What this chapter is about
This chapter explores what happens when three straight lines cross each other. You already know that two lines can meet at one point. Now imagine a third line joining in. The pattern of angles and points that forms is interesting and useful.
When three lines intersect, they can create triangles, or they might all pass through the same point. You will learn about the angles formed at these crossings. You will also discover special relationships between these angles, such as vertically opposite angles and angles on a straight line.
After studying this chapter, you should be able to identify angle pairs formed by intersecting lines, calculate unknown angles using angle relationships, and understand how three lines can divide a flat surface into regions.
Key ideas
- When two lines intersect, they form two pairs of vertically opposite angles. Vertically opposite angles are always equal.
- Angles on a straight line add up to 180 degrees. These are called supplementary angles or a linear pair.
- Three lines can intersect in different ways: all three meeting at one single point, or forming a triangle by meeting at three different points.
- When three lines meet at one point, six angles are formed around that point. The sum of all angles around a point is 360 degrees.
- If three lines form a triangle, the sum of the interior angles of that triangle is always 180 degrees.
- Adjacent angles share a common arm and a common vertex. They sit next to each other without overlapping.
- Vertically opposite angles are formed when two lines cross. They face each other across the intersection point.
Formulas and facts to remember
- Vertically opposite angles are equal: If lines cross at a point, the angles facing each other are the same size.
- Linear pair: Two adjacent angles on a straight line add up to 180°.
- Angles around a point: All angles meeting at a single point add up to 360°.
- Sum of angles in a triangle: The three interior angles of any triangle add up to 180°.
- If angle A and angle B are vertically opposite: A = B.
- If angles P and Q form a linear pair: P + Q = 180°.
Worked examples
Example 1: Two lines cross at point O. One of the angles formed is 65°. Find all four angles.
Step 1: The angle vertically opposite to 65° is also 65°.
Step 2: The angles next to 65° form a linear pair with it. So, each of these angles = 180° − 65° = 115°.
Step 3: The four angles are 65°, 115°, 65°, 115°.
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Example 2: Three lines meet at a single point. Two of the six angles formed are 40° and 70°. These two angles are next to each other. Find the angle on the opposite side of the 40° angle.
Step 1: Vertically opposite angles are equal.
Step 2: The angle opposite to 40° is also 40°.
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Example 3: Three lines form a triangle. Two angles of the triangle measure 55° and 75°. Find the third angle.
Step 1: Sum of angles in a triangle = 180°.
Step 2: Third angle = 180° − 55° − 75° = 50°.
Common mistakes
- Confusing adjacent angles with vertically opposite angles → Remember, vertically opposite angles face each other across the point; adjacent angles sit side by side.
- Forgetting that a straight line means 180°, not 360° → Use 360° only for angles around a full point, and 180° for angles on one side of a line.
- Adding all four angles at an intersection and expecting 180° → Four angles around a point sum to 360°, not 180°.
- Thinking any two angles at an intersection are equal → Only vertically opposite pairs are equal; adjacent pairs are supplementary.
- Mixing up interior angles of a triangle with angles around a point → Triangle interior angles sum to 180°; angles around a point sum to 360°.
Quick revision
- Vertically opposite angles are equal.
- Adjacent angles on a straight line add up to 180°.
- All angles around a point add up to 360°.
- Three lines can meet at one point (forming 6 angles) or at three points (forming a triangle).
- The three angles inside any triangle add up to 180°.