What if you were presented with two angles that are on the same side of the transversal and on the interior of the two lines? How would you describe these angles and what could you conclude about their measures? After completing this Concept, you'll be able to answer these questions and apply same side interior angle theorems to find the measure of unknown angles.
Watch This
CK-12 Same Side Interior Angles
Watch the portions of this video dealing with same side interior angles.
James Sousa: Angles and Transversals
Then watch this video.
James Sousa: Proof that Consecutive Interior Angles Are Supplementary
Finally, watch this video.
James Sousa: Proof of Consecutive Interior Angles Converse
Guidance
Same side interior angles are two angles that are on the same side of the transversal and on the interior of the two lines.
Same Side Interior Angles Theorem: If two parallel lines are cut by a transversal, then the same side interior angles are supplementary.
If \begin{align*}l || m\end{align*}, then \begin{align*}m\angle 1 + m\angle 2 = 180^\circ\end{align*}.
Converse of the Same Side Interior Angles Theorem: If two lines are cut by a transversal and the same side interior angles are supplementary, then the lines are parallel.
If then \begin{align*}l || m\end{align*}.
Example A
Find \begin{align*}z\end{align*}.
\begin{align*}z+116^\circ=180^\circ\end{align*} so \begin{align*}z = 64^\circ\end{align*} by Same Side Interior Angles Theorem.
Example B
Is \begin{align*}l || m\end{align*}? How do you know?
These angles are Same Side Interior Angles. So, if they add up to \begin{align*}180^\circ\end{align*}, then \begin{align*}l || m\end{align*}.
\begin{align*}130^\circ + 67^\circ = 197^\circ\end{align*}, therefore the lines are not parallel.
Example C
Give two examples of same side interior angles in the diagram below:
There are MANY examples of same side interior angles in the diagram. Two are \begin{align*}\angle 6\end{align*} and \begin{align*} \angle 10\end{align*}, and \begin{align*}\angle 8\end{align*} and \begin{align*}\angle 12\end{align*}.
CK-12 Same Side Interior Angles
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Guided Practice
1. Find the value of \begin{align*}x\end{align*}.
2. Find the value of \begin{align*}y\end{align*}.
3. Find the value of \begin{align*}x\end{align*} if \begin{align*}m\angle 3 = (3x +12)^\circ\end{align*} and \begin{align*}\ m\angle 5 = (5x + 8)^\circ\end{align*}.
Answers:
1. The given angles are same side interior angles. Because the lines are parallel, the angles add up to \begin{align*}180^\circ\end{align*}.
\begin{align*}(2x+43)^\circ + (2x-3)^\circ & = 180^\circ\\ (4x+40)^\circ & = 180^\circ\\ 4x & = 140\\ x & =35\end{align*}
2. \begin{align*}y\end{align*} is a same side interior angle with the marked right angle. This means that \begin{align*}90^\circ +y=180\end{align*} so \begin{align*}y=90\end{align*}.
3. These are same side interior angles so set up an equation and solve for \begin{align*}x\end{align*}. Remember that same side interior angles add up to \begin{align*}180^\circ\end{align*}.
\begin{align*}(3x+12)^\circ+(5x+8)^\circ&=180^\circ\\ (8x+20)^\circ&=180^\circ\\ 8x&=160\\x&=20\end{align*}
Explore More
For questions 1-2, determine if each angle pair below is congruent, supplementary or neither.
- \begin{align*}\angle 5\end{align*} and \begin{align*}\angle 8\end{align*}
- \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*}
- Are the lines below parallel? Justify your answer.
In 4-5, use the given information to determine which lines are parallel. If there are none, write none. Consider each question individually.
- \begin{align*}\angle AFD\end{align*} and \begin{align*}\angle BDF\end{align*} are supplementary
- \begin{align*}\angle DIJ\end{align*} and \begin{align*}\angle FJI\end{align*} are supplementary
For 6-8, what does the value of \begin{align*}x\end{align*} have to be to make the lines parallel?
- \begin{align*}m\angle 3 = (3x+25)^\circ\end{align*} and \begin{align*}m\angle 5 = (4x-55)^\circ\end{align*}
- \begin{align*}m\angle 4 = (2x+15)^\circ\end{align*} and \begin{align*}m\angle 6 = (3x-5)^\circ\end{align*}
- \begin{align*}m\angle 3 = (x+17)^\circ\end{align*} and \begin{align*}m\angle 5 = (3x-5)^\circ\end{align*}
For 9-10, determine whether the statement is true or false.
- Same side interior angles are on the same side of the transversal.
- Same side interior angles are congruent when lines are parallel.