Math resources Geometry

Angles in parallel lines

Same side interior angles

# Same side interior angles

Here you will learn about same side interior angles. You’ll learn how to recognize same side interior angles, and apply this understanding to solve problems.

Students will first learn about same side interior angles as part of geometry in 8 th grade.

## What are same side interior angles?

Same side interior angles, also known as consecutive interior angles or co-interior angles, are pairs of angles that are on the same side of a transversal line and inside the two lines that the transversal intersects.

The two angles that occur on the same side of the transversal are supplementary, so they always add up to 180^{\circ}.

The two interior angles are only equal when they are both 90^{\circ}.

You can find the angle measure of one of the same side interior angles by subtracting the other from 180^{\circ}.

You can often spot same side interior angles by drawing a C shape.

## Common Core State Standards

How does this relate to 8 th grade math and high school math?

• Grade 8 – Geometry (8.G.A.5)
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

• High School – Geometry – Congruence (HS.G.CO.C.9)
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

## How to calculate with same side interior angles

In order to find a missing angle in parallel lines:

1. Highlight the angle(s) that you already know.
2. Use same side interior angles to find a missing angle.
3. Use basic angle facts if needed to calculate other missing angles.

Steps 2 and 3 may be done in either order and may need to be repeated. Step 3 may not always be required.

## Same side interior angles examples

### Example 1: same side interior angles

1. Highlight the angle(s) that you already know.

The given angle is 113^{\circ}.

2Use same side interior angles to find a missing angle.

Here, you can label the co-interior angle on the diagram as 60^{\circ} as 113+67=180.

This gives you the measure of the missing angle:

x=67^{\circ}

### Example 2: same side interior angles

Highlight the angle(s) that you already know.

Use same side interior angles to find a missing angle.

Use basic angle facts if needed to calculate other missing angles.

### Example 3: same side interior angles

Highlight the angle(s) that you already know.

Use same side interior angles to find a missing angle.

### Example 4: same side interior angles

Highlight the angle(s) that you already know.

Use same side interior angles to find a missing angle.

Use basic angle facts if needed to calculate other missing angles.

### Example 5: same side interior angles with algebra

Highlight the angle(s) that you already know.

Use same side interior angles to find a missing angle.

### Example 6: same side interior angles with algebra

Highlight the angle(s) that you already know.

Use same side interior angles to find a missing angle.

Use basic angle facts if needed to calculate other missing angles.

### Teaching tips for same side interior angles

• Begin by reviewing key angle relationships, such as complementary angles, supplementary angles, vertical angles, and adjacent angles. (Review can be given using a study guide or video lesson.) This helps students understand how same side interior angles fit into these broader categories, particularly their supplementary nature when lines are parallel.

• Start with a clear diagram showing two parallel lines cut by a transversal. You can use graph paper to make parallel lines easy to identify. Highlight the same side interior angles with different colors to help students easily identify them.

### Easy mistakes to make

• Mixing up angle facts
There are a lot of angle facts and it is easy to mistake alternate angles with corresponding angles. To prevent this from occurring, think about same side interior angles being inside the C shape.

• Using a protractor to measure an angle
Using a protractor to measure an angle. Most diagrams are not to scale and so using a protractor will not result in a correct answer unless it is a coincidence.

• Confusing with polygon angles
Mistaking angles in polygons, like those in a parallelogram, for same side interior angles. Same side interior angles specifically refer to angles formed by a transversal intersecting two lines, not just any interior angles within polygons.

### Practice same side interior angles questions

1. Calculate the value of \theta.

\theta =136^{\circ}

\theta =44^{\circ}

\theta =143^{\circ}

\theta =37^{\circ}

\theta and 44^{\circ} are same side interior angles therefore,

\theta = 180-44=136^{\circ}.

2. Calculate the value of \theta.

\theta =55^{\circ}

\theta =70^{\circ}

\theta =125^{\circ}

\theta =62.5^{\circ}

Using same side interior angles, you can calculate 180-125=55^{\circ}.

Using same side interior angles again, you can see that \theta=180-55=125^{\circ}.

3. Given the information in the diagram, calculate the size of angle \theta.

\theta =110^{\circ}

\theta =40^{\circ}

\theta =70^{\circ}

\theta =140^{\circ}

Using same side interior angles, you can calculate 180-110=70^{\circ}.

Since it is an isosceles triangle, the other angle at the bottom of the triangle is 70^{\circ}   as well.

Then, using angles in a triangle, \theta=180-(70+70)=40^{\circ}.

4. Calculate the value of \theta.

\theta =85^{\circ}

\theta =90^{\circ}

\theta =95^{\circ}

\theta =83^{\circ}

(7+\theta)^{\circ} and 88^{\circ} are same side interior therefore,

\begin{aligned}(7+ \theta) +88&=180 \\\\ \theta&=180-(88+7) \\\\ \theta&=85^{\circ} \end{aligned}

5. Find the value of \theta.

\theta =68^{\circ}

\theta =44^{\circ}

\theta =112^{\circ}

\theta =22^{\circ}

Since the triangle is an isosceles triangle, you know that the other angle at the top of the triangle is 68^{\circ}.

(\theta+68)^{\circ} and 68^{\circ} are same side interior angles therefore,

\begin{aligned}(\theta+68)+68&=180 \\\\ \theta&=180-(68+68) \\\\ \theta&=44^{\circ} \end{aligned}

6. By calculating the value for x, find the size of each angle labeled.

42^{\circ} and 38^{\circ}

30^{\circ} and 38^{\circ}

80^{\circ} and 100^{\circ}

78^{\circ} and 102^{\circ}

3x+12 and 2x+18 are same side interior angles, so they add up to 180^{\circ}. Therefore, you can write

\begin{aligned}3x+12+2x+18&=180 \\\\ 5x+30&=180 \\\\ 5x&=150 \\\\ x&=30 \end{aligned}

x=30 so

\begin{aligned} 3x+12&=3\times30+12=102^{\circ} \\\\ 2x+18&=2 \times 30+18=78^{\circ} \end{aligned}

## Same side interior angles FAQs

What are same side interior angles?

Same side interior angles are pairs of angles that lie on the same side of a transversal and are located between the two lines that the transversal intersects. These angles are also known as consecutive interior angles.

When are same side interior angles supplementary?

Same side interior angles are supplementary (their measures add up to 180^{\circ} ) when the two lines intersected by the transversal are parallel. This is a key property used to identify whether two lines are parallel.

How are same side interior angles different from alternate interior angles?

Same side interior angles are on the same side of the transversal, whereas alternate interior angles are on opposite sides of the transversal but still between the two intersected lines. Alternate interior angles are congruent when the lines are parallel, while same side interior angles are supplementary.

How are same side interior angles different from same side exterior angles?

Same side interior angles are inside the two lines intersected by a transversal, while same side exterior angles are outside the two lines. Both are on the same side of the transversal and are supplementary when the lines are parallel.

What is the converse of the same side interior angles theorem?

The converse states that if a pair of same side interior angles are supplementary, then the lines intersected by the transversal are parallel.

How are same side interior angles similar to linear pairs?

Same side interior angles and linear pairs are both supplementary, adding up to 180^{\circ}, but they differ in position. Linear pairs are adjacent angles formed by intersecting lines, while same side interior angles are non-adjacent angles on the same side of a transversal crossing two lines.

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