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# Equilateral Triangles

## Properties of triangles with three equal sides.

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Equilateral Triangles

What if your parents want to redo the bathroom? Below is the tile they would like to place in the shower. The blue and green triangles are all equilateral. What type of polygon is dark blue outlined figure? Can you determine how many degrees are in each of these figures? Can you determine how many degrees are around a point?

### Equilateral Triangles

By definition, all sides in an equilateral triangle have exactly the same length.

#### Investigation: Constructing an Equilateral Triangle

Tools Needed: pencil, paper, compass, ruler, protractor

1. Because all the sides of an equilateral triangle are equal, pick a length to be all the sides of the triangle. Measure this length and draw it horizontally on your paper.

2. Put the pointer of your compass on the left endpoint of the line you drew in Step 1. Open the compass to be the same width as this line. Make an arc above the line.

3. Repeat Step 2 on the right endpoint.

4. Connect each endpoint with the arc intersections to make the equilateral triangle.

Use the protractor to measure each angle of your constructed equilateral triangle. What do you notice?

From the Base Angles Theorem, the angles opposite congruent sides in an isosceles triangle are congruent. So, if all three sides of the triangle are congruent, then all of the angles are congruent or 60\begin{align*}60^\circ\end{align*} each.

Equilateral Triangles Theorem: All equilateral triangles are also equiangular. Also, all equiangular triangles are also equilateral.

#### Solving for Variables

1. Find the value of x\begin{align*}x\end{align*}.

Because this is an equilateral triangle 3x1=11\begin{align*}3x-1=11\end{align*}. Now, we have an equation, solve for x\begin{align*}x\end{align*}.

3x13xx=11=12=4\begin{align*}3x-1 &= 11\\ 3x &= 12\\ x &= 4\end{align*}

2. Find the values of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}.

Let’s start with y\begin{align*}y\end{align*}. Both sides are equal, so set the two expressions equal to each other and solve for y\begin{align*}y\end{align*}.

5y13yy=2y+11=12=4\begin{align*}5y-1 &= 2y+11\\ 3y &= 12\\ y &= 4\end{align*}

For x\begin{align*}x\end{align*}, we need to use two (2x+5)\begin{align*}(2x + 5)^\circ\end{align*} expressions because this is an isosceles triangle and that is the base angle measurement. Set all the angles equal to 180\begin{align*}180^\circ\end{align*} and solve.

(2x+5)+(2x+5)+(3x5)(7x+5)7xx=180=180=175=25\begin{align*}(2x+5)^\circ+(2x+5)^\circ+(3x-5)^\circ &= 180^\circ\\ (7x+5)^\circ &= 180^\circ\\ 7x &= 175^\circ\\ x &= 25^\circ\end{align*}

#### Measuring the Sides of an Equilateral Triangle

Two sides of an equilateral triangle are 2x+5\begin{align*}2x+5\end{align*} units and x+13\begin{align*}x+13\end{align*} 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\begin{align*}x\end{align*}.

2x+5x=x+13=8\begin{align*} 2x+5 &= x+13 \\ x &= 8\end{align*}

To figure out how long each side is, plug in 8\begin{align*}8\end{align*} for x\begin{align*}x\end{align*} in either of the original expressions. 2(8)+5=21\begin{align*}2(8)+5=21\end{align*}. Each side is 21\begin{align*}21\end{align*} units.

Bathroom Tile Problem Revisited

Let’s focus on one tile. First, these triangles are all equilateral, so this is an equilateral hexagon (6 sided polygon). Second, we now know that every equilateral triangle is also equiangular, so every triangle within this tile has 360\begin{align*}360^\circ\end{align*} angles. This makes our equilateral hexagon also equiangular, with each angle measuring 120\begin{align*}120^\circ\end{align*}. Because there are 6 angles, the sum of the angles in a hexagon are 6.120\begin{align*}6.120^\circ\end{align*} or 720\begin{align*}720^\circ\end{align*}. Finally, the point in the center of this tile, has 660\begin{align*}660^\circ\end{align*} angles around it. That means there are 360\begin{align*}360^\circ\end{align*} around a point.

### Examples

#### Example 1

Find the measure of y\begin{align*}y\end{align*}.

The marking show that all angles are congruent. Since all three angles must add up to 180\begin{align*}180^\circ\end{align*} this means that each angle must equal 60\begin{align*}60^\circ\end{align*}. Write and solve an equation:

8y+48yy=60=56=7\begin{align*}8y +4 & = 60\\ 8y & = 56 \\ y & =7\end{align*}

#### Example 2

Fill in the proof:

Given: Equilateral RST\begin{align*}\triangle RST\end{align*} with

RT¯¯¯¯¯¯¯ST¯¯¯¯¯¯¯RS¯¯¯¯¯¯¯\begin{align*}\overline{RT} \cong \overline{ST} \cong \overline{RS}\end{align*}

Prove: RST\begin{align*}\triangle RST\end{align*} is equiangular

Statement Reason
1. 1. Given
2. 2. Base Angles Theorem
3. 3. Base Angles Theorem
4. 4. Transitive PoC
5. RST\begin{align*}\triangle RST\end{align*} is equiangular 5.
Statement Reason
1. RT¯¯¯¯¯¯¯ST¯¯¯¯¯¯¯RS¯¯¯¯¯¯¯\begin{align*}\overline{RT} \cong \overline{ST} \cong \overline{RS}\end{align*} 1. Given
2. RS\begin{align*}\angle{R} \cong \angle{S}\end{align*} 2. Base Angles Theorem
3. TR\begin{align*}\angle{T} \cong \angle{R}\end{align*} 3. Base Angles Theorem
4. TS\begin{align*}\angle{T} \cong \angle{S}\end{align*} 4. Transitive PoC
5. RST\begin{align*}\triangle RST\end{align*} is equiangular 5. Definition of equiangular.

#### Example 3

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

### Review

The following triangles are equilateral triangles. Solve for the unknown variables.

1. Find the measures of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}.

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