Question 1

Are  $a, b$  and  $c$  parallel to  $p, q$  and  $r$  respectively? Why or why not?

Accepted Answer

Line segments  $a, b$  and  $c$  are parallel to Line segments  $p, q$  and  $r$  respectively because:
-  $a$  and  $p$  lie on parallel lines.
-  $b$  and  $q$  are both perpendicular to these lines, so they are parallel to each other.
-  $c$  and  $r$  are parallel because the triangular folds create lines parallel to the same diagonal. — The reasoning is based on the properties of parallel and perpendicular lines. If two lines are perpendicular to the same line, they are parallel to each other. Also, folds creating triangles parallel to the diagonal ensure parallelism.

Question 2

Draw some lines perpendicular to the lines given on the dot paper in the Figure.

Accepted Answer

Draw lines perpendicular to the given lines on the dot paper as shown in the textbook figure. The perpendicular lines intersect the given lines at right angles (90 degrees). — Perpendicular lines meet at right angles. Using the dot paper, lines are drawn so that they intersect the given lines forming 90° angles.

Question 3

In the given Fig.5.11, mark the parallel lines using the notation given above (single arrow, double arrow, etc). Mark the angle between perpendicular lines with a square symbol.

(a) How did you spot the perpendicular lines?
(b) How did you spot the parallel lines?

Accepted Answer

(a) The vertical and horizontal lines on the grid paper meet at a 90-degree angle (a right angle), so they are perpendicular.

(b) Parallel lines are identified by lines that always remain the same distance apart. — Perpendicular lines intersect at right angles, which can be seen on grid paper. Parallel lines never meet and maintain equal distance, which can be visually identified.

Question 4

In the dot paper following, draw different sets of parallel lines. The line segments can be of different lengths but should have dots as endpoints.

Accepted Answer

Draw various sets of parallel lines on the dot paper, ensuring the line segments have dots as endpoints. The lines should be parallel, i.e., they never meet and are equidistant. — Parallel lines are drawn by keeping the distance between them constant and ensuring they do not intersect. Using dots as endpoints helps in precise drawing.

Question 5

Using your sense of how parallel lines look, try to draw lines parallel to the line segments on this dot paper.

(a) Did you find it challenging to draw some of them?
(b) Which ones?
(c) How did you do it?

Accepted Answer

(a) Yes, it was challenging.
(b) Line segments e, f, h, and g were challenging.
(c) Parallel lines were drawn by keeping them equidistant from the given line segments. — Drawing parallel lines requires maintaining equal distance from the original line. Some segments are harder due to orientation or spacing, requiring careful measurement.

Question 6

In Fig 5.13, which line is parallel to line  $a$  — line  $b$  or line  $c$ ? How do you decide this?

Accepted Answer

Line  $c$  is parallel to line  $a$  because these two lines are always at the same distance apart, which is the defining property of parallel lines. — Parallel lines maintain a constant distance and never intersect. By observing the figure, line  $c$  satisfies this condition with line  $a$ , whereas line  $b$  does not.

Question 7

⑤ Is it possible for all the eight angles to have different measurements? Why, why not?

What about five different angles — 6, 5, 4, 3 and 2?

Accepted Answer

No, it is not possible for all eight angles to have different measurements because vertically opposite angles are equal:
-  $	riangle 1 = 	riangle 3$
-  $	riangle 2 = 	riangle 4$
-  $	riangle 5 = 	riangle 7$
-  $	riangle 6 = 	riangle 8$

Regarding five different angles — 6, 5, 4, 3 and 2, it is also not possible because  $	riangle 2 = 	riangle 4$. — Vertically opposite angles formed by intersecting lines are equal, reducing the number of distinct angle measures possible.

Question 8

Can you draw a line parallel to  $l$ , that goes through point A? How will you do it with the tools from your geometry box? Describe your method.

Accepted Answer

Tools required: Ruler, Set-squares (right-angled triangle), Pencil, eraser

Steps:
1. Place the set square so that one side is along the line  $l$ .
2. Hold the ruler against the other side of the set square so that the ruler won't move.
3. Slide the set square along the ruler until one side reaches point A.
4. Draw a line along the edge of the set square through point A.

This new line is parallel to line  $l$  and passes through point A. — Using set squares and a ruler ensures the line drawn through point A is parallel to line  $l$  by maintaining the same angle and distance.

Question 9

Why are lines  $l$  and  $m$  parallel to each other?

Accepted Answer

Lines  $l$  and  $m$  are parallel because:
- Line  $t$  is constructed perpendicular to  $l$  passing through point A.
- Line  $m$  is constructed perpendicular to  $t$  through point A.
- Since both  $l$  and  $m$  are perpendicular to the same line  $t$ , they are parallel to each other.
- Corresponding angles formed are equal (each  $90^{	ext{o}}$ ). — Two lines perpendicular to the same line are parallel. The construction uses perpendicular lines to establish parallelism.

Question 10

Find the angles marked below.

(angles a, b, c, d, e, f, g, h, i, j as shown in figures)

Accepted Answer

The angles are:
-  $	riangle a = 48^{	ext{o}}$
-  $	riangle b = 52^{	ext{o}}$
-  $	riangle c = 81^{	ext{o}}$
-  $	riangle d = 99^{	ext{o}}$
-  $	riangle e = 69^{	ext{o}}$
-  $	riangle f = 48^{	ext{o}}$
-  $	riangle g = 122^{	ext{o}}$
-  $	riangle h = 75^{	ext{o}}$
-  $	riangle i = 54^{	ext{o}}$
-  $	riangle j = 97^{	ext{o}}$ — Using properties of parallel lines and angles, and possibly supplementary and vertically opposite angles, the values are calculated as above.

Question 11

Find the angle represented by  $a$ .

(i)
(ii)
(iii)
(iv) (four figures shown)

Accepted Answer

(i)  $138^{	ext{o}}$
(ii)  $118^{	ext{o}}$
(iii)  $105^{	ext{o}}$
(iv)  $23^{	ext{o}}$ — By using angle properties such as supplementary angles, alternate interior angles, and corresponding angles, the values of  $a$  in each figure are found.

Question 12

In the figures below, what angles do  $x$  and  $y$  stand for?

(i)
(ii) (two figures shown)

Accepted Answer

(i)  $	riangle x = 25^{	ext{o}}$,  $	riangle y = 155^{	ext{o}}$
(ii)  $	riangle x = 25^{	ext{o}}$ — Using angle sum properties and linear pair angles, the values of  $x$  and  $y$  are calculated.

Question 13

In Figure,  $	riangle ABC = 45^{	ext{o}}$  and  $	riangle IKJ = 78^{	ext{o}}$ . Find angles  $	riangle GEH$ ,  $	riangle HEF$ ,  $	riangle FED$ .

Accepted Answer

Given:
-  $	riangle ABC = 45^{	ext{o}}$
-  $	riangle IKJ = 78^{	ext{o}}$

Using properties of parallel lines and angles:
-  $	riangle GEH = 45^{	ext{o}}$
-  $	riangle HEF = 57^{	ext{o}}$
-  $	riangle FED = 78^{	ext{o}}$ — Angles are found using alternate interior angles, corresponding angles, and angle sum properties in triangles.

Question 14

In the Figure, AB is parallel to CD, and CD is parallel to EF. Also, EA is perpendicular to AB. If  $	riangle BEF = 55^{	ext{o}}$ , find the values of  $x$  and  $y$ .

Accepted Answer

Given:
- AB || CD
- CD || EF
- EA ⟂ AB
-  $	riangle BEF = 55^{	ext{o}}$

Since AB || CD || EF, corresponding angles are equal.
Therefore,  $	riangle x = 	riangle y = 125^{	ext{o}}$ (since 180° - 55° = 125°). — Using the properties of parallel lines and the fact that EA is perpendicular to AB, the angles  $x$  and  $y$  are calculated as corresponding angles.

Question 15

What is the measure of angle  $	riangle$  NOP in the given figure?

[Hint: Draw lines parallel to LM and PQ through points N and O]

Accepted Answer

By drawing lines parallel to LM and PQ through points N and O, and using properties of parallel lines and angles, the measure of  $	riangle NOP$  is found to be  $108^{	ext{o}}$. — Using parallel lines and transversal angle properties, the angle  $	riangle NOP$  is calculated as 108°.

Intersecting

Intersecting — Study Notes

5.1 Across the Line

5.2 Perpendicular Lines

5.3 Between Lines

Practice Questions — Intersecting

All 8 Chapters in Ganita Prakash