4.5: Equilateral Triangles

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Properties of triangles with three equal sides.

Equilateral Triangle Theorem: All equilateral triangles are also equiangular. Furthermore, all equiangular triangles are also equilateral.

f-d_8f7eede1204302418e1424f330dc88db79a35babad10dfc3d161f618+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{1}\)

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 \(\overline{AB}\cong \overline{BC}\cong \overline{AC}\).

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 \(y\)?

Example \(\PageIndex{1}\)

Fill in the proof:

Given: Equilateral \(\Delta RST\) with

\(\overline{RT}\cong \overline{ST}\cong \overline{RS}\)

Prove: \(\Delta RST\) is equiangular

f-d_9ddf3fb72ca471f7e81104a9c522b71b3f419c20b14f5684e902a7c4+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{2}\)

Solution

*Statement*	*Reason*
1.	1. Given
2.	2. Base Angles Theorem
3.	3. Base Angles Theorem
4.	4. Transitive \(PoC\)
5. \(\Delta RST\) is equiangular	5.

*Statement*	*Reason*
1. \(RT\overline{AB}\cong ST\overline{AB}\cong RS\overline{AB}\)	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. \(\Delta RST\) is equiangular	5. Definition of equiangular.

Example \(\PageIndex{2}\)

True or false: All equilateral triangles are isosceles triangles.

Solution

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.

Example \(\PageIndex{3}\)

Find the value of \(x\).

f-d_95e02acc8d554078a2d006667bd2925fa2f21c2b786bcb58dd134570+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{3}\)

Solution

Because this is an equilateral triangle \(3x−1=11\). Solve for \(x\).

\(\bgin{align*} 3x−1&=11 \\3x&=12 \\ x&=4 \end{align*}\)

Example \(\PageIndex{4}\)

Find the values of \(x\) and \(y\).

f-d_757dfef69dbac7f1274f934a7c4376800b17932f95c014c09206e2cf+IMAGE_TINY+IMAGE_TINY.png — Figure \(\PageIndex{3}\)

Solution

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 \(\PageIndex{5}\)

Two sides of an equilateral triangle are \(2x+5\) units and \(x+13\) units. How long is each side of this triangle?

Solution

The two given sides must be equal because this is an equilateral triangle. Write and solve the equation for \(x\).

\(\egin{align*}2x+5 &=x+13 \\ x&=8 \end{align*}\)

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.