Properties of Angles and Lines Class 6 ICSE — Chapter 18
This chapter on the properties of angles and lines class 6 ICSE shows what happens when straight lines meet and when a third line cuts across parallel lines. You learn the rules for adjacent angles, vertically opposite angles, and every angle pair a transversal creates. Because the exercises ask for a reason with each answer, each rule below carries the name you should write.
Download Properties of Angles and Lines Class 6 ICSE Mathematics Chapter 18 PDF
Download the Properties of Angles and Lines Class 6 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 6 pages, 5 sections, 12 questions.
What is inside Chapter 18
6 pages · 5 sections · 8 worked examples · 12 exercise questions
| Section | Title |
|---|---|
| 18.1 | PROPERTIES OF ADJACENT AND VERTICALLY OPPOSITE ANGLES |
| 18.2 | PARALLEL LINES |
| 18.3 | CONCEPT OF TRANSVERSAL LINES |
| 18.4 | ANGLES FORMED BY TWO LINES AND THEIR TRANSVERSAL LINE |
| 18.5 | WHEN TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL |
- EXERCISE 18(A): 5 questions
- Revision Exercise (Chapter 18): 7 questions
18.1 PROPERTIES OF ADJACENT AND VERTICALLY OPPOSITE ANGLES
When two straight lines intersect at a point, four angles form around that point. Two angles that share a side are adjacent angles. The two angles directly across from each other are vertically opposite angles. These two ideas appear throughout the Class 6 ICSE syllabus.
Property 1 states two facts about this arrangement. First, every pair of adjacent angles adds to 180^\circ, because together they lie on a straight line. Second, vertically opposite angles are always equal. So for crossing lines you can write \angle APD + \angle BPD = 180^\circ and \angle APC = \angle BPD.
Property 2 links the sum to a straight line. If two adjacent angles add to 180^\circ, their outer arms lie on one straight line. The converse also holds: if the exterior arms form a straight line, the two angles add to 180^\circ.
Worked pattern. Two lines AB and CD cross at P, and \angle BPD = 62^\circ. Because adjacent angles are supplementary:
\angle APD = 180^\circ - 62^\circ = 118^\circ
Then \angle APC = \angle BPD = 62^\circ, since vertically opposite angles are equal. Write the rule name in brackets after each answer.
Collinearity check. Suppose \angle AOB = 110^\circ and \angle BOC = 70^\circ. Their sum is 110^\circ + 70^\circ = 180^\circ, so the exterior arms OA and OC lie on one straight line. Had the sum been anything else, the arms would not be collinear.
18.2 PARALLEL LINES
Two straight lines are parallel when they never meet, no matter how far they are extended in either direction. Diagrams mark parallel lines with matching arrows drawn on both lines, pointing the same way.
The gap between two parallel lines stays the same everywhere. It neither widens nor narrows, which is why the lines never touch. This constant distance is the reason parallel lines can never intersect.
18.3 CONCEPT OF TRANSVERSAL LINES
A transversal is a line that cuts across two or more other lines. The lines it crosses may be parallel or non-parallel; the name applies either way. In the chapter figures, a single line such as PQ is labelled the transversal.
18.4 ANGLES FORMED BY TWO LINES AND THEIR TRANSVERSAL LINE
A transversal cutting two lines creates eight angles, numbered 1 to 8 in the figure. These angles fall into named pairs, and each pair obeys its own rule once the lines are parallel. The table below summarises the six classifications from this ICSE chapter.
| Angle pair | Angle numbers | Rule when lines are parallel |
|---|---|---|
| Exterior angles | 1, 2, 7, 8 | positions outside the two lines |
| Interior angles | 3, 4, 5, 6 | positions between the two lines |
| Exterior alternate | 2 and 8; 1 and 7 | equal |
| Interior alternate | 3 and 5; 4 and 6 | equal |
| Corresponding | 1 and 5; 2 and 6; 3 and 7; 4 and 8 | equal |
| Co-interior (allied) | 3 and 6; 4 and 5 | sum is 180^\circ |
| Exterior allied | 2 and 7; 1 and 8 | sum is 180^\circ |
Alternate angles lie on opposite sides of the transversal. Corresponding angles lie on the same side, one interior and one exterior. Allied angles sit together on the same side of the transversal, inside or outside the parallel lines. The previous chapter on angles builds the naming skills you reuse here.
18.5 WHEN TWO PARALLEL LINES ARE CUT BY A TRANSVERSAL
Once the two lines are known to be parallel, five rules fix every angle. Alternate and corresponding pairs are equal; allied pairs add to 180^\circ. This ICSE chapter expects a stated reason with every value you find.
- Exterior alternate: \angle 1 = \angle 7, \angle 2 = \angle 8
- Interior alternate: \angle 3 = \angle 5, \angle 4 = \angle 6
- Corresponding: \angle 1 = \angle 5, \angle 2 = \angle 6, \angle 3 = \angle 7, \angle 4 = \angle 8
- Co-interior allied: \angle 4 + \angle 5 = 180^\circ, \angle 3 + \angle 6 = 180^\circ
- Exterior allied: \angle 2 + \angle 7 = 180^\circ, \angle 1 + \angle 8 = 180^\circ
Worked pattern. Parallel lines are cut by a transversal and \angle 1 = 75^\circ. Then \angle 7 = 75^\circ because exterior alternate angles are equal. Next, \angle 8 = 180^\circ - 75^\circ = 105^\circ because exterior allied angles are supplementary. Finally, \angle 3 = 75^\circ as corresponding angles, so \angle 6 = 180^\circ - 75^\circ = 105^\circ by the co-interior rule.
Memory aid. Picture the letters the pairs trace. Corresponding angles form an F shape, alternate angles form a Z shape, and allied angles form a C or U shape. If the pair looks like F or Z, they are equal; if it looks like C or U, they add to 180^\circ.
Common mistakes. Students mix up alternate and corresponding pairs. Alternate angles sit on opposite sides of the transversal; corresponding angles sit on the same side. Another slip is treating allied angles as equal — they are supplementary, not equal. Always check which side of the transversal each angle lies on before naming the pair.
What to practise from Properties of Angles and Lines Class 6 ICSE
- EXERCISE 18(A) — 5 questions on intersecting lines, collinearity of exterior arms, and finding the value of x.
- Revision Exercise (Chapter 18) — 7 questions on parallel lines cut by a transversal, including reasoning questions such as whether corresponding angles are always equal when two lines are cut by a transversal.
These questions drill naming angle pairs, finding missing angles, and giving reasons. You can browse the full set of chapters in the Class 6 ICSE maths book. For the official Class 6 syllabus, check the CISCE website.
Properties of Angles and Lines Class 6 ICSE frequently asked questions
What are the properties of angles and lines in class 6 ICSE?
Two straight lines that intersect give adjacent angles that add to 180^\circ and vertically opposite angles that are equal. Parallel lines never meet, and a transversal cutting parallel lines creates equal alternate and corresponding angles plus allied angles that add to 180^\circ.
How do I find missing angles when two parallel lines are cut by a transversal?
Name the angle pair first, then apply its rule. Equal pairs — alternate and corresponding — give the value directly. Allied pairs need subtraction from 180^\circ. Write the rule name as your reason.
What is the difference between corresponding angles and alternate angles?
Corresponding angles lie on the same side of the transversal, one inside and one outside the parallel lines. Alternate angles lie on opposite sides of the transversal. Both pairs are equal when the lines are parallel.
When are the exterior arms of two adjacent angles in a straight line?
Exactly when the two angles add to 180^\circ. That sum is both the test and the result: a straight line forces the sum, and the sum forces the straight line.
Are corresponding angles always equal when a transversal cuts two lines?
Only when the two lines are parallel. If the lines are not parallel, the corresponding angles differ. This is why the chapter’s reasoning questions check whether the lines are parallel before applying the rule.
Reference: ICSE Class 6 Mathematics textbook material, chapter 18: Properties of Angles and Lines — CISCE-aligned.