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# Equilateral Triangles

## Properties of triangles with three equal sides.

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Equilateral Triangles

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

Watch this video first.

Now watch this video.

Finally, watch this video.

### 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 $60^\circ$ 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 $\overline{AB} \cong \overline{BC} \cong \overline{AC}$ , then $\angle A \cong \angle B \cong \angle C$ . Conversely, if $\angle A \cong \angle B \cong \angle C$ , then $\overline{AB} \cong \overline{BC} \cong \overline{AC}$ .

#### 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 $y=7$ .

#### 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.

### Guided Practice

1. Find the measure of $y$ .

2. Fill in the proof:

Given : Equilateral $\triangle RST$ with

$\overline{RT} \cong \overline{ST} \cong \overline{RS}$

Prove : $\triangle RST$ is equiangular

Statement Reason
1. 1. Given
2. 2. Base Angles Theorem
3. 3. Base Angles Theorem
4. 4. Transitive PoC
5. $\triangle RST$ is equiangular 5.

3. True or false: All equilateral triangles are isosceles triangles.

1. The markings show that all angles are congruent. Since all three angles must add up to $180^\circ$ this means that each angle must equal $60^\circ$ . Write and solve an equation:

$8y +4 & = 60\\8y & = 56 \\y & =7$

2.

Statement Reason
1. $\overline{RT} \cong \overline{ST} \cong \overline{RS}$ 1. Given
2. $\angle{R} \cong \angle{S}$ 2. Base Angles Theorem
3. $\angle{T} \cong \angle{R}$ 3. Base Angles Theorem
4. $\angle{T} \cong \angle{S}$ 4. Transitive PoC
5. $\triangle RST$ is equiangular 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. Find the measures of $x$ and $y$ .