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Equilateral Triangles
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
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Printed: November 21, 2012
AUTHORS
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
EDITOR
Annamaria Farbizio
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C ONCEPT
Concept 1. Equilateral Triangles
1
Equilateral Triangles
Here you’ll learn the definition of an equilateral triangle as well as an important theorem about equilateral triangles:
Equilateral triangles are always equiangular.
What if you were presented with an equilateral triangle and told that its sides measure x, y, and 8? What could you
conclude about x and y? After completing this Concept, you’ll be able to apply important properties about equilateral
triangles to help you solve problems like this one.
Watch This
MEDIA
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CK-12 Equilateral Triangles
Watch this video first.
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James Sousa:Constructing anEquilateralTriangle
Now watch this video.
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James Sousa:EquilateralTriangles Theorem
Finally, watch this video.
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James Sousa:Using the Properties ofEquilateralTriangles
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Guidance
All sides in an equilateral triangle have the same length. One important property of equilateral triangles is that
all of their angles are congruent (and thus 60◦ each). This is called the Equilateral Triangle Theorem and can be
derived from the Base Angles Theorem.
Equilateral Triangle Theorem: All equilateral triangles are also equiangular. Furthermore, all equiangular triangles are also equilateral.
If AB ∼
= BC ∼
= AC, then 6 A ∼
=6 B∼
= 6 C. Conversely, if 6 A ∼
=6 B∼
= 6 C, then AB ∼
= BC ∼
= AC.
Example A
Find the value of x.
Solution: Because this is an equilateral triangle 3x − 1 = 11. Solve for x.
3x − 1 = 11
3x = 12
x=4
Example B
Find the values of x and y.
The markings show that this is an equilateral triangle since all sides are congruent. This means all sides must equal
10. We have x = 10 and y + 3 = 10 which means that y = 7.
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Concept 1. Equilateral Triangles
Example C
Two sides of an equilateral triangle are 2x + 5 units and x + 13 units. How long is each side of this triangle?
The two given sides must be equal because this is an equilateral triangle. Write and solve the equation for x.
2x + 5 = x + 13
x=8
To figure out how long each side is, plug in 8 for x in either of the original expressions. 2(8) + 5 = 21. Each side is
21 units.
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CK-12 Equilateral Triangles
Guided Practice
1. Find the measure of y.
2. Fill in the proof:
Given: Equilateral 4RST with
RT ∼
= ST ∼
= RS
Prove: 4RST is equiangular
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TABLE 1.1:
Statement
1.
2.
3.
4.
5. 4RST is equiangular
Reason
1. Given
2. Base Angles Theorem
3. Base Angles Theorem
4. Transitive PoC
5.
3. True or false: All equilateral triangles are isosceles triangles.
Answers:
1. The markings show that all angles are congruent. Since all three angles must add up to 180◦ this means that each
angle must equal 60◦ . Write and solve an equation:
8y + 4 = 60
8y = 56
y=7
2.
TABLE 1.2:
Statement
1. RT ∼
= ST ∼
= RS
6
2. 6 R ∼
S
=
3. 6 T ∼
=6 R
4. 6 T ∼
=6 S
5. 4RST is equiangular
Reason
1. Given
2. Base Angles Theorem
3. Base Angles Theorem
4. Transitive PoC
5. Definition of equiangular.
3. This statement is true. The definition of an isosceles triangle is a triangle with at least two congruent sides. Since
all equilateral triangles have three congruent sides, they fit the definition of an isosceles triangle.
Practice
The following triangles are equilateral triangles. Solve for the unknown variables.
1.
4
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Concept 1. Equilateral Triangles
2.
3.
4.
5.
5
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6.
7.
8.
9.
6
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Concept 1. Equilateral Triangles
10.
11.
12.
13.
7
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14.
15. Find the measures of x and y.
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