Parallel Lines and Transversals — Questions and Answers
Welcome to HSLC Guru. This page gives the complete, worked solutions with figures for ASSEB (Assam State School Education Board) Class 7 New Mathematics, Chapter 5 Parallel Lines and Transversals — every “Work it Out” box, the paper-folding activities, the solved examples and Exercise 5.
Summary
When two lines meet at a point they are intersecting lines, and the meeting point is their point of intersection. If two intersecting lines form a 90° (right) angle they are perpendicular lines. A pair of lines in the same plane that never meet, however far they are extended, are parallel lines.
A line that cuts two or more lines of a plane at two or more distinct points is a transversal. A transversal cutting two lines forms eight angles, among which we identify adjacent angles, vertically opposite angles, linear pairs, interior and exterior angles, corresponding angles and alternate angles.
When a transversal cuts two parallel lines, the corresponding angles are equal and the alternate angles are equal. Two angles whose sum is 90° are complementary angles, and two angles whose sum is 180° are supplementary angles.
Summary: This ASSEB Class 7 New Mathematics Chapter 5 (Parallel Lines and Transversals) question-answer set explains intersecting, perpendicular and parallel lines, transversals and the eight angles they form — adjacent, vertically opposite, linear pair, interior, exterior, corresponding and alternate angles — with complementary and supplementary angles, and gives fully worked solutions with figures for every “Work it Out” box and Exercise 5.
Key Concepts and Figures
When a transversal $t$ cuts two lines $p$ and $q$, the eight angles formed are numbered 1 to 8 below. Here $\angle 3, \angle 4, \angle 5, \angle 6$ are interior angles (between the two lines) and $\angle 1, \angle 2, \angle 7, \angle 8$ are exterior angles.
Corresponding angles: a pair of angles in the same relative position on the same side of the transversal, e.g. $\angle 1$ and $\angle 5$. For parallel lines $\angle 1 = \angle 5$.
Alternate angles: two interior (or two exterior) angles on opposite sides of the transversal, e.g. $\angle 3$ and $\angle 5$. For parallel lines $\angle 3 = \angle 5$.
Textbook Questions and Answers
Work it Out 5.1
In the figure, lines $x$ and $y$ intersect at a point and form the four angles $\angle a, \angle b, \angle c, \angle d$.
Question: From the figure identify the adjacent angles, linear pairs of angles and vertically opposite angles. (a) If $\angle a = 65°$, find $\angle b, \angle c$ and $\angle d$. (b) If $\angle b = 115°$, find $\angle d$.
Answer: Adjacent angle pairs: ($\angle a, \angle b$), ($\angle b, \angle c$), ($\angle c, \angle d$), ($\angle d, \angle a$). Each of these pairs is also a linear pair because their sum is 180°. Vertically opposite pairs: ($\angle a, \angle c$) and ($\angle b, \angle d$).
(a) $\angle a$ and $\angle b$ form a linear pair, so $\angle b = 180° − 65° = 115°$. $\angle a$ and $\angle c$ are vertically opposite, so $\angle c = \angle a = 65°$. $\angle a$ and $\angle d$ form a linear pair, so $\angle d = 180° − 65° = 115°$. Thus $\angle b = 115°,\ \angle c = 65°,\ \angle d = 115°$.
(b) $\angle b$ and $\angle d$ are vertically opposite, so $\angle d = \angle b = 115°$.
Activities 1 and 2 (paper folding for parallel and perpendicular lines)
Activity 1: Fold a rectangular/square sheet horizontally and draw a line along the crease. This line is perpendicular to the vertical sides and parallel to the horizontal sides. Each extra fold increases the number of parallel creases — the more folds, the more parallel lines.
Answer: The horizontal crease is perpendicular to the vertical sides and parallel to the horizontal sides. A line drawn along a later vertical fold is perpendicular to the earlier horizontal lines and parallel to the vertical sides.
Activity 2: Fold a square along each diagonal and draw lines $p$ and $q$ along the creases. Answer: The diagonals of a square are perpendicular, so $p \perp q$. Folding each vertex to the point of intersection gives lines $a, b, c, d$ where $a \parallel c$ and $b \parallel d$, and these two pairs are perpendicular to each other.
Work it Out 5.2
1. On the dotted paper, draw at least four lines joining three dots and try to draw perpendiculars at each point of the lines.
Answer: Using a scale and a set square, at any point of a line draw another line making 90° with it — that gives a perpendicular. Repeat at different points to get perpendiculars everywhere (drawing activity).
2. Draw a zig-zag path of 5 straight lines, then draw a second zig-zag path each section of which is parallel to the corresponding section of the first.
Answer: For each section of the first path, draw a segment in the same direction and at a fixed distance; then each section of the second path is parallel to the corresponding section of the first (drawing activity).
3. Identify the pairs of parallel and perpendicular lines in the figures; mark parallel lines with arrows (>, >>) and perpendicular lines with a right-angle sign.
Answer: Lines that never meet even when extended are marked as parallel with arrows; lines that meet at 90° are marked perpendicular with a small square (⌐). We check by extending the lines and testing whether the angle between them is 90°.
4. Using a scale and a set square, draw two lines parallel to the given line $x$ passing through the points $m$ and $n$.
Answer: Place one arm of the set-square’s right angle on line $x$, put the scale against the other arm, and slide the set square along the scale until it touches $m$ (and $n$); drawing along that arm gives a line parallel to $x$.
5. Draw a new line parallel to each line segment on the dot-paper. (a) Was it easy for all segments? If not, for which ones? (b) How did you solve it?
Answer: Horizontal and vertical segments are easy; slanted (oblique) segments are harder. Using the scale-and-set-square method, slide the set square along the direction of the segment to draw a parallel line — this solves the difficulty.
Solved Examples
Example 1. $t$ is a transversal of lines $l$ and $m$, with $\angle p = 125°$. (a) If $l \parallel m$, find $\angle q$. (b) If $\angle r = 75°$, will $l$ and $m$ be parallel?
Solution: (a) Since $l \parallel m$ and $\angle p, \angle q$ are corresponding angles, $\angle q = \angle p = 125°$. (b) $\angle q$ and $\angle r$ form a linear pair, so $\angle q = 180° − 75° = 105°$; but the corresponding angle $\angle p = 125° \ne 105°$, so $l$ and $m$ are not parallel.
Example 2. $r$ is a transversal of the parallel lines $p$ and $q$. If $\angle d = 55°$, find all the other angles (top intersection $a, b, c, d$; bottom intersection $e, f, g, h$).
Solution: $\angle b = \angle d = 55°$ (vertically opposite). $\angle c = 180° − 55° = 125°$ (linear pair). $\angle a = \angle c = 125°$ (vertically opposite). $\angle g = \angle c = 125°$ (corresponding). $\angle h = \angle d = 55°$ (corresponding). $\angle f = \angle h = 55°$ (vertically opposite). $\angle e = \angle g = 125°$ (vertically opposite).
Example 3. $s$ and $t$ are two transversals of the parallel lines $m$ and $n$. If $\angle 1 = 60°$ and $\angle 10 = 50°$, find the other angles.
Solution: $\angle 3 = \angle 1 = 60°$ (vertically opposite). $\angle 2 = 180° − 60° = 120°$ (linear pair); $\angle 4 = \angle 2 = 120°$ (vertically opposite). $\angle 11 = \angle 1 = 60°$ (corresponding). Since $\angle 9, \angle 10, \angle 11$ together make a straight angle, $\angle 9 = 180° − (50° + 60°) = 70°$. $\angle 8 = \angle 9 = 70°$ (corresponding); $\angle 6 = \angle 8 = 70°$ (vertically opposite). $\angle 5 = 180° − 70° = 110°$ (linear pair); $\angle 7 = \angle 5 = 110°$ (vertically opposite).
Work it Out 5.3
1. In the figures $l \parallel m$ and $t$ is the transversal. Find the value of $\angle x$.
Answer: (a) The 120° at the top and $\angle x$ are corresponding angles; corresponding angles of parallel lines are equal, so $\angle x = 120°$.
(b) The given 60° and $\angle x$ are alternate angles (a “Z”); alternate angles are equal, so $\angle x = 60°$.
(c) The lower 120° and $\angle x$ are alternate interior angles, so $\angle x = 120°$.
2. In the figures $l$ and $m$ are two lines and $n$ is the transversal. Which are parallel? Justify.
Answer: (a) The interior angles on the same side of the transversal are 57° and 123°, and 57° + 123° = 180°, so $l \parallel m$ (parallel).
(b) The marked 72° and 98° are alternate angles; for parallel lines they must be equal, but 72° ≠ 98°, so the lines are not parallel.
(c) Both marked angles are 75° and 75°, which are equal alternate angles, so $l \parallel m$ (parallel).
(d) 54° and 126° are co-interior angles; 54° + 126° = 180°, so $l \parallel m$ (parallel).
Work it Out 5.4
1. From the angles below, find the pairs of complementary angles: (a) 50° (b) 20° (c) 35° (d) 70° (e) 40° (f) 30°. Also state which have no complement.
Answer: Complementary angles add up to 90°. Pairs: (a) 50° and (e) 40° [50° + 40° = 90°]; (b) 20° and (d) 70° [20° + 70° = 90°]. Among the given angles, (c) 35° and (f) 30° have no complement.
2. What is the complement of each angle? (a) 35° (b) 45° (c) 25° (d) 60°.
Answer: Complement = 90° − angle. (a) 55° (b) 45° (c) 65° (d) 30°.
3. Write a pair of complementary angles that are equal in measure.
Answer: If both are equal, each is $90° \div 2 = 45°$. So the pair is 45° and 45°.
4. What is the supplement of each angle? (a) 60° (b) 70° (c) 85° (d) 55° (e) 50°.
Answer: Supplement = 180° − angle. (a) 120° (b) 110° (c) 95° (d) 125° (e) 130°.
5. Find an angle that is equal to its own supplement.
Answer: Let the angle be $x$. Then $x = 180° − x$ ⟹ $2x = 180°$ ⟹ $x = 90°$. So the angle is 90°.
6. From the figure (five rays at a point forming $\angle a, \angle b, \angle c, \angle d, \angle e$), identify which are complementary and which are supplementary.
Answer: A vertical ray makes a right angle with the horizontal line ($\angle a = 90°$). On the upper straight angle $\angle a + \angle b + \angle c = 180°$, so $\angle b + \angle c = 90°$ — that is, $\angle b$ and $\angle c$ are complementary. Below the line $\angle d$ and $\angle e$ together make a straight angle, $\angle d + \angle e = 180°$ — that is, $\angle d$ and $\angle e$ are supplementary.
Exercise 5
1. In the following figures $AB \parallel CD$; find $\angle x, \angle y, \angle z$.
Answer: (a) The two lines from the apex to $CD$ form a triangle on $CD$. Since the exterior angle at the right foot is 100°, the interior angle there $= 180° − 100° = 80°$. With $AB \parallel CD$ and the left slant as transversal, $\angle x = 70°$ (alternate angles). In the triangle $\angle y = 180° − (70° + 80°) = 30°$. On the straight line $AB$ at the apex, $70° + \angle y + \angle z = 180°$, so $\angle z = 180° − 70° − 30° = 80°$. Thus $\angle x = 70°,\ \angle y = 30°,\ \angle z = 80°$.
(b) Lines from $B$ and $D$ meet at $P$ with $\angle ABP = 65°$ and $\angle BPD = 30°$. Drawing a line through $P$ parallel to $AB$ and $CD$ gives $\angle ABP = \angle BPD + \angle CDP$, so $\angle x = \angle CDP = 65° − 30° = 35°$.
(c) The triangle from the apex to $CD$ has base angles 80° and 40°. With $AB \parallel CD$, alternate angles give $\angle x = 80°$ and $\angle z = 40°$. At the apex $\angle x + \angle y + \angle z = 180°$, so $\angle y = 180° − 80° − 40° = 60°$. Thus $\angle x = 80°,\ \angle y = 60°,\ \angle z = 40°$.
(d) [Hint: extend $BE$ to meet $CD$ at $F$.] $BF$ is a transversal of $AB \parallel CD$, so the alternate angle $\angle EFD = \angle ABE = 55°$. In triangle $EFD$, $\angle FED = 180° − 55° − 25° = 100°$. Since $B, E, F$ are collinear, $\angle x = \angle BED = 180° − 100° = 80°$. (Equivalently, drawing a parallel through $E$ gives $\angle x = 55° + 25° = 80°$.)
2. Find $\angle x$ from the given figure ($AB \parallel CD$, with a reflex angle $x$ at the middle point).
Answer: Draw a line through the middle point parallel to $AB$ and $CD$; the 45° (top) and 30° (bottom) become alternate angles. The inner angle $= 45° + 30° = 75°$, so the marked reflex angle $\angle x = 360° − 75° = 285°$.
3. Which is a pair of interior alternate angles? (a) $\angle 1, \angle 3$ (b) $\angle 2, \angle 3$ (c) $\angle 2, \angle 5$ (d) $\angle 2, \angle 4$.
Answer: (b) $\angle 2, \angle 3$. $\angle 2$ is interior at the top intersection and $\angle 3$ is interior at the bottom intersection, lying on opposite sides of the transversal — so they are alternate interior angles.
4. If $a \parallel b$ and $c$ is the transversal, then $\angle y = ?$ (a) 90° (b) 125° (c) 55° (d) 180°.
Answer: (b) 125°. The given 55° and $\angle y$ are on the same side, so $\angle y = 180° − 55° = 125°$.
5. If $a \parallel b$ and $c$ is the transversal, then $\angle y = ?$ (a) 90° (b) 25° (c) 55° (d) 35°.
Answer: (c) 55°. The given 55° and $\angle y$ are alternate (or corresponding) angles, hence equal — $\angle y = 55°$.
6. If $l \parallel m$ and $c$ is the transversal, find $\angle x$. (a) 50° (b) 130° (c) 120° (d) 100°.
Answer: (b) 130°. The top 130° and $\angle x$ are alternate exterior angles, so $\angle x = 130°$.
7. In the figure $u$ and $v$ are parallel lines; $r$ and $s$ are transversals (meeting at a point on $u$). Given 44° and 128°, find all the angles.
Answer: At the point on $u$: $\angle c = 44°$ (vertically opposite to the 44°). The linear pair of 128° is $180° − 128° = 52°$, which gives the inclination, so $\angle d = 52°$ and $\angle a = 52°$; $\angle b = 180° − 44° − 52° = 84°$ and $\angle e = 84°$. At the left intersection on $v$: $\angle f = 84°$, $\angle h = 84°$, $\angle g = 96°$, $\angle i = 96°$. At the right intersection on $v$ (with the given 128°): $\angle k = 128°$, $\angle j = 52°$, $\angle l = 52°$.
In short — $\angle a = 52°,\ \angle b = 84°,\ \angle c = 44°,\ \angle d = 52°,\ \angle e = 84°,\ \angle f = 84°,\ \angle g = 96°,\ \angle h = 84°,\ \angle i = 96°,\ \angle j = 52°,\ \angle k = 128°,\ \angle l = 52°$. (Check: apex 44° + left foot 84° + right foot 52° = 180°.)
8. Find $\angle a, \angle b, \angle c$ from the figure (two parallel vertical lines, a horizontal line perpendicular to both, and a transversal making 38°).
Answer: The transversal makes 38° with the vertical lines. $\angle c$ and 38° lie on the same straight vertical line, so $\angle c = 180° − 38° = 142°$. The vertical lines are parallel, so by corresponding angles the transversal makes 38° with the right vertical as well: $\angle b = 38°$. Since the horizontal line is perpendicular to the vertical, $\angle a = 90° − 38° = 52°$.
9. In the figure $AB \parallel ED$, $ED \parallel FG$ and $EF \parallel CD$. If $\angle 2 = 50°$ and $\angle 4 = 65°$, find $\angle 1, \angle 3, \angle 5$.
Answer: $FG \parallel ED$ with transversal $EF$; $\angle 1$ and $\angle 2$ are co-interior angles, so $\angle 1 = 180° − \angle 2 = 180° − 50° = 130°$. $EF \parallel CD$ with transversal $ED$; $\angle 3$ and $\angle 2$ are alternate angles, so $\angle 3 = \angle 2 = 50°$. At $C$, $\angle 4 = 65°$ is the angle between $CD$ and $CB$; using $AB \parallel ED$ and $CD \parallel EF$, $\angle 5 = \angle 3 + \angle 4 = 50° + 65° = 115°$. Thus $\angle 1 = 130°,\ \angle 3 = 50°,\ \angle 5 = 115°$.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. When two lines intersect, the vertically opposite angles are — (a) equal (b) complementary (c) supplementary (d) 90°.
Answer: (a) equal.
2. Two adjacent angles whose sum is 180° form a — (a) vertically opposite pair (b) linear pair (c) complementary pair (d) corresponding pair.
Answer: (b) linear pair.
3. How many angles are formed when a transversal cuts two lines? (a) 4 (b) 6 (c) 8 (d) 12.
Answer: (c) 8.
4. When a transversal cuts two parallel lines, the corresponding angles are — (a) equal (b) supplementary (c) complementary (d) unequal.
Answer: (a) equal.
5. If the complement of an angle is 35°, the angle is — (a) 145° (b) 65° (c) 55° (d) 45°.
Answer: (c) 55° ($90° − 35° = 55°$).
6. If the supplement of an angle is 110°, the angle is — (a) 70° (b) 80° (c) 20° (d) 90°.
Answer: (a) 70° ($180° − 110° = 70°$).
7. If two lines are perpendicular, the angle between them is — (a) 45° (b) 60° (c) 90° (d) 180°.
Answer: (c) 90°.
8. When a transversal cuts parallel lines, the alternate angles are — (a) equal (b) sum 90° (c) sum 180° (d) unequal.
Answer: (a) equal.
9. The angle that equals its own supplement is — (a) 45° (b) 60° (c) 90° (d) 180°.
Answer: (c) 90°.
10. The angles that lie between the two lines cut by a transversal are called — (a) exterior angles (b) interior angles (c) adjacent angles (d) vertically opposite angles.
Answer: (b) interior angles.
Fill in the Blanks
1. A pair of lines that never meet is called ____ lines. Answer: parallel.
2. Two lines that form a 90° angle are called ____ lines. Answer: perpendicular.
3. Vertically opposite angles are always ____. Answer: equal.
4. If the sum of two angles is 90°, one is the ____ angle of the other. Answer: complementary.
5. When a transversal cuts two parallel lines, the corresponding angles are ____. Answer: equal.
True or False
1. The sum of vertically opposite angles is always 180°. Answer: False (vertically opposite angles are equal).
2. Parallel lines lie in the same plane. Answer: True.
3. A transversal cutting two lines forms six angles. Answer: False (it forms eight angles).
4. A 90° angle has no complement. Answer: True (its complement would be 0°, which is not an angle).
5. The alternate angles formed by a transversal cutting parallel lines are equal. Answer: True.
Short Answer Questions
1. What is a transversal? Answer: A line that cuts two or more lines of a plane at two or more distinct points is called a transversal (or oblique) line.
2. If an angle is 40°, find its complement and its supplement. Answer: Complement $= 90° − 40° = 50°$; supplement $= 180° − 40° = 140°$.
3. When a transversal cuts two parallel lines, one of a pair of co-interior angles is 65°. Find the other. Answer: Co-interior angles add up to 180°, so the other $= 180° − 65° = 115°$.
4. What is the difference between corresponding angles and alternate angles? Answer: Corresponding angles lie in the same relative position on the same side of the transversal; alternate angles lie on opposite sides of the transversal (as interior or exterior angles). For parallel lines both types are equal.
Key Terms
| Term | Meaning |
|---|---|
| Intersecting lines | Two lines that meet at a point |
| Point of intersection | The point where two lines meet |
| Perpendicular lines | Two lines that form a 90° angle |
| Parallel lines | A pair of lines that never meet |
| Transversal | A line cutting two or more lines |
| Adjacent angles | Angles with a common vertex and a common arm |
| Linear pair | Adjacent angles whose sum is 180° |
| Vertically opposite angles | Opposite (equal) angles at an intersection |
| Corresponding angles | Angles in the same position on the same side of a transversal |
| Alternate angles | Angles on opposite sides of a transversal |
| Interior / Exterior angles | Angles inside / outside the two lines |
| Complementary angles | Two angles whose sum is 90° |
| Supplementary angles | Two angles whose sum is 180° |