What this chapter is about
This chapter helps you understand how lines behave when they are drawn on a flat surface. You will learn about lines that cross each other (intersecting lines) and lines that never meet no matter how far you extend them (parallel lines).
Understanding parallel and intersecting lines is useful in everyday life. Railway tracks, edges of your notebook, and ladder steps are examples of parallel lines. The letter X, crossroads, and scissor blades show intersecting lines. Builders, artists and engineers use these ideas constantly.
After studying this chapter, you should be able to identify parallel and intersecting lines around you. You will also learn about the angles formed when lines cross and what happens when a third line cuts through parallel lines.
Key ideas
- Two lines on a flat surface are called parallel lines if they never meet, even when extended forever. We write AB ∥ CD to say line AB is parallel to line CD.
- Two lines that cross each other at exactly one point are called intersecting lines. The point where they meet is called the point of intersection.
- When two lines intersect, they form four angles. The angles opposite to each other are called vertically opposite angles, and they are always equal.
- A transversal is a line that cuts across two or more lines at different points.
- When a transversal crosses two parallel lines, it creates eight angles. Some special pairs of these angles are equal, and some add up to 180°.
- Corresponding angles are in the same position at each intersection point. When the two lines are parallel, corresponding angles are equal.
- Alternate angles are on opposite sides of the transversal and between the parallel lines. When the two lines are parallel, alternate angles are equal.
- Co-interior angles (also called same-side interior angles) are on the same side of the transversal and between the parallel lines. When the two lines are parallel, they add up to 180°.
Formulas and facts to remember
- Vertically opposite angles are equal.
- If two parallel lines are cut by a transversal: corresponding angles are equal.
- If two parallel lines are cut by a transversal: alternate interior angles are equal.
- If two parallel lines are cut by a transversal: co-interior angles add up to 180°.
- If a transversal makes equal corresponding angles (or equal alternate angles) with two lines, then those two lines are parallel.
Worked examples
Example 1: Two lines cross at point P. One of the angles formed is 65°. Find the other three angles.
Step 1: The angle vertically opposite to 65° is also 65°.
Step 2: The two remaining angles are next to 65°. Since angles on a straight line add up to 180°, each of these is 180° − 65° = 115°.
Answer: The four angles are 65°, 115°, 65°, 115°.
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Example 2: A transversal crosses two parallel lines. One of the angles formed is 70°. It is above the first parallel line and to the left of the transversal. Find the corresponding angle at the second parallel line.
Step 1: Corresponding angles are in the same position at both intersection points.
Step 2: Since the lines are parallel, the corresponding angle at the second line is also 70°.
Answer: 70°.
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Example 3: A transversal cuts two parallel lines. An angle on the left side of the transversal, between the two parallel lines, measures 110°. Find the alternate interior angle.
Step 1: The alternate interior angle is between the parallel lines but on the right side of the transversal.
Step 2: For parallel lines, alternate interior angles are equal.
Answer: The alternate interior angle is 110°.
Common mistakes
- Confusing alternate angles with corresponding angles → Draw the transversal and mark positions carefully; alternate angles are on opposite sides.
- Thinking vertically opposite angles add up to 180° → They are equal, not supplementary.
- Forgetting that angle rules for parallel lines do not work if the lines are not parallel → Always check whether the lines are truly parallel before using the rules.
- Adding alternate angles instead of recognising they are equal → Remember: alternate angles are equal; co-interior angles add to 180°.
Quick revision
- Parallel lines never meet; intersecting lines meet at one point.
- Vertically opposite angles are equal.
- When a transversal cuts parallel lines, corresponding angles are equal and alternate angles are equal.
- Co-interior angles between parallel lines add up to 180°.
- To prove lines are parallel, show that corresponding or alternate angles are equal.