Parallel and Intersecting Lines | CBSE Class 7 Maths Notes

This note covers intersecting lines, linear pairs, vertically opposite angles, perpendicular and parallel lines, paper folding, transversals, corresponding and alternate angles, interior angles on the same side of a transversal, constructions, and angle calculations.

What happens when two lines intersect?

A plane surface is a flat surface, such as a sheet of paper, a table top or a blackboard. A line is understood to extend in both directions. A line segment is a part of a line with two endpoints, which mark its ends.

Two lines on a plane intersect when they meet at a point. Their meeting point is the point of intersection. When two lines cross, four angles are formed. An angle is the opening between two rays, called its arms, with a common starting point. A ray starts at a point and extends in one direction.

The symbol ∠ means “angle”, and ° means “degrees”, the unit used here to measure angles. Letters such as l and m name lines. Lower-case letters beside an intersection label its angles; ∠a means the angle labelled a.

Line l runs horizontally and line m slopes upwards to the right. The angles are labelled a above-left, b above-right, c below-right and d below-left of the intersection.

See Fig. 5.2 in your NCERT textbook

Property: A linear pair adds to 180°

Adjacent angles are neighbouring angles with a common corner and arm. A linear pair consists of adjacent angles whose other arms form a straight line. A straight angle measures 180°, so the two angles in a linear pair add to 180°.

With the labels just described, the linear pairs are a with b, b with c, c with d, and d with a. The symbol + means addition, while = means that the quantities on its two sides are equal.

Worked example 1. Two lines cross with angles a, b, c and d in order around the intersection. If ∠a = 120°, find the other three angles.

Answer: ∠b = 180° − 120° = 60°. Here − means subtraction. Next, ∠c = 180° − 60° = 120°, and ∠d = 180° − 120° = 60°. Each calculation uses a linear pair.

Property: Vertically opposite angles are equal

Vertically opposite angles are the opposite, rather than neighbouring, angles formed by two intersecting lines. Here a and c form one pair, and b and d form the other. This relationship does not depend on the particular value 120°.

1. The linear pairs a with b and a with d both total 180°.
2. Removing the same angle a from those equal totals leaves ∠b = ∠d.
3. The linear pairs b with a and b with c also total 180°.
4. Removing angle b from those totals gives ∠a = ∠c.

This reasoning establishes the result for any such intersection. A proof is a mathematical justification showing why a result holds, instead of checking only particular measurements.

How do measurement and perpendicular lines relate to angle rules?

Why might a measured result differ?

An ideal line in geometry has no thickness. A pencil line has thickness, and this can affect where an angle measurement is read. A protractor is an instrument used to measure angles. Improper use of it can also cause measurement errors.

Consequently, measured linear pairs may sometimes fail to total exactly 180°, and measured vertically opposite angles may sometimes appear unequal. Such differences do not overturn the reasoning based on straight angles. The measurements still come very close to the predicted values.

Drawing several pairs of intersecting lines and measuring their angles helps reveal the pattern. The proof explains why that pattern belongs to ideal lines. Geometry can therefore be useful in physics, art, engineering and architecture even though actual drawings cannot reproduce ideal lines exactly.

What makes an intersection perpendicular?

Definition: Perpendicular lines intersect at right angles. A right angle measures 90°. If all four angles at an intersection are equal, each must be a right angle.

The word perpendicular describes the relationship between the two lines. It is not restricted to a drawing with one horizontal line and one vertical line. The angle at which the lines meet is the feature to check.

Worked example 2. Two intersecting lines form four equal angles. Find the measure of each angle and name the relationship between the lines.

Answer: Any adjacent pair forms a straight angle of 180°. Since the two angles in that pair are equal, each measures 90°. All four angles are therefore right angles, and the lines are perpendicular.

Two sloping lines labelled l and m cross, with a small square marking a right angle between them.

See Fig. 5.4 in your NCERT textbook

A square angle mark records a right angle without requiring its measure to be written beside every angle. In a perpendicular intersection, the linear-pair and vertically-opposite-angle rules still hold.

What are parallel lines, and how are they marked?

Definition: Parallel lines lie in the same plane and do not meet, however far they are extended in either direction.

Both parts of the definition matter. Lines must share a plane, and extending them must not produce an intersection. A line on a table and a line on a board may never meet, but that alone does not make them parallel.

Two drawn line segments may stop before the lines containing them would meet. Therefore, seeing a gap between segments on a page is not enough to establish parallelism. Imagine extending the lines beyond both ends of the visible drawing.

How do the markings distinguish relationships?

Matching arrow marks on lines identify a set of parallel lines. If a diagram contains another set, double arrow marks distinguish it from the set with single arrow marks. A square at an angle indicates a perpendicular relationship.

Matching single arrows mark one parallel set, while double arrows distinguish another set. A separate right-angle drawing carries a square corner and the label 90°.

See Fig. 5.9 in your NCERT textbook

Parallel lines are often used in artwork and shading. Drawings on dot paper provide practice in recognising and constructing them. The visible segments can have different lengths; their lengths do not replace the test of how the lines extend.

Appearance can be difficult to judge. Parallel-line illusions invite a closer look at arrangements that do not seem to contain parallel lines. For a mathematical decision, use the given markings and the angle relationships developed below.

How can paper folding help explore line relationships?

A crease is the line left by a fold in paper. Folding a square sheet creates lines that can be traced with a pencil and ruler. Its opposite edges are parallel, while adjacent edges, which meet at a corner, are perpendicular.

What do horizontal and vertical folds show?

1. Begin with a plain square sheet, such as a suitable piece of newspaper.
2. Fold it horizontally in half and examine the new crease in relation to the horizontal edges.
3. Make another horizontal fold in the folded sheet, then repeat and inspect the resulting parallel creases.
4. Make a vertical fold and compare this crease with the horizontal lines.
5. Fold along a diagonal, the segment joining opposite corners, and explore making a parallel crease.

The horizontal creases help connect the idea of parallel lines with the shape of the sheet. The vertical crease provides a way to compare perpendicular directions. Examine the relationship between each new crease and the existing lines, instead of merely counting folds.

How does folding towards a centre line work?

Another activity begins by folding the square in the middle and unfolding it. Fold the edges towards the centre line, the crease through the middle, and unfold them again. Then fold the top-right and bottom-left corners onto the creased line.

These corner folds form triangles, shapes with three sides. Keep the triangles from crossing the crease lines. Compare the segments created by the folds to decide which are parallel. This activity combines careful construction with explaining the relationships seen in the result.

On rectangular dot paper, horizontal, vertical and 45° directions are easier to draw. Drawing a line parallel to one with a different orientation is slightly harder. Here orientation means the direction in which a line lies on the page.

These activities also show why visual judgement needs support. A fold or drawing helps suggest a relationship, but an angle-based explanation gives a reason for accepting it. The same distinction separates measuring several examples from proving a general result.

What is a transversal, and how are its angles named?

Definition: A transversal is a line that crosses a pair of lines at separate points, forming two sets of four angles.

Let l and m name the two lines, and t name the transversal. The lines l and m do not need to be parallel for t to be called a transversal. The intersections create eight angles in total.

Transversal t crosses lines l and m. At the upper intersection, angles 1, 2, 3 and 4 occupy the upper-left, upper-right, lower-right and lower-left positions. Angles 5, 6, 7 and 8 occupy matching positions at the lower intersection.

See Fig. 5.14 in your NCERT textbook

In this numbering, ∠1 means “angle labelled 1”; the number is a label, not its degree measure. For example, writing ∠1 does not assert that the angle measures one degree. An actual measure needs the degree symbol.

Why are there at most four distinct measures?

Distinct measures means different numerical angle sizes. Vertically opposite angles are equal at each intersection. Thus ∠1 = ∠3 and ∠2 = ∠4 at the first intersection, while ∠5 = ∠7 and ∠6 = ∠8 at the second.

These four equal pairs mean that the eight angles can have at most four distinct measures. It is impossible for all eight to have different measures. This conclusion uses vertically opposite angles, so it does not require l and m to be parallel.

Which angles correspond?

Corresponding angles occupy matching positions at the two intersections. Their name describes their positions. Whether they are equal is a separate question about the relationship between the two lines crossed by the transversal.

How do corresponding angles establish parallelism?

Property: Corresponding angles connect equality and parallelism

When a transversal crosses parallel lines, corresponding angles are equal. The relationship also works in the other direction: if a transversal forms equal corresponding angles with two lines, those lines are parallel. The starting information determines which direction to use.

If parallelism is given, equality helps calculate an unknown angle. If parallelism is being tested, compare corresponding angles first. When the lines are not parallel, the corresponding angles formed by a transversal can never be equal.

How can the relationship be explored?

1. Draw a line l and a transversal t meeting at a point called X.
2. Measure an angle labelled a at X; take the illustrated case in which ∠a = 60°.
3. Mark another point, Y, on the transversal t.
4. Through Y, draw a line m so that its corresponding angle b also measures 60°, using tracing paper or a protractor.

The constructed lines appear to be parallel. The equality of their corresponding angles supplies the criterion for parallelism. At the first intersection, the adjacent angle is 120°, so the construction is arranged to produce the same two distinct measures at the second intersection.

Worked example 3. A transversal t crosses lines l and m. Corresponding angles a and b both measure 60°. What is their adjacent linear-pair angle at each intersection, and what follows about l and m?

Answer: Each adjacent angle measures 180° − 60° = 120°. Because the corresponding angles are equal, l and m are parallel. The angles formed have the two distinct measures 60° and 120°.

To explore the reverse starting point, draw parallel lines with a transversal. Trace one angle, then place the tracing over its corresponding angle. The angles match. A protractor can be used to check the other corresponding pairs as well.

Note: Do not call two angles equal merely because they correspond. Establish that the lines are parallel, or use an explicitly given equality to establish parallelism.

How can parallel lines be constructed and justified?

How does a ruler and set square help?

A set square is a triangular drawing instrument with a right-angle corner. Place it against a ruler, then slide it while keeping the ruler fixed. This preserves the angle made by the drawing edge and the line along the ruler.

Start with a line l. Using the right-angle corner, draw two lines perpendicular to l at different positions. View l as a transversal of those new lines. Each corresponding angle measures 90°, so the new lines are parallel.

Worked example 4. Two lines are drawn perpendicular to the same line l at different points on a sheet. Explain why the two new lines are parallel.

Answer: Use l as the transversal. Its corresponding angles with the new lines are both 90°. Equal corresponding angles establish that the two new lines are parallel.

The position of the set square changes during sliding, but the relevant angle stays the same. Parallel lines can also be drawn using its long side. Check equality of corresponding angles to explain why the resulting lines are parallel.

How can a parallel crease pass through a given point?

Let A name a point away from a given crease l. The construction must satisfy two requirements: the new crease must pass through A, and it must be parallel to l. Making an arbitrary parallel crease does not fulfil the first requirement.

1. Begin with the given crease l and the marked point A.
2. Fold a crease perpendicular to l and passing through A; call it t.
3. Through A, fold another crease perpendicular to t; call this one m.
4. Compare the right angles that t makes with l and m to justify that l and m are parallel.

The sequence shows horizontal crease l, point A above it, vertical crease t through A, and a final horizontal crease m through A. Square angle marks show the perpendicular relationships.

See Fig. 5.24 in your NCERT textbook

The construction and its explanation belong together. The folds locate the required line, while the equal corresponding right angles establish its parallel relationship with the original crease.

What are alternate angles, and why are they equal for parallel lines?

The interior region lies between the two lines crossed by a transversal. The alternate angles considered here are inside this region, at different intersections and on opposite sides of the transversal. First identify their positions, then check whether parallelism is given.

Parallel lines l and m are crossed by t. At the upper intersection a and b lie above l, with d and c below it. At the lower intersection e and f lie above m, with h and g below it.

See Fig. 5.25 in your NCERT textbook

For these labels, d and f are an alternate pair, as are c and e. The letters label the angles shown in this arrangement. Angle b corresponds to f, while angle d is vertically opposite to b.

Property: Alternate angles on parallel lines are equal

1. Take parallel lines l and m crossed by transversal t.
2. Angles b and f are corresponding, so they are equal.
3. Angles b and d are vertically opposite, so they are equal.
4. It follows that angles d and f are equal.

The reasoning combines two established relationships. One uses the parallel lines, and the other uses the intersection at the upper line. Because no particular measure is needed, the equality holds irrespective of the size of the selected angle.

Worked example 5. Parallel lines l and m are crossed by t. Angle f is 120°, angle b corresponds to f, and angle d is vertically opposite to b. Find the alternate angle d.

Answer: ∠b = 120° by equality of corresponding angles on parallel lines. Then ∠d = 120° because vertically opposite angles are equal. Thus the alternate angles d and f have the same measure.

This route is useful when an alternate pair is difficult to spot directly. Find the corresponding angle first, then its vertically opposite partner. State the parallel-line condition when using alternate-angle equality; the position-based name alone is not an equality rule.

How can angle rules be combined in worked problems?

How can one angle determine the other seven?

In each numbered example below, angles 1, 2, 3 and 4 occur in order around the first intersection. Angles 5, 6, 7 and 8 occupy the respective matching positions at the second intersection. Thus 1 corresponds to 5, 2 to 6, and so on.

Worked example 6. Parallel lines l and m are crossed by transversal t. Using the numbering just defined, ∠6 = 135°. Find all the other angles.

Answer: ∠2 = 135° by corresponding angles. ∠8 = 135° by vertically opposite angles, and ∠4 = 135° by correspondence with angle 8. The adjacent angle ∠5 = 180° − 135° = 45°. Angles 1, 3 and 7 also measure 45°.

How can unequal corresponding angles disprove parallelism?

Worked example 7. Transversal t crosses lines l and m. Angles a and b form a linear pair. Angle b corresponds to angle f. Given ∠a = 120° and ∠f = 70°, decide whether l and m are parallel.

Answer: ∠b = 180° − 120° = 60°. The corresponding angles b and f measure 60° and 70°, so they are unequal. Therefore l and m are not parallel.

The given angles a and f are not the corresponding pair used in this test. The intermediate angle b, meaning the angle found along the way, connects the linear-pair information to the corresponding-angle test. Naming that connection makes the conclusion checkable.

Property: Interior angles on the same side total 180°

When a transversal crosses parallel lines, the two interior angles on the same side of the transversal add to 180°. In the numbered arrangement above, angles 3 and 6 form such a pair. They are not an alternate pair.

Worked example 8. Parallel lines l and m are crossed by t. Angle 3 measures 50°, angle 2 forms a linear pair with angle 3, and angle 6 corresponds to angle 2. Find angle 6.

Answer: ∠2 = 180° − 50° = 130°. Since corresponding angles on parallel lines are equal, ∠6 = 130°. The interior angles 3 and 6 consequently total 180°.

The general explanation follows the same route: angles 2 and 3 total 180°, and angle 6 equals angle 2. Replacing angle 2 by its equal angle 6 shows why the same-side interior pair has a total of 180°.

How are parallel-line rules used in a four-sided figure?

A quadrilateral is a figure with four sides. Let A, B, C and D name its corners, also called vertices, in boundary order. The notation AB names the segment joining A to B. In a three-letter angle name, the middle letter names the corner.

Thus ∠DAC is the angle at A between segments AD and AC. Segment AC is a diagonal, joining opposite corners. Naming angles carefully matters because the diagonal splits the angle at A into two smaller angles.

A is upper-left, B upper-right, C lower-right and D lower-left. Matching arrows mark AB parallel to CD and AD parallel to BC. Diagonal AC is drawn, with ∠DAC labelled 65° and ∠ADC labelled 60°.

See Fig. 5.29 in your NCERT textbook

How do we select the relevant transversal?

The same segment may act as a transversal for a chosen pair of parallel lines. To find the whole angle at A, use AB and CD as the parallel pair and AD as their transversal. To find the angle at C, use AD and BC with CD.

Worked example 9. Quadrilateral ABCD has AB parallel to CD and AD parallel to BC. Diagonal AC lies inside the figure. Given ∠DAC = 65° and ∠ADC = 60°, find ∠CAB, ∠ABC and ∠BCD.

Answer: The same-side interior angles ∠ADC and ∠DAB total 180°, so ∠DAB = 120°. Since AC divides ∠DAB, ∠CAB = 120° − 65° = 55°. With AD parallel to BC and CD as transversal, ∠BCD = 180° − 60° = 120°. With AB as transversal, ∠ABC = 180° − 120° = 60°.

The three requested results are therefore 55°, 60° and 120°, respectively. The 65° value is part of the angle at A, so it must not be substituted for the whole angle ∠DAB in the same-side interior relationship.

For a clear solution, identify the parallel pair before each use of the 180° rule. Then distinguish the full corner angle from the part cut off by the diagonal. This keeps the subtraction tied to the correct angles.

Glossary

• Plane surface — A flat surface on which the line relationships in this chapter are considered.
• Line segment — A part of a line bounded by two endpoints marking its ends.
• Intersecting lines — Lines that meet at a point and form angles at their intersection.
• Linear pair — Two adjacent angles whose other arms form a straight line, giving a total of 180°.
• Vertically opposite angles — The opposite angles formed by two intersecting lines, equal to one another.
• Proof — A mathematical justification that explains why a result holds beyond particular measurements.
• Right angle — An angle measuring 90°, such as each angle at a perpendicular intersection.
• Perpendicular lines — Two lines that intersect at right angles, forming four angles of 90°.
• Parallel lines — Lines in the same plane that never meet, however far they are extended.
• Transversal — A line crossing a pair of lines at separate points and creating eight angles.
• Corresponding angles — Angles in matching positions at the two intersections made by a transversal.
• Alternate angles — Here, interior angles at different intersections on opposite sides of a transversal.
• Same-side interior angles — Angles inside the two lines and on the same side of their transversal.
• Diagonal — A segment joining opposite corners of a square or of the four-sided figure considered here.

Common errors and misconceptions

• Misconception: Segments that do not meet on the page must be parallel. Correct: Consider the lines extended in both directions. Their visible portions may end before an intersection.
• Misconception: Any lines that never meet are parallel. Correct: They must also lie in the same plane. Non-intersection alone omits part of the definition.
• Misconception: Vertically opposite angles form a linear pair. Correct: Opposite angles are equal. A linear pair consists of neighbouring angles whose sum is 180°.
• Misconception: Corresponding angles are equal whenever a transversal appears. Correct: Their equality is tied to parallel lines. On non-parallel lines, corresponding angles cannot be equal.
• Misconception: Alternate angles and same-side interior angles use the same rule. Correct: For parallel lines, alternate angles are equal; same-side interior angles add to 180°.
• Misconception: A slightly unequal measurement disproves the vertically-opposite-angle rule. Correct: Instrument use and the thickness of drawn lines can affect measurements. The proof concerns ideal lines.
• Misconception: The angle ∠DAC is the whole angle ∠DAB when AC lies inside it. Correct: ∠DAC is only one part; add ∠CAB to obtain the whole angle.

Exam-style questions with model answers

Q1. State the two conditions that define a pair of parallel lines. [2 marks]

1. The two lines must lie in the same plane.
2. They must not meet, however far they are extended in either direction.

Q2. Lines l and m cross. Angles a, b, c and d occur in that order around the intersection. If ∠a = 120°, find b, c and d, giving reasons. [3 marks]

1. Angle b is adjacent to a in a linear pair. Their total is 180°, so ∠b = 180° − 120° = 60°.
2. Angle c is vertically opposite to a. Vertically opposite angles are equal, so ∠c = 120°.
3. Angle d is vertically opposite to b. Using the value already found for b gives ∠d = 60°.

Q3. A transversal t crosses lines l and m. Angles a and b form a linear pair, and b corresponds to f. Given ∠a = 120° and ∠f = 70°, are l and m parallel? Explain. [3 marks]

1. Use the linear pair to obtain ∠b = 180° − 120° = 60°.
2. The relevant corresponding angles are b and f. They measure 60° and 70°, respectively, so they are unequal.
3. Corresponding angles would be equal if l and m were parallel. Since this necessary equality fails, the two lines are not parallel.

Q4. Parallel lines l and m are crossed by t. Angle 3 is 50°, angle 2 forms a linear pair with angle 3, and angle 6 corresponds to angle 2. Find angle 6 and the sum of angles 3 and 6. [3 marks]

1. Angles 2 and 3 form a linear pair, so ∠2 = 180° − 50° = 130°.
2. Since l and m are parallel, the corresponding angles 2 and 6 are equal. Therefore ∠6 = 130°.
3. The requested sum is 50° + 130° = 180°. These are the interior angles on the same side of the transversal in this arrangement.

Q5. Parallel lines l and m are crossed by t. Angles 1, 2, 3, 4 occur in order around the first intersection; 5, 6, 7, 8 occupy matching positions at the second. Given ∠6 = 135°, find the other seven angles with reasons. [5 marks]

1. Angle 2 corresponds to angle 6. Because the two lines are parallel, these angles are equal, giving ∠2 = 135°.
2. Angle 8 is vertically opposite to angle 6, so ∠8 = 135°. Angle 4 corresponds to angle 8, giving ∠4 = 135° as well.
3. Angles 5 and 6 form a linear pair. Subtracting the given angle from a straight angle gives ∠5 = 180° − 135° = 45°.
4. Angle 7 is vertically opposite to angle 5, so ∠7 = 45°. This completes the unknown measures at the second intersection.
5. Angle 1 corresponds to angle 5 and angle 3 corresponds to angle 7. Thus ∠1 = 45° and ∠3 = 45°, completing all seven requested angles.

Q6. In quadrilateral ABCD, the vertices are in boundary order, AB is parallel to CD, and AD is parallel to BC. Diagonal AC lies inside the figure. Given ∠DAC = 65° and ∠ADC = 60°, find ∠CAB, ∠BCD and ∠ABC with reasons. [5 marks]

1. Use AB and CD as the parallel lines, with AD as their transversal. The interior angles ∠ADC and ∠DAB lie on the same side and therefore total 180°.
2. Since ∠ADC = 60°, subtraction gives ∠DAB = 180° − 60° = 120°. This is the whole angle at A.
3. Diagonal AC divides that whole angle into ∠DAC and ∠CAB. Hence ∠CAB = 120° − 65° = 55°.
4. Next use AD parallel to BC, with CD as transversal. Angles ∠ADC and ∠BCD total 180°, giving ∠BCD = 120°.
5. Use AD parallel to BC again, now with AB as transversal. Angles ∠DAB and ∠ABC total 180°, giving ∠ABC = 60°.

Q7. A crease l and a point A away from l are given on a sheet. Describe how to construct a parallel crease through A by folding twice, and justify the construction. [4 marks]

1. Make the first new crease through A and perpendicular to l. Name this crease t so that its role can be tracked.
2. Make a second crease through A, this time perpendicular to t. Name the second crease m.
3. View t as a transversal of l and m. Since both are perpendicular to t, the corresponding angles are both 90°.
4. Equal corresponding angles establish that l and m are parallel. The second fold passes through A, fulfilling the required point condition.

Key takeaways

• Two intersecting lines form four angles. Adjacent linear pairs total 180°, while vertically opposite angles have equal measures.
• Perpendicular lines meet at right angles, so all four angles at their intersection measure 90°.
• Parallel lines must lie in the same plane and remain separate however far they are extended.
• A transversal crossing two lines forms eight angles, with at most four distinct measures because opposite angles are equal.
• Corresponding angles are equal for parallel lines; equal corresponding angles can also establish that two lines are parallel.
• For parallel lines, alternate angles are equal and interior angles on the same side of a transversal total 180°.
• A parallel line through a given point can be constructed using two successive perpendiculars and justified through corresponding angles.
• Angle calculations should identify the parallel pair, the transversal and the exact relationship before using a numerical value.

Test yourself

Why does a linear pair total 180°?

The adjacent angles together form a straight angle, which measures 180°.

Two intersecting lines form four equal angles. What are their measures?

Each angle measures 90°, so the two lines are perpendicular.

Why is non-intersection alone insufficient to define parallel lines?

The lines must also lie in the same plane; both conditions belong to the definition.

How many distinct angle measures can a transversal across two lines produce at most?

At most four distinct measures, because each intersection contains two equal pairs of vertically opposite angles.

Corresponding angles made by a transversal are both 60°. What follows about the two lines?

The two lines are parallel because their corresponding angles are equal.

On parallel lines, angle f is 120° and angle d is its alternate angle. Find d.

Angle d measures 120° because alternate angles on parallel lines are equal.

Two interior angles lie on the same side of a transversal across parallel lines. One measures 50°. Find the other.

The other measures 130°, since the two angles must total 180°.

Why might protractor readings of vertically opposite angles differ slightly?

Improper instrument use or the thickness of drawn lines can affect measured values.