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6.6. Theorems Involving Similarity
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6.6 Theorems Involving Similarity
Here you will use similar triangles to prove new theorems about triangles.
Can you find any similar triangles in the picture below?
Watch This
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http://www.youtube.com/watch?v=YLtQmuTY5rI James Sousa: Triangle Proportionality Theorem
Guidance
If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional. There are many theorems about triangles that you can prove using similar triangles.
1. Triangle Proportionality Theorem: A line parallel to one side of a triangle divides the other two sides of the
triangle proportionally. This theorem and its converse will be explored and proved in Example A, Example B,
and the practice exercises.
2. Triangle Angle Bisector Theorem: The angle bisector of one angle of a triangle divides the opposite side of
the triangle into segments proportional to the lengths of the other two sides of the triangle. This theorem will
be explored and proved in Example C.
3. Pythagorean Theorem: For a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2 . This theorem
will be explored and proved in the Guided Practice problems.
Example A
Prove that ∆ADE ∼ ∆ABC.
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Chapter 6. Similarity
Solution: The two triangles share 6 A. Because DEkBC, corresponding angles are congruent. Therefore, 6 ADE ∼
=
6 ABC. The two triangles have two pairs of congruent angles. Therefore, ∆ADE ∼ ∆ABC by AA ∼.
Example B
Use your result from Example A to prove that
AB
AD
=
AC
AE .
EC
AE .
AB
Therefore, AD
Then, use algebra to show that
DB
AD
=
AC
Solution: ∆ADE ∼ ∆ABC which means that corresponding sides are proportional.
= AE
. Now, you
can use algebra to show that the second proportion must be true. Remember that AB = AD + DB and AC = AE + EC.
AB
AD
AD + DB
→
AD
DB
→ 1+
AD
DB
→
AD
AC
AE
AE + EC
=
AE
EC
= 1+
AE
EC
=
AE
=
You have now proved the triangle proportionality theorem: a line parallel to one side of a triangle divides the
other two sides of the triangle proportionally.
Example C
Consider ∆ABC with AE the angle bisector of 6 BAC and point D constructed so that DCkAE. Prove that
EB
BA
=
EC
CA .
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6.6. Theorems Involving Similarity
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Solution: By the triangle proportionality theorem,
EC
BA
EB
EC
=
BA
AD .
Multiply both sides of this proportion by
EC
BA .
EB
BA
EC
·
=
·
EC AD
BA
EB EC
→
=
BA AD
Now all you need to show is that AD = CA in order to prove the desired result.
•
•
•
•
Because AE is the angle bisector of 6 BAC, 6 BAE ∼
= 6 EAC.
∼
Because DCkAE, 6 BAE = 6 BDC (corresponding angles).
Because DCkAE, 6 EAC ∼
= 6 DCA (alternate interior angles).
∼
Thus, 6 BDC = 6 DCA by the transitive property.
Therefore, ∆ADC is isosceles because its base angles are congruent and it must be true that AD ∼
= CA. This means
that AD = CA. Therefore:
EB
BA
=
EC
CA
This proves the triangle angle bisector theorem: the angle bisector of one angle of a triangle divides the opposite
side of the triangle into segments proportional to the lengths of the other two sides of the triangle.
Concept Problem Revisited
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Chapter 6. Similarity
There are three triangles in this picture: ∆BAC, ∆BCD, ∆CAD. All three triangles are right triangles so they have one
set of congruent angles (the right angle). ∆BAC and ∆BCD share 6 B, so ∆BAC ∼ ∆BCD by AA ∼. Similarly, ∆BAC
and ∆CAD share 6 C, so ∆BAC ∼ ∆CAD by AA ∼. By the transitive property, all three triangles must be similar to
one another.
Vocabulary
A similarity transformation is one or more rigid transformations followed by a dilation.
Two figures are similar if a similarity transformation will carry one figure to the other. Similar figures will always
have corresponding angles congruent and corresponding sides proportional.
The Pythagorean Theorem states that for a right triangle with legs a and b and hypotenuse c, a2 + b2 = c2 .
An angle bisector divides an angle into two congruent angles.
Guided Practice
The large triangle above has sides a, b, and c. Side c has been divided into two parts: y and c − y. In the Concept
Problem Revisited you showed that the three triangles in this picture are similar.
1. Explain why
a
c
=
2. Explain why
b
c
=
c−y
a .
y
b.
3. Use the results from #1 and #2 to show that a2 + b2 = c2 .
Answers:
1. When triangles are similar, corresponding sides are proportional. Carefully match corresponding sides and you
see that ac = c−y
a .
2. When triangles are similar, corresponding sides are proportional. Carefully match corresponding sides and you
see that bc = by .
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3. Cross multiply to rewrite each equation. Then, add the two equations together.
a c−y
=
→ a2 = c2 − cy
c
a
b y
= → b2 = cy
c b
→ a2 + b2 = c2 − cy + cy
→ a2 + b2 = c2
You have just proved the Pythagorean Theorem using similar triangles.
Practice
Solve for x in each problem.
1.
2.
3.
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Chapter 6. Similarity
4.
5.
6.
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6.6. Theorems Involving Similarity
7.
Use the picture below for #8-#10.
8. Solve for x.
9. Solve for z.
10. Solve for y.
Use the picture below for #11-#13.
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11. Assume that
Chapter 6. Similarity
b
a
= dc . Use algebra to show that
b+a
a
=
d+c
c .
12. Prove that ∆Y ST ∼ ∆Y XZ
13. Prove that ST kXZ
14. Prove that a segment that connects the midpoints of two sides of a triangle will be parallel to the third side of the
triangle.
15. Prove the Pythagorean Theorem using the picture below.
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6.7. Applications of Similar Triangles
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6.7 Applications of Similar Triangles
Here you will solve problems involving similar triangles and learn about special right triangles.
Michael is 6 feet tall and is standing outside next to his younger sister. He notices that he can see both of their
shadows and decides to measure each shadow. His shadow is 8 feet long and his sister’s shadow is 5 feet long. How
tall is Michael’s sister?
Watch This
MEDIA
Click image to the left for more content.
http://www.youtube.com/watch?v=LhEe0kB4QIs James Sousa: Indirect Measurement Using Similar Triangles
Guidance
If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional. There are three criteria for proving that triangles are similar:
1. AA: If two triangles have two pairs of congruent angles, then the triangles are similar.
2. SAS: If two sides of one triangle are proportional to two sides of another triangle and their included angles
are congruent, then the triangles are similar.
3. SSS: If three sides of one triangle are proportional to three sides of another triangle, then the triangles are
similar.
Once you know that two triangles are similar, you can use the fact that their corresponding sides are proportional
and their corresponding angles are congruent to solve problems. In the Examples and practice, you will consider
many different applications of similar triangles.
Example A
a) Prove that the two triangles below are similar.
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