4.5: Equilateral Triangles
- Page ID
- 2174
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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](https://k12.libretexts.org/@api/deki/files/1360/f-d_8f7eede1204302418e1424f330dc88db79a35babad10dfc3d161f618%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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](https://k12.libretexts.org/@api/deki/files/1361/f-d_9ddf3fb72ca471f7e81104a9c522b71b3f419c20b14f5684e902a7c4%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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](https://k12.libretexts.org/@api/deki/files/1362/f-d_95e02acc8d554078a2d006667bd2925fa2f21c2b786bcb58dd134570%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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](https://k12.libretexts.org/@api/deki/files/1363/f-d_757dfef69dbac7f1274f934a7c4376800b17932f95c014c09206e2cf%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
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.
Review
The following triangles are equilateral triangles. Solve for the unknown variables.
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Figure \(\PageIndex{17}\) - Find the measures of \(x\) and \(y\).
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Review (Answers)
To see the Review answers, open this PDF file and look for section 4.11.
Additional Resources
Interactive Element
Video: Equilateral Triangles Principles - Basic
Activities: Equilateral Triangles Discussion Questions
Study Aids: Equilateral Triangles Discussion Questions
Practice: Equilateral Triangles
Real World: Equilateral Triangles