Corresponding Angles
When a straight line (called a transversal) cuts across two parallel lines, it creates several pairs of angles. One important pair is called corresponding angles.
Corresponding angles are in the same position at each intersection. Think of them as being in matching corners — both on the top-left, or both on the bottom-right, for example.
In Class 7 NCERT Maths, you will learn to identify corresponding angles and use the property that corresponding angles are equal when the lines are parallel.
What is Corresponding Angles - Grade 7 Maths (Lines and Angles)?
Definition: When a transversal cuts two lines, corresponding angles are the angles that are on the same side of the transversal and in the same position (both above or both below) at each intersection point.
Key property:
If two parallel lines are cut by a transversal, then each pair of corresponding angles is equal.
The converse is also true: if a transversal cuts two lines and the corresponding angles are equal, then the lines are parallel.
Corresponding Angles Formula
Identifying corresponding angle pairs:
When a transversal cuts two lines at points P and Q, it creates 8 angles (4 at each point). The corresponding angle pairs are:
- Angle 1 and Angle 5 (both above the line, on the left of the transversal)
- Angle 2 and Angle 6 (both above the line, on the right of the transversal)
- Angle 3 and Angle 7 (both below the line, on the left of the transversal)
- Angle 4 and Angle 8 (both below the line, on the right of the transversal)
Property: If the lines are parallel, then:
- Angle 1 = Angle 5
- Angle 2 = Angle 6
- Angle 3 = Angle 7
- Angle 4 = Angle 8
Types and Properties
Related angle pairs formed by a transversal:
- Corresponding angles: Same position at each intersection (equal if lines are parallel).
- Alternate interior angles: On opposite sides of the transversal, between the two lines (equal if lines are parallel).
- Alternate exterior angles: On opposite sides of the transversal, outside the two lines (equal if lines are parallel).
- Co-interior angles (same-side interior): On the same side of the transversal, between the two lines (add up to 180° if lines are parallel).
Solved Examples
Example 1: Finding a Corresponding Angle
Problem: Two parallel lines are cut by a transversal. One of the corresponding angles is 65°. Find the other.
Solution:
Corresponding angles are equal when lines are parallel.
The other corresponding angle = 65°.
Answer: The corresponding angle is 65°.
Example 2: Finding an Unknown Angle
Problem: In the figure, line AB is parallel to line CD. A transversal makes an angle of 110° at AB. Find the corresponding angle at CD.
Solution:
Since AB is parallel to CD, corresponding angles are equal.
Corresponding angle at CD = 110°.
Answer: The angle is 110°.
Example 3: Using Corresponding Angles and Linear Pair
Problem: Lines l and m are parallel. A transversal makes an angle of 72° with line l (above, left of transversal). Find all four angles at line m.
Solution:
Let the angle at l (above-left) = 72°.
Corresponding angle at m (above-left) = 72° (corresponding angles).
Above-right at m = 180° − 72° = 108° (linear pair).
Below-left at m = 108° (vertically opposite to above-right).
Below-right at m = 72° (vertically opposite to above-left).
Answer: The four angles at m are 72°, 108°, 108°, 72°.
Example 4: Checking if Lines Are Parallel
Problem: A transversal cuts two lines. At one line, the angle is 85°. The corresponding angle at the other line is 80°. Are the lines parallel?
Solution:
For lines to be parallel, corresponding angles must be equal.
85° is not equal to 80°.
Answer: The lines are not parallel.
Example 5: Finding x Using Corresponding Angles
Problem: Two parallel lines are cut by a transversal. The corresponding angles are (3x + 10)° and 70°. Find x.
Solution:
Corresponding angles are equal (lines are parallel).
3x + 10 = 70
3x = 70 − 10 = 60
x = 60 / 3 = 20
Check: 3(20) + 10 = 70°. Correct!
Answer: x = 20.
Example 6: Corresponding Angles in F-Shape
Problem: Corresponding angles form an F-shape (or reversed F-shape) pattern. If one angle of the F is 125°, what is the corresponding angle?
Solution:
In the F-pattern, the angles at the two arms of the F are corresponding angles. If the lines are parallel, they are equal.
Answer: The corresponding angle is 125°.
Example 7: Real-Life Example: Railway Tracks
Problem: Railway tracks are parallel lines. A road crosses the tracks at 55°. At what angle does the road meet the second track?
Solution:
The road is a transversal cutting two parallel tracks. The angles at both tracks are corresponding angles.
Corresponding angle = 55°.
Answer: The road meets the second track also at 55°.
Example 8: Finding Both Angles When Sum Is Given
Problem: Two parallel lines are cut by a transversal. A pair of corresponding angles are x° and x°. The angle adjacent to one of them is (180 − x)°. If the adjacent angle is 120°, find x.
Solution:
Adjacent angles form a linear pair: x + 120 = 180
x = 180 − 120 = 60°
So both corresponding angles = 60°.
Answer: x = 60°.
Real-World Applications
Real-life examples of corresponding angles:
- Railway tracks: When a road crosses parallel railway tracks, the angles formed at each track are corresponding angles and are equal.
- Buildings: Horizontal floors of a building are parallel. A staircase (transversal) makes equal corresponding angles with each floor.
- Ladders: A ladder leaning against a wall with parallel horizontal markings creates corresponding angles.
- Design and architecture: Parallel lines in window frames, tiles, and brick patterns create corresponding angles.
- Navigation: Sailors use the concept of parallel lines and transversals to chart courses.
Key Points to Remember
- Corresponding angles are in the same position at each intersection when a transversal cuts two lines.
- If the two lines are parallel, corresponding angles are equal.
- If corresponding angles are equal, the lines are parallel (converse).
- There are 4 pairs of corresponding angles when a transversal cuts two lines.
- Corresponding angles form an F-shape pattern.
- This property is used along with linear pair, vertically opposite angles, and alternate angles to find unknown angles.
- Corresponding angles are always on the same side of the transversal.
Practice Problems
- Two parallel lines are cut by a transversal. One angle is 78°. Find its corresponding angle.
- A transversal makes an angle of 55° with one of two parallel lines. Find all 8 angles formed.
- Corresponding angles are (2x + 15)° and 75°. The lines are parallel. Find x.
- If corresponding angles formed by a transversal are 95° and 85°, are the lines parallel? Why?
- In a figure, identify which pairs of angles are corresponding angles.
- A ladder makes a 60° angle with the ground. If the ground is horizontal and a shelf above is parallel to the ground, what angle does the ladder make with the shelf?
Frequently Asked Questions
Q1. What are corresponding angles?
Corresponding angles are pairs of angles formed when a transversal crosses two lines. They are in the same position (same side, same level) at each intersection point. For example, both are above the line on the left side of the transversal.
Q2. Are corresponding angles always equal?
Corresponding angles are equal only when the two lines are parallel. If the lines are not parallel, corresponding angles are not equal.
Q3. How do I identify corresponding angles?
Look for the F-shape (or reversed/upside-down F-shape). The angles at the arms of the F are corresponding angles. They are on the same side of the transversal and in the same position relative to their line.
Q4. How many pairs of corresponding angles are there?
When a transversal cuts two lines, there are 4 pairs of corresponding angles. At each intersection, there are 4 angles, and each one has a corresponding partner at the other intersection.
Q5. What is the difference between corresponding and alternate angles?
Corresponding angles are on the same side of the transversal and in the same position. Alternate angles are on opposite sides of the transversal. Both are equal when the lines are parallel.
Q6. Can corresponding angles be used to prove lines are parallel?
Yes. If a transversal cuts two lines and the corresponding angles are equal, then the lines must be parallel. This is the converse of the corresponding angles property.
Related Topics
- Alternate Interior Angles
- Co-Interior Angles
- Transversal and Parallel Lines
- Vertically Opposite Angles
- Complementary Angles
- Supplementary Angles
- Adjacent Angles
- Linear Pair of Angles
- Angles on a Straight Line
- Angles at a Point
- Alternate Exterior Angles
- Proving Lines are Parallel
- Word Problems on Lines and Angles
