3.5: Alternate Interior Angles
- Page ID
- 2188
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Angles on opposite sides of a transversal, but inside the lines it intersects.
Alternate interior angles are two angles that are on the interior of \(l\) and \(m\), but on opposite sides of the transversal.
![f-d_b313250db66fc135445ed3dd3c50506c7d5bf3c4faf0bb904d13e994+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/1212/f-d_b313250db66fc135445ed3dd3c50506c7d5bf3c4faf0bb904d13e994%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
![f-d_a3314f04c041c4e4e7cd77c12a9dadec3926b523462deb2726d9c47a+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/1213/f-d_a3314f04c041c4e4e7cd77c12a9dadec3926b523462deb2726d9c47a%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
If \(l\parallel m\), then \(\angle 1\cong \angle 2\)
Converse of Alternate Interior Angles Theorem: If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.
If
![f-d_7adc3b2e74fc08ba224a323728ef32dca9f0b0042fec18750610e3c4+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/1214/f-d_7adc3b2e74fc08ba224a323728ef32dca9f0b0042fec18750610e3c4%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
then \(l\parallel m\).
What if you were presented with two angles that are on the interior of two parallel lines cut by a transversal but on opposite sides of the transversal? How would you describe these angles and what could you conclude about their measures?
For Examples \(\PageIndex{1}\) and \(\PageIndex{2}\), use the given information to determine which lines are parallel. If there are none, write none. Consider each question individually.
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Example \(\PageIndex{1}\)
\(\angle EAF\cong \angle FJI\)
Solution
None
Example \(\PageIndex{2}\)
\(\angle EFJ\cong \angle FJK\)
Solution
\(\overleftarrow{CG} \parallel \overleftarrow{HK}\)
Example \(\PageIndex{3}\)
Find the value of \(x\).
![f-d_ca38ab0710d052576dd2f77a26c9dbd478f26fe51964cfe9e67c547c+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/1216/f-d_ca38ab0710d052576dd2f77a26c9dbd478f26fe51964cfe9e67c547c%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
Solution
The two given angles are alternate interior angles and equal.
\(\begin{align*} (4x−10)^{\circ} &=58^{\circ}\\ 4x &=68 \\ x &=17 \end{align*}\)
Example \(\PageIndex{4}\)
True or false: alternate interior angles are always congruent.
Solution
This statement is false, but is a common misconception. Remember that alternate interior angles are only congruent when the lines are parallel.
Example \(\PageIndex{5}\)
The angles are alternate interior angles, and must be equal for \(a\parallel b\). Set the expressions equal to each other and solve.
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Solution
\(\begin{align*} 3x+16^{\circ} &=5x−54^{\circ} \\ 70&=2x \\ 35 &=x\end{align*}\)
To make \(a\parallel b\), \(x=35\).
Review
- Is the angle pair \(\angle 6\) and \(\angle 3\) congruent, supplementary or neither?
Figure \(\PageIndex{7}\) - Give two examples of alternate interior angles in the diagram:
Figure \(\PageIndex{8}\)
For 3-4, find the values of \(x\).
-
Figure \(\PageIndex{9}\)
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Figure \(\PageIndex{10}\)
For question 5, use the picture below. Find the value of \(x\).
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- \(m \angle 4=(5x−33)^{\circ}\), \(m \angle 5=(2x+60)^{\circ}\)
- Are lines \(l\) and \(m\) parallel? If yes, how do you know?
Figure \(\PageIndex{12}\)
For 7-10, what does the value of \(x\) have to be to make the lines parallel?
![f-d_963dbdfc7b826d8117ff8f1171f6c6bf93e5c8d514fd8352bb797b67+IMAGE_TINY+IMAGE_TINY.png](https://k12.libretexts.org/@api/deki/files/1224/f-d_963dbdfc7b826d8117ff8f1171f6c6bf93e5c8d514fd8352bb797b67%252BIMAGE_TINY%252BIMAGE_TINY.png?revision=1&size=bestfit&width=450)
- \(m \angle 4=(3x−7)^{\circ}\) and \(m \angle 5=(5x−21)^{\circ}\)
- \(m \angle 3=(2x−1)^{\circ}\) and \(m \angle 6=(4x−11)^{\circ}\)
- \(m \angle 3=(5x−2)^{\circ}\) and \(m \angle 6=(3x)^{\circ}\)
- \(m \angle 4=(x−7)^{\circ}\) and \(m \angle 5=(5x−31)^{\circ}\)
Review (Answers)
To see the Review answers, open this PDF file and look for section 3.4.
Resources
Vocabulary
Term | Definition |
---|---|
alternate interior angles | Alternate interior angles are two angles that are on the interior of two different lines, but on the opposite sides of the transversal. |
alternate exterior angles | Alternate exterior angles are two angles that are on the exterior of two different lines, but on the opposite sides of the transversal. |
Additional Resources
Video: Alternate Interior Angles Principles - Basic
Activities: Alternate Interior Angles Discussion Questions
Study Aids: Angles and Transversals Study Guide
Practice: Alternate Interior Angles
Real World: Alternate Interior Angles