What this chapter is about
This chapter takes you deeper into the world of geometry by exploring ideas that connect shapes, angles, lines and their properties. You have already studied basic shapes like triangles and quadrilaterals. Now you will see how certain geometric principles work together — how parallel lines create angle relationships, how triangles can be proved congruent, and how symmetry appears in everyday objects.
The chapter helps you think like a mathematician by asking why certain results are true, not just what the results are. You will learn to reason step by step, using facts you already know to discover new facts. This skill of logical reasoning is useful far beyond geometry.
After studying this chapter, you should be able to recognise angle relationships formed by parallel lines and a transversal, understand when two triangles are congruent, identify lines of symmetry, and use these ideas to solve problems about unknown angles and lengths.
Key ideas
- When a line (called a transversal) crosses two parallel lines, it creates pairs of equal angles: corresponding angles are equal, and alternate interior angles are equal.
- Co-interior angles (also called same-side interior angles) formed by a transversal and two parallel lines add up to 180°.
- Two triangles are congruent when they have exactly the same shape and size — every side and angle of one matches a side and angle of the other.
- You can prove triangles congruent using rules: SSS (three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal), or RHS (right angle, hypotenuse and one side equal in right triangles).
- A figure has line symmetry if you can fold it along a line and both halves match perfectly. That fold line is called the axis of symmetry.
- Some figures have more than one axis of symmetry: a square has four, an equilateral triangle has three, and a rectangle has two.
- Point symmetry exists when every part of a figure has a matching part at an equal distance on the opposite side of a central point.
Formulas and facts to remember
- Corresponding angles are equal when lines are parallel: if line l ∥ line m and a transversal cuts them, then each pair of corresponding angles are equal.
- Alternate interior angles are equal: angles on opposite sides of the transversal, between the parallel lines, are equal.
- Co-interior angles are supplementary: they add up to 180°.
- SSS rule: if three sides of one triangle equal three sides of another, the triangles are congruent.
- SAS rule: if two sides and the angle between them in one triangle equal those in another, the triangles are congruent.
- ASA rule: if two angles and the side between them in one triangle equal those in another, the triangles are congruent.
- RHS rule: in right triangles, if the hypotenuse and one other side are equal, the triangles are congruent.
Worked examples
Example 1: Finding angles with parallel lines
Two parallel lines are cut by a transversal. One of the angles formed is 65°. Find the angle that is co-interior to it.
Solution: Co-interior angles add up to 180°. Unknown angle = 180° − 65° = 115°.
Example 2: Proving triangles congruent
Triangle PQR has PQ = 5 cm, QR = 7 cm, and angle Q = 50°. Triangle XYZ has XY = 5 cm, YZ = 7 cm, and angle Y = 50°. Are the triangles congruent?
Solution: We have two sides and the included angle (the angle between those two sides) equal in both triangles. PQ = XY = 5 cm, QR = YZ = 7 cm, angle Q = angle Y = 50°. By the SAS rule, triangle PQR is congruent to triangle XYZ.
Example 3: Counting axes of symmetry
Ravi draws a regular hexagon (six equal sides). How many axes of symmetry does it have?
Solution: A regular hexagon can be folded along a line through each pair of opposite vertices (3 such lines) and also along a line through the midpoints of each pair of opposite sides (3 more lines). Total axes of symmetry = 3 + 3 = 6.
Common mistakes
- Confusing alternate angles with co-interior angles → remember alternate angles are on opposite sides of the transversal and are equal; co-interior angles are on the same side and add to 180°.
- Using the wrong congruence rule, like trying SSA (two sides and a non-included angle) → SSA does not guarantee congruence; always check the angle is between the two sides for SAS.
- Thinking every four-sided figure has four axes of symmetry → only a square does; a rectangle has two, a parallelogram has none.
- Forgetting that congruence means both shape and size match → similar triangles have the same shape but may differ in size; congruent triangles are identical in both.
- Assuming all equal angles mean congruent triangles → you also need at least one pair of equal sides to prove congruence.
Quick revision
- Parallel lines cut by a transversal give equal corresponding and alternate angles, and supplementary co-interior angles.
- Triangles are congruent if they satisfy SSS, SAS, ASA or RHS.
- A line of symmetry divides a figure into two mirror-image halves.
- A square has 4 axes of symmetry; an equilateral triangle has 3; a circle has infinitely many.
- Congruent means same shape and same size.