CLASS 7 · GANITA PRAKASH PART I · CHAPTER 5
Explore intersecting and parallel lines, perpendicularity, transversals and angle relationships.
37 worked answers · short steps first, explanations when you need them
Textbook pages 106–108
Question 1
If two lines shared two points, the unique straight line through those points would be both lines.
No. Two distinct straight lines can have at most one intersection point.
Question 2
Adjacent angles form a straight angle and sum to 180°.
Opposite angles are equal.
The four angles alternate 120°, 60°, 120°, 60°; the other three are 60°, 120° and 60°.
Question 3
Use vertical opposition and straight-line angles.
∠a=∠c; ∠b=∠d. Adjacent pairs (a,b), (b,c), (c,d), (d,a) each sum to 180°.
Question 4
Angles around a point total 360°.
Divide by four.
Each angle is 90°, so the lines are perpendicular.
Textbook pages 109–114
Question 1
Compare direction and spacing of the labelled segments.
a, i and h are parallel; c∥g; d∥f; e∥b.
Question 2
Opposite sides have the same direction; adjacent sides meet at right angles.
Each new fold splits every existing interval in half.
Opposite edges are parallel; adjacent edges are perpendicular. Including the original top and bottom edges, the counts after 1, 2 and 3 successive midpoint folds are 3, 5 and 9 parallel horizontal lines. In general, after n folds the count is 2^n+1. The new vertical crease is perpendicular to them.
Question 3
Compare the corresponding fold directions.
Each pair has the same orientation; the constructed crease lines are equally directed.
Yes: a∥p, b∥q and c∥r in the printed construction.
Question 4
Check whether the lines meet at a square corner.
A square corner measures 90°.
The horizontal and vertical grid lines are perpendicular because they meet at 90°.
Question 5
Place one set-square edge along l.
Hold a ruler firmly against another edge.
Slide the set square along the ruler until the first edge passes through A, then draw.
The line through A is parallel to l because the set square keeps the same angle with the guide as it slides.
Question 6
Check which candidate keeps the same direction and spacing as a.
Line c is parallel to a.
Textbook pages 115–122
Question 1
At each crossing, vertical opposite angles are equal.
The two angles at one crossing repeat in opposite pairs.
No. There are at most four distinct measures among the eight. In the second set, ∠2=∠4, so all five cannot be different.
Question 2
Match angles that occupy the same corner position at their intersections.
(1,5), (2,6), (3,7), (4,8).
Question 3
Equal corresponding angles imply the lines are parallel.
If they are unequal, the lines cannot be parallel.
Equal: the lines are parallel. Unequal: they are not parallel.
Question 4
Each makes a 90° angle with t.
These are equal corresponding angles on transversal t.
l∥m.
Question 5
The corresponding angle ∠b equals ∠f.
∠d is vertically opposite ∠b.
∠d=120°.
Question 6
Corresponding and vertical opposite angles match ∠6.
Adjacent angles are supplementary: 180°−135°.
∠2=∠4=∠6=∠8=135°; ∠1=∠3=∠5=∠7=45°.
Question 7
The angle adjacent to ∠a is 180°−120°=60°.
Compare this corresponding angle with 70°.
No. The corresponding angles are 60° and 70°, so they are not equal.
Question 8
The angle adjacent to ∠3 is 180°−50°=130°.
That angle corresponds to ∠6.
∠6=130°. Same-side interior angles add to 180°.
Question 9
Along transversal AD, interior angles give ∠DAB=180°−60°=120°.
Split ∠DAB into 65°+∠CAB.
Along transversal CD, ∠ADC+∠BCD=180°.
Use the other parallel pair for ∠ABC.
∠CAB=55°; ∠ABC=60°; ∠BCD=120°.
Textbook pages 123–125
Question 1
Use vertical opposite angles for matching pairs.
Use linear pairs for angles on a straight line.
Use the parallel-line angle relationships where marked.
a=48°; b=52°; c=81°; d=99°; e=69°; f=48°; g=122°; h=75°; i=54°; j=97°.
Question 2
For each figure, identify whether a forms a linear pair, equals a vertical angle, or corresponds to a marked parallel-line angle.
Question 3
(i) The two intersecting lines give vertically opposite x angles. The linear pair next to x is y.
(ii) Use the straight-line angle in the second drawing.
Question 4
Transfer the given angle measures along the parallel segments as shown.
Use a straight angle at E to find the middle angle.
∠GEH=45°; ∠HEF=57°; ∠FED=78°.
Question 5
Use the common transversal to identify corresponding angles.
The marked angles are supplementary to 55°.
x=y=125°.
Question 6
Draw the suggested parallel guides through N and O.
Copy corresponding/alternate angles onto the new intersections.
Combine the resulting angles at O.
∠NOP=108°.
Textbook pages 106–125
Question intersecting-angle-pattern
Compare opposite angles at each crossing.
Add neighbouring angles that make a straight line.
Opposite angles are equal, and adjacent angles add to 180°. Small differences in measured drawings come from drawing and protractor errors.
Question figure-5-5-observations
Name the two segments you are comparing.
Check whether they share an endpoint, cross, or stay apart when extended.
Answers depend on the selected pair. For example, FG and FH meet at endpoint F and form 115.3°. State the pair each time; other pairs have different relationships.
Question figure-5-10-construction
At the point where your lines meet, make a right angle.
Use a square corner to check the 90° angle.
Many drawings are correct. Any line that meets a given line at 90° is perpendicular to it.
Question figure-5-11-markings
Group lines with the same direction and spacing.
Mark a square where horizontal and vertical grid lines meet.
Use matching arrow marks for parallel families. The horizontal and vertical grid lines are perpendicular because they meet at 90°.
Question figure-5-12-parallel-dots
Choose a direction between two dots.
Draw another segment in the same direction; its length may differ.
Many answers are possible. For instance, two horizontal segments on different rows are parallel, even if they have different lengths.
Question figure-5-13-copies
Keep the same direction or slope as the original segment.
Use a ruler through suitable grid points to draw each copy.
Many drawings are possible. The book notes e, f, g and h are harder to copy; keep each new segment in the same direction as its model.
Question activity-3-corresponding
Make one angle 60° at the first crossing.
Copy the corresponding 60° angle at the second crossing.
At each crossing, the other angle is 180°−60°=120°.
Draw equal corresponding angles, such as 60° at both crossings. Then all eight angles are either 60° or 120°, and the crossed lines are parallel.
Question figure-5-20-transversal
Choose any angle, such as 60°, at the first crossing.
Copy it into the matching position at the second crossing.
Many constructions work. Copying a 60° corresponding angle makes the two lines parallel.
Question figure-5-21-set-square
Draw a guide line.
Slide the set square along a fixed ruler and draw two lines perpendicular to the guide.
Both constructed lines make 90° corresponding angles with the guide, so they are parallel to each other.
Question figure-5-22-set-square
Keep the ruler fixed as a guide.
Slide without turning the set square and draw through new positions.
The new lines remain parallel because sliding preserves their direction. A transversal check gives equal corresponding angles.
Question paper-fold-parallel
Fold crease t through A perpendicular to l.
Fold crease m through A perpendicular to t.
The crease m is parallel to l because both are perpendicular to t.
Question parallel-illusion
Follow the long lines, ignoring the short crossing marks.
Check their spacing at several places.
The long lines remain parallel and equally spaced. The repeated crossing strokes can make our eyes think the lines bend or spread.