What if you were presented with two angles that are on the same side of a transversal, but inside the 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 using your knowledge of same side interior angles.
Watch This
CK-12 Foundation: Chapter3SameSideInteriorAnglesA
Watch the portions of this video dealing with same side interior angles.
James Sousa: Angles and Transversals
James Sousa: Proof that Consecutive Interior Angles Are Supplementary
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. \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 5\end{align*} are same side interior angles.
Same Side Interior Angles Theorem: If two parallel lines are cut by a transversal, then the same side interior angles are supplementary.
So, if \begin{align*}l \ || \ m\end{align*} and both are cut by \begin{align*}t\end{align*}, then \begin{align*}m \angle 3 + m \angle 5 = 180^\circ\end{align*} and \begin{align*}m \angle 4 + m \angle 6 = 180^\circ\end{align*}.
Converse of the Same Side Interior Angles Theorem: If two lines are cut by a transversal and the consecutive interior angles are supplementary, then the lines are parallel.
Example A
Using the picture above, list all the pairs of same side interior angles.
Same Side Interior Angles: \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 6\end{align*}, \begin{align*}\angle 5\end{align*} and \begin{align*}\angle 3\end{align*}.
Example B
Find \begin{align*}m \angle 2\end{align*}.
Here, \begin{align*}m \angle 1 = 66^\circ\end{align*} because they are alternate interior angles. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are a linear pair, so they are supplementary.
\begin{align*}m\angle 1 + m\angle 2 & = 180^\circ\\ 66^\circ + m\angle 2 & = 180^\circ\\ m\angle 2 & = 114^\circ\end{align*}
This example shows why if two parallel lines are cut by a transversal, the same side interior angles are supplementary.
Example C
Find the measure of \begin{align*}x\end{align*}.
The given angles are same side interior angles. The lines are parallel, therefore the angles add up to \begin{align*}180^\circ\end{align*}. Write an equation.
\begin{align*}(2x+43)^\circ + (2x-3)^\circ & = 180^\circ\\ (4x+40)^\circ & = 180^\circ\\ 4x & = 140^\circ\\ x & = 35^\circ\end{align*}
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter3SameSideInteriorAnglesB
Vocabulary
Same Side Interior Angles are two angles that are on the same side of the transversal and on the interior of the two lines. Two angles are supplementary if they add to \begin{align*}180^\circ\end{align*}.
Guided Practice
1. Is \begin{align*}l \ || \ m\end{align*}? How do you know?
2. Find the value of \begin{align*}x\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. These 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*}113^\circ + 67^\circ = 180^\circ\end{align*}, therefore \begin{align*}l \ || \ m\end{align*}.
2. 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*}
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*}
Interactive Practice
Practice
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 6-11, 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*}
- \begin{align*}m\angle 4 = (3x+12)^\circ\end{align*} and \begin{align*}m\angle 6 = (4x-1)^\circ\end{align*}
- \begin{align*}m\angle 3 = (2x+14)^\circ\end{align*} and \begin{align*}m\angle 5 = (3x-2)^\circ\end{align*}
- \begin{align*}m\angle 4 = (5x+16)^\circ\end{align*} and \begin{align*}m\angle 6 = (7x-4)^\circ\end{align*}
For 12-13, 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.
For questions 14-15, use the picture below.
- What is the same side interior angle with \begin{align*}\angle 3\end{align*}?
- Are the lines parallel? Explain.