Co-Interior Angles
When a transversal crosses two lines, the angles that are between the two lines and on the same side of the transversal are called co-interior angles. They are also called same-side interior angles, allied angles, or consecutive interior angles.
If the two lines are parallel, co-interior angles are supplementary — their sum is always 180°.
Think of the letter U — co-interior angles sit inside the U shape formed between the parallel lines on the same side of the transversal.
What is Co-Interior Angles - Grade 7 Maths (Lines and Angles)?
Definition: Co-interior angles are the pair of interior angles that lie on the same side of the transversal when it cuts two lines.
Theorem: If two parallel lines are cut by a transversal, each pair of co-interior angles is supplementary (sum = 180°).
Converse: If the co-interior angles add up to 180°, the lines are parallel.
- Co-interior angles are also called allied angles or same-side interior angles.
- They form a U-shape (or C-shape) between the lines.
Co-Interior Angles Formula
For parallel lines cut by a transversal:
Co-interior Angle 1 + Co-interior Angle 2 = 180°
There are 2 pairs of co-interior angles when a transversal crosses two lines.
Types and Properties
Identifying Co-interior Angles:
- Both angles are between the two lines (interior).
- Both are on the same side of the transversal.
- There are exactly 2 pairs of co-interior angles.
Comparison:
- Co-interior angles: same side, supplementary (sum 180°).
- Alternate interior angles: opposite sides, equal.
- Corresponding angles: same position at each intersection, equal.
Solved Examples
Example 1: Finding the Co-interior Angle
Problem: Lines l ∥ m. A transversal makes an interior angle of 70° on one side at line l. Find the co-interior angle at line m.
Solution:
- Co-interior angles sum to 180°.
- Angle at m = 180° − 70° = 110°.
Answer: The co-interior angle is 110°.
Example 2: Solving with Variables
Problem: Two parallel lines are cut by a transversal. The co-interior angles are (4x + 20)° and (2x + 40)°. Find both angles.
Solution:
- (4x + 20) + (2x + 40) = 180
- 6x + 60 = 180
- 6x = 120
- x = 20
- First angle = 4(20) + 20 = 100°
- Second angle = 2(20) + 40 = 80°
- Check: 100 + 80 = 180 ✓
Answer: The angles are 100° and 80°.
Example 3: Checking Parallel Lines
Problem: A transversal makes co-interior angles of 95° and 85° with two lines. Are the lines parallel?
Solution:
- Sum = 95 + 85 = 180°
- Since co-interior angles sum to 180°, the lines are parallel.
Answer: Yes, the lines are parallel.
Example 4: Ratio Problem
Problem: Co-interior angles of parallel lines are in the ratio 3:2. Find both angles.
Solution:
- Let angles = 3x and 2x
- 3x + 2x = 180°
- 5x = 180°
- x = 36°
- Angles: 108° and 72°
Answer: The co-interior angles are 108° and 72°.
Example 5: Not Parallel
Problem: Two lines are cut by a transversal making co-interior angles of 100° and 70°. Are the lines parallel?
Solution:
- Sum = 100 + 70 = 170° ≠ 180°
Answer: The lines are NOT parallel.
Example 6: Finding All Angles
Problem: Lines a ∥ b, transversal t. One angle is 125°. Find all co-interior angles.
Solution:
- The angle at line a = 125° (interior, one side).
- Co-interior angle at b (same side) = 180° − 125° = 55°.
- On the other side: the angle at a = 180° − 125° = 55° (linear pair).
- Co-interior angle at b (other side) = 180° − 55° = 125°.
Answer: Co-interior pairs: 125° and 55° on one side, 55° and 125° on the other.
Real-World Applications
Real-world uses:
- Architecture: Parallel walls cut by slanting supports create co-interior angles used in design.
- Navigation: When a boat crosses two parallel shipping lanes, co-interior angles help calculate the correct heading.
- Ladder placement: A ladder against a wall between parallel floor and ceiling creates co-interior angles.
Key Points to Remember
- Co-interior angles are between two lines on the same side of the transversal.
- If the lines are parallel, co-interior angles sum to 180°.
- If co-interior angles sum to 180°, the lines are parallel (converse).
- They are also called same-side interior angles, allied angles, or consecutive interior angles.
- They form a U-shape or C-shape.
- There are exactly 2 pairs of co-interior angles.
- Do NOT confuse with alternate interior angles (opposite sides, equal).
Practice Problems
- Parallel lines have co-interior angles x° and 120°. Find x.
- Co-interior angles are (3a − 10)° and (2a + 30)°. Lines are parallel. Find a.
- Are lines parallel if co-interior angles are 88° and 92°?
- Co-interior angles are in ratio 7:5. Find them.
- One co-interior angle is double the other. Find both.
Frequently Asked Questions
Q1. What are co-interior angles?
Co-interior angles are the pair of angles between two lines and on the same side of a transversal. When the lines are parallel, they add up to 180°.
Q2. What is another name for co-interior angles?
Same-side interior angles, allied angles, or consecutive interior angles. All refer to the same angle pair.
Q3. How are co-interior angles different from alternate interior angles?
Co-interior angles are on the same side of the transversal and are supplementary (sum 180°). Alternate interior angles are on opposite sides and are equal. Both properties hold only when the lines are parallel.
Q4. Can co-interior angles both be 90°?
Yes. If the transversal is perpendicular to both parallel lines, each co-interior angle is 90° and their sum is 180°.
Related Topics
- Corresponding Angles
- Alternate Interior Angles
- Transversal and Parallel Lines
- Supplementary Angles
- Complementary Angles
- Adjacent Angles
- Linear Pair of Angles
- Vertically Opposite Angles
- Angles on a Straight Line
- Angles at a Point
- Alternate Exterior Angles
- Proving Lines are Parallel
- Word Problems on Lines and Angles
