruent as are
<B and <C.
Therefore: ABE ~ DCE
AA~
Yes, they are similar because
LM || OP (given) which creates
congruent corresponding angles:
<L and <O are congruent as are
<M and <P.
Therefore: LMN ~ OPN
AA~
Let’s practice some more.
Yes, MNL~ QPO
The sides are all in the same proportion,
having the scale factor of 2.
SSS~
Not similar.
There is no way of knowing if the
corresponding angles are congruent,
or if the corresponding sides are
proportional. Not enough information
is given for us to make a certain
determination.
Next… Similarity in right triangles
Please get:
♥One piece of colored paper
♥Straight edge
♥Scissors
♥Your compass
♥Your pencil
1. Draw a diagonal
5. Cut out the three triangles.
2. You have created two right triangles =)
6. How can you match the angles of the
triangles to show that all three triangles
are similar?
3. In one triangle, draw the altitude from
the right angle to the hypotenuse
4. Number the angles as shown
1
7. Explain how you know the matching
angles are congruent.
7
6 5
8
2
9
4
3
Essential Understanding:
When you draw the altitude to the hypotenuse of a right
triangle, you form three pairs of similar right triangles.
7
6 5
1
8
2
9
4
3
Angles 2, 8 and 9 are all congruent – right angles are congruent
Angles 1 and 4 are congruent – alternate interior angles in a parallelogram are
congruent
Angles 3 and 7 are congruent – alternate interior angles in a parallelogram are
congruent
Essential Understanding:
When you draw the altitude to the hypotenuse of a right
triangle, you form three pairs of similar right triangles.
What similarity statement can you write relating
the three triangles in the diagram?
X
XYZ ~ YWZ ~ XWY
Z
W
Y
Geometric Mean
Example
A proportion in which the means are equal.
What is the geometric mean of 6 and 15?
6=x
x 15
x2 = 90
x = √90
9 ▪ 10
3▪3▪2▪5
x = 3 √10
a=x
x b
Your turn:
What is the
geometric mean
of 4 and 10?
4=x
x 10
x2 = 40
x = √40
4 ▪ 10
2▪2▪2▪5
x = 2 √10
What is the
geometric mean
of 3 and 16?
3=x
x 16
x2 = 48
x = √48
6▪8
2▪3▪2▪2▪2
x = 4 √3
Solve for x and y…. (Pull the triangles apart)
X
12 = x
x 9
y=3
9 y
3
W
x = 6 √3
y
Y
X
y = 3 √3
9
Z
x
12
Y
x
X
y
Y
ZW
x
3
9
Z
W
y
Y
Solve for x and y…. (Pull the triangles apart)
50
40
X = 20
x
Y = 10√5
y
X
50
Y
y
X
x
Y
Z W
10
40
Z
W
y
x
Y
Your assignment
7.3 and 7.4 Practice
worksheets
Cross off: 7 & 8 on pg 23
10, 14, 15, 16, 17 on pg 24
and
32 – 38 on pg 34