Download Geometry - Chapter 18 Similar Triangles Key Concepts

Geometry - Chapter 18
Similar Triangles
Key Concepts
This lesson is devoted to similar triangles. We are going to learn lots of new theorems
and applications on how to solve and identify similar triangles. This lesson is a lot like
the lesson where we proved that triangles are congruent. It relates also to our last lesson,
so we should breeze right through it!
Section 18-1
Like congruent triangles, there are lots of theorems and corollaries to help us identify
similar triangles. That is what section 18-1 is all about. Understanding the theorems is a
crucial part of this section. Let’s try a few problems to help us better understand the
properties of similar triangles.
Theorem 18-1
AAA Similarity Theorem
If the angles of one triangle are congruent to the angles of another triangle,
then the triangles are similar.
Corollary 18-2
AA Similarity Corollary
If two angles of one triangle are congruent to two angles of a second
triangle, then the triangles are similar.
Corollary 18-3
Right-Triangle Similarity Corollary
If two right triangles have an acute angle of one congruent to an acute
angle of the other, then the triangles are similar.
Corollary 18-4
Parallel Line Similarity Corollary
If a line parallel to one side of a triangle determines a second triangle, then
the second triangle will be similar to the original triangle.
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Problem:
Q
Given m SQ = 12 and TU SR
Find TS.
Solution:
∆QTU and ∆QSR are similar triangles.
Therefore, all of their corresponding sides
have a ratio of 1 to 4 since we know that two
of their corresponding sides (TU and SR) have this
ratio (2 to 8 reduces to 1 to 4). So, if SQ = 12, then
TQ = 3. Then
TS = 12 – 3 = 9.
T
2
S
U
R
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Problem:
Given:
Prove:
PL
ML
=
PO MN
L
P
∆LMP is similar to ∆LNO
O
Solution:
We need to set up a two-column proof.
Statements
1.
M
PL
LM
=
PO MN
N
Reasons
1. Given
2. PM ON
2. By Corollary 17-4 which says, if a line
divides two sides of a triangle proportionally
then the line is parallel to the third side.
3. ∆LMP is similar to ∆LNO
3. By Corollary 18-4 which says, if a line
is parallel to one side of a triangle
determines a second triangle, then the
second triangle will be similar to the original
triangle.
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Section 18-2
In this section, we are going to study similar triangles a little more deeply. Like
discovering whether two triangles are congruent, there are theorems that help us to
discover if two triangles are similar. It is important to look over the theorems and
corollaries first, then try the practice problems before you get started on your submission
problems.
Theorem 18-5
SAS Similarity Theorem
If an angle of one triangle is congruent to an angle of another, and the
sides including these angles are proportional, then the triangles are similar.
Theorem 18-6
SSS Similarity Theorem
If the corresponding sides of two triangles are proportional, then the
triangles are similar.
Corollary 18-7
Proportional Perimeters Corollary
The perimeters of two similar triangles are proportional to any pair of
corresponding sides.
Corollary 18-8
Proportional Altitudes Corollary
The altitudes of similar triangles are proportional to any pair of
corresponding sides.
Corollary 18-9
Proportional Medians Corollary
The medians of similar triangles are proportional to any pair of
corresponding sides.
Problem:
If m∠A = 35
Find m∠1, m∠2& m∠3 ,
B
3
D
2
1
Solution:
C
A
m∠3 = 180 − 90 − 35 = 55
m∠2 = 180 − 90 − 55 = 35
m∠1 = 90 − 35 = 55
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That is the end of the first semester. It was kind of fun, wasn’t it? Now it’s time to get
ready for your semester exam. You should review all of your submissions to be sure you
understand how to do ALL the problems. Ask for help if you don’t understand! Review
all your theorems, corollaries, postulates, and definitions. If you’ve been keeping up with
your lessons and submissions all semester, you won’t have any difficulty with this exam.
Good luck!
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