GSEB Class 7 Maths Solutions Chapter 5 Lines and Angles InText Questions

Try These (Page 94)

Question 1. List ten figures around you and identify the acute, obtuse and right angles found in them.
Answer: This is a hands-on activity. You should look around your own environment and point out examples of various angles. For instance, a book's corner forms a right angle, the tip of a knife might show an acute angle, and an opened laptop screen could display an obtuse angle. Exploring your surroundings will help you recognize these geometric shapes naturally.
In simple words: This is an activity. Look at things around you and find different types of angles.

Exam Tip: For observation-based questions, provide concrete examples from everyday life to show clear understanding of the concepts.

Think, Discuss and Write (Page 95)

Question 1. Can two acute angles be complement to each other?
Answer: Yes. For instance, \( 30^\circ \) is an acute angle, and \( 60^\circ \) is also an acute angle. Since \( 30^\circ + 60^\circ = 90^\circ \), these two acute angles can be complementary to one another.
In simple words: Yes, two small angles can add up to 90 degrees. For example, 30 and 60 degrees.

Exam Tip: Remember that complementary angles must sum up to exactly \( 90^\circ \). Acute angles are always less than \( 90^\circ \).

Question 2. Can two obtuse angles be complement to each other?
Answer: No. An obtuse angle measures more than \( 90^\circ \). Therefore, the total of two obtuse angles cannot be equal to \( 90^\circ \), meaning they cannot be complementary to each other.
In simple words: No, because two big angles, each bigger than 90 degrees, will always add up to more than 90 degrees.

Exam Tip: Understand the definition of an obtuse angle (greater than \( 90^\circ \)) to quickly determine if it can form a complementary pair.

Question 3. Can two right angles be complement to each other?
Answer: No. A right angle measures exactly \( 90^\circ \). Therefore, two right angles would sum to \( 90^\circ + 90^\circ = 180^\circ \), which means they cannot be complementary to one another.
In simple words: No, because a right angle is 90 degrees. Two right angles would make 180 degrees, not 90 degrees.

Exam Tip: Clearly state the measure of a right angle (\( 90^\circ \)) and the sum required for complementary angles (\( 90^\circ \)) to explain why this is not possible.

Try These (Page 95)

Question 1. Which pairs of following angles are complementary?
(i) 70°, 20°
(ii) 75°, 25°
(iii) 48°, 52°
(iv) 35°, 55°
Answer:
(i) Since \( 70^\circ + 20^\circ = 90^\circ \).
\( \implies \) Angles \( 70^\circ \) and \( 20^\circ \) are complementary.
(ii) Since \( 75^\circ + 25^\circ = 100^\circ \) and \( 100^\circ \ne 90^\circ \).
\( \implies \) Angles \( 75^\circ \) and \( 25^\circ \) are not complementary.
(iii) Since \( 48^\circ + 52^\circ = 100^\circ \) and \( 100^\circ \ne 90^\circ \).
\( \implies \) Angles \( 48^\circ \) and \( 52^\circ \) are not complementary.
(iv) Since \( 35^\circ + 55^\circ = 90^\circ \).
\( \implies \) The angles \( 35^\circ \) and \( 55^\circ \) are complementary.
In simple words: Add the two angles together. If they make 90 degrees, they are complementary. If they don't, then they are not.

Exam Tip: To determine if angles are complementary, always sum their measures. If the total is exactly \( 90^\circ \), they are. Otherwise, they are not.

Question 2. What is the measure of the complement of each of the following angles?
(i) 45°
(ii) 65°
(iii) 41°
(iv) 54°
Answer:
(i) Let the complement of \( 45^\circ \) be \( x \).
\( \implies x + 45^\circ = 90^\circ \) or \( x = 90^\circ - 45^\circ = 45^\circ \).
\( \implies \) The complement of \( 45^\circ \) is \( 45^\circ \).
(ii) Let the complement of \( 65^\circ \) be \( p \).
\( \implies p + 65^\circ = 90^\circ \) or \( p = 90^\circ - 65^\circ = 25^\circ \).
\( \implies \) The complement of \( 65^\circ \) is \( 25^\circ \).
(iii) Let the complement of \( 41^\circ \) be \( m \).
\( \implies m + 41^\circ = 90^\circ \) or \( m = 90^\circ - 41^\circ = 49^\circ \).
\( \implies \) Thus, the complement of \( 41^\circ \) is \( 49^\circ \).
(iv) Let the complement of \( 54^\circ \) be \( y \).
\( \implies y + 54^\circ = 90^\circ \) or \( y = 90^\circ - 54^\circ = 36^\circ \).
\( \implies \) Thus, the complement of \( 54^\circ \) is \( 36^\circ \).
In simple words: To find the complement of an angle, subtract that angle from 90 degrees. The number you get is the complementary angle.

Exam Tip: The sum of an angle and its complement must always be \( 90^\circ \). Set up an equation \( \text{angle} + \text{complement} = 90^\circ \) to solve.

Question 3. The difference in the measures of two complementary angles is 12°. Find the measures of the angles.
Answer: Let one of the angles be \( x \). Their difference is \( 12^\circ \), so the other angle will be \( x + 12^\circ \). Since the sum of the measures of two complementary angles is \( 90^\circ \):
\( x + (x + 12^\circ) = 90^\circ \)
\( \implies 2x = 90^\circ - 12^\circ \)
\( \implies 2x = 78^\circ \)
Dividing both sides by 2, we obtain:
\( x = \frac { 78^\circ }{ 2 } \)
\( \implies x = 39^\circ \)
Therefore, one angle is \( 39^\circ \). And the other angle is \( 39^\circ + 12^\circ = 51^\circ \).
In simple words: If two angles add up to 90 degrees and are 12 degrees different, then one angle is 39 degrees and the other is 51 degrees.

Exam Tip: When dealing with two unknown angles with a given sum and difference, represent one angle as \( x \) and the other as \( x + \text{difference} \). Then, form an equation based on their sum.

Think, Discuss and Write (Page 96)

Question 1. Can two obtuse angles be supplementary?
Answer: No. An obtuse angle's measure is always greater than \( 90^\circ \). Therefore, the sum of two angles, where each is more than \( 90^\circ \), will be greater than \( 180^\circ \). Thus, two obtuse angles cannot be supplementary.
In simple words: No, because an obtuse angle is bigger than 90 degrees. So, two obtuse angles added together will be much bigger than 180 degrees.

Exam Tip: Supplementary angles must sum to \( 180^\circ \). An obtuse angle alone is already more than halfway to \( 180^\circ \).

Question 2. Can two acute angles be supplementary?
Answer: No. An acute angle's measure is always less than \( 90^\circ \). Consequently, the sum of two acute angles will be less than \( 180^\circ \). Thus, two acute angles cannot be supplementary.
In simple words: No, because an acute angle is smaller than 90 degrees. Two small angles added together will always be less than 180 degrees.

Exam Tip: Keep in mind that for supplementary angles, the sum must be exactly \( 180^\circ \). Two angles each smaller than \( 90^\circ \) cannot achieve this sum.

Question 3. Can two right angles be supplementary?
Answer: Yes, because the total measure of two right angles is \( 90^\circ + 90^\circ = 180^\circ \). Since their sum is \( 180^\circ \), two right angles are supplementary.
In simple words: Yes, because two right angles are 90 degrees each. When you add 90 and 90, you get 180 degrees, which is what supplementary angles need.

Exam Tip: Remember the precise definition: two angles are supplementary if their sum is \( 180^\circ \). A right angle perfectly fits this condition when paired with another right angle.

Try These (Page 96)

Question 1. Find the pairs of supplementary angles in the following figures:
(i) Angles 110° and 50°
(ii) Angles 105° and 65°
(iii) Angles 50° and 130°
(iv) Angles 45° and 45°
Answer:
(i) Measures of the given angles are \( 110^\circ \) and \( 50^\circ \). Since \( 110^\circ + 50^\circ = 160^\circ \) and \( 160^\circ \ne 180^\circ \), \( 110^\circ \) and \( 50^\circ \) are not a pair of supplementary angles.
(ii) Measures of the given angles are \( 105^\circ \) and \( 65^\circ \). Since \( 105^\circ + 65^\circ = 170^\circ \) and \( 170^\circ \ne 180^\circ \), \( 105^\circ \) and \( 65^\circ \) are not a pair of supplementary angles.
(iii) Measures of the given angles are \( 50^\circ \) and \( 130^\circ \). Since \( 50^\circ + 130^\circ = 180^\circ \), \( 50^\circ \) and \( 130^\circ \) are a pair of supplementary angles.
(iv) Measures of the given angles are \( 45^\circ \) and \( 45^\circ \). Since \( 45^\circ + 45^\circ = 90^\circ \) and \( 90^\circ \ne 180^\circ \), \( 45^\circ \) and \( 45^\circ \) are not a pair of supplementary angles.
In simple words: To check for supplementary angles, simply add them up. If the total is 180 degrees, they are supplementary. Otherwise, they are not.

Exam Tip: Always sum the given angles and compare the result to \( 180^\circ \). This is the key step to identify supplementary pairs.

Question 2. What will be the measure of the supplement of each one of the following angles?
(i) 100°
(ii) 90°
(iii) 55°
(iv) 125°
Answer:
(i) Let the supplement of \( 100^\circ \) be \( x \).
\( \implies 100^\circ + x = 180^\circ \)
\( \implies x = 180^\circ - 100^\circ = 80^\circ \).
The measure of the supplement of \( 100^\circ \) is \( 80^\circ \).
(ii) Let the supplement of \( 90^\circ \) be \( x \).
\( \implies x + 90^\circ = 180^\circ \)
\( \implies x = 180^\circ - 90^\circ = 90^\circ \).
The measure of the supplement of \( 90^\circ \) is \( 90^\circ \).
(iii) Let the supplement of \( 55^\circ \) be \( m \).
\( \implies 55^\circ + m = 180^\circ \)
\( \implies m = 180^\circ - 55^\circ = 125^\circ \).
The supplement of \( 55^\circ \) is \( 125^\circ \).
(iv) Let the supplement of \( 125^\circ \) be \( y \).
\( \implies y + 125^\circ = 180^\circ \)
\( \implies y = 180^\circ - 125^\circ = 55^\circ \).
The supplement of \( 125^\circ \) is \( 55^\circ \).
In simple words: To find the supplement of an angle, subtract that angle from 180 degrees. The remaining value is the supplementary angle.

Exam Tip: Always remember that an angle and its supplement add up to \( 180^\circ \). To find the supplement, simply subtract the given angle from \( 180^\circ \).

Question 3. Among two supplementary angles the measure of the larger angle is 44° more than the measure of the smaller. Find their measures.
Answer: Let the smaller angle be \( x \). Therefore, the measure of the larger angle will be \( (x + 44^\circ) \). Since the two angles are supplementary, their sum is \( 180^\circ \):
\( x + (x + 44^\circ) = 180^\circ \)
\( \implies 2x + 44^\circ = 180^\circ \)
\( \implies 2x = 180^\circ - 44^\circ \)
\( \implies 2x = 136^\circ \)
\( \implies x = \frac { 136^\circ }{ 2 } \)
\( \implies x = 68^\circ \)
Therefore, the smaller angle is \( 68^\circ \). The larger angle is \( 68^\circ + 44^\circ = 112^\circ \).
In simple words: If two angles add to 180 degrees and one is 44 degrees bigger than the other, the angles are 68 degrees and 112 degrees.

Exam Tip: Define both angles using a single variable based on the given relationship. Then, set up an equation where their sum equals \( 180^\circ \) for supplementary angles.

Try These (Page 97)

Question 1. Are the angles marked 1 and 2 adjacent? If they are not adjacent, say, 'why'.
Answer:
(i) Yes, \( \angle 1 \) and \( \angle 2 \) are adjacent angles.
(ii) Yes, \( \angle 1 \) and \( \angle 2 \) are adjacent angles.
(iii) No, \( \angle 1 \) and \( \angle 2 \) are not adjacent angles because they have no common vertex.
(iv) No, \( \angle 1 \) and \( \angle 2 \) are not adjacent angles because \( \angle 1 \) is a part of \( \angle 2 \).
(v) Yes, \( \angle 1 \) and \( \angle 2 \) are adjacent angles.
In simple words: Adjacent angles share a common side and vertex, and they don't overlap. If they don't have these, then they are not adjacent.

Exam Tip: For angles to be adjacent, they must share a common vertex, a common arm (side), and their non-common arms must lie on opposite sides of the common arm.

Question 2. In the given figure, are the following adjacent angles?
(a) \( \angle AOB \) and \( \angle BOC \)
(b) \( \angle BOD \) and \( \angle BOC \) Justify your answer.
Answer:
(a) Yes, \( \angle AOB \) and \( \angle BOC \) are adjacent angles because they have a common vertex O and their outer arms (OA and OC) are on either side of the common arm OB.
(b) No, because \( \angle BOC \) is a part of \( \angle BOD \). For angles to be adjacent, they should not overlap, and one should not be inside the other.
In simple words: Angles are adjacent if they are next to each other, sharing a point and a side, without one angle being inside another.

Exam Tip: Understand the conditions for adjacent angles: common vertex, common arm, and non-common arms on opposite sides. If one angle is included within another, they are not adjacent.

Think, Discuss and Write (Page 98)

Question 1. Can two adjacent angles be supplementary?
Answer: Yes, in the adjoining figure, \( \angle AOB \) and \( \angle BOC \) are adjacent angles. Also, \( \angle AOB + \angle BOC = 180^\circ \). Therefore, \( \angle AOB \) and \( \angle BOC \) are supplementary.
In simple words: Yes, two angles that are next to each other can add up to 180 degrees, like angles on a straight line.

Exam Tip: A linear pair is a classic example of two adjacent angles that are supplementary. Ensure you can identify such pairs in diagrams.

Question 2. Can two adjacent angles be complementary?
Answer: Yes, in the figure, \( \angle PQR \) and \( \angle RQS \) are adjacent angles. Also \( \angle PQR + \angle RQS = 90^\circ \). This means \( \angle PQR \) and \( \angle RQS \) are complementary angles.
In simple words: Yes, two angles next to each other can add up to 90 degrees.

Exam Tip: When angles form a right angle (\( 90^\circ \)) and share a common vertex and arm, they are adjacent and complementary.

Question 3. Can two obtuse angles be adjacent angles?
Answer: Yes, in the figure, \( \angle BOC \) and \( \angle COD \) are obtuse angles, and they are adjacent angles. They share the common vertex O and the common arm OC, with their non-common arms OB and OD on opposite sides of OC.
In simple words: Yes, two angles bigger than 90 degrees can be next to each other, sharing a side and a corner.

Exam Tip: Adjacency only requires shared vertex and arm, not a specific sum of angle measures. So, two obtuse angles can indeed be adjacent.

Question 4. Can an acute angle be adjacent to an obtuse angle?
Answer: Yes, in the figure, \( \angle 1 \) and \( \angle 2 \) are adjacent angles. Here, \( \angle 1 \) is an acute angle and \( \angle 2 \) is an obtuse angle. They share a common vertex and a common arm, making them adjacent.
In simple words: Yes, a small angle and a big angle can be next to each other, sharing a side and a corner.

Exam Tip: The type of angle (acute, obtuse, right) does not prevent angles from being adjacent. Adjacency depends solely on their positional relationship.

Think, Discuss and Write (Page 99)

Question 1. Can two acute angles form a linear pair?
Answer: No. The sum of the measures of two acute angles is less than \( 180^\circ \), because the measure of each acute angle is less than \( 90^\circ \). A linear pair must add up to exactly \( 180^\circ \).
In simple words: No, because two small angles will always add up to less than 180 degrees, which is needed for a linear pair.

Exam Tip: A linear pair consists of two adjacent angles whose non-common arms form a straight line, summing to \( 180^\circ \).

Question 2. Can two obtuse angles form a linear pair?
Answer: No. The sum of the measures of two obtuse angles is more than \( 180^\circ \). Since each obtuse angle is greater than \( 90^\circ \), their combined total will always exceed \( 180^\circ \), meaning they cannot form a linear pair.
In simple words: No, because two big angles will always add up to more than 180 degrees, so they cannot form a straight line.

Exam Tip: An obtuse angle is already greater than \( 90^\circ \). Two such angles would always exceed the \( 180^\circ \) required for a linear pair.

Question 3. Can two right angles form a linear pair?
Answer: Yes, because the sum of two right angles is \( 90^\circ + 90^\circ = 180^\circ \). When two right angles are adjacent and their non-common arms form a straight line, they constitute a linear pair.
In simple words: Yes, two right angles (90 degrees each) add up to 180 degrees, so they can form a straight line.

Exam Tip: Two right angles forming a linear pair is a common scenario, as their sum perfectly equals \( 180^\circ \).

Try These (Page 99)

Question 1. Check which of the following pairs of angles form a linear pair.
(i) 140°, 40°
(ii) 60°, 90°
(iii) 90°, 80°
(iv) 115°, 65°
Answer:
(i) Yes. Since \( 140^\circ + 40^\circ = 180^\circ \), this pair of angles forms a linear pair.
(ii) No. Since \( 60^\circ + 90^\circ = 150^\circ \) and \( 150^\circ \ne 180^\circ \), this pair of angles does not form a linear pair.
(iii) No. Since \( 90^\circ + 80^\circ = 170^\circ \) and \( 170^\circ \ne 180^\circ \), this pair of angles does not form a linear pair.
(iv) Yes. Since \( 115^\circ + 65^\circ = 180^\circ \), this pair of angles forms a linear pair.
In simple words: Add the two angles together. If they make 180 degrees, they are a linear pair. If they don't, then they are not.

Exam Tip: For a pair of angles to form a linear pair, they must be adjacent and their sum must be exactly \( 180^\circ \).

Try These (Page 101)

Question 1. In the given figure, if \( \angle 1 = 30^\circ \), find \( \angle 2 \) and \( \angle 3 \).
Answer: \( \angle 3 \) and \( \angle 1 \) are vertically opposite angles, so \( \angle 3 = \angle 1 \).
But \( \angle 1 = 30^\circ \) [Given]
\( \implies \angle 3 = 30^\circ \).
Again, \( \angle 3 \) and \( \angle 2 \) form a linear pair.
\( \implies \angle 3 + \angle 2 = 180^\circ \)
\( \implies 30^\circ + \angle 2 = 180^\circ \)
\( \implies \angle 2 = 180^\circ - 30^\circ = 150^\circ \).
Thus, \( \angle 2 = 150^\circ \) and \( \angle 3 = 30^\circ \).
In simple words: Since angle 1 is 30 degrees, angle 3 is also 30 degrees because they are opposite. Angle 3 and angle 2 make a straight line, so angle 2 is 180 minus 30, which is 150 degrees.

Exam Tip: Remember two key angle relationships: vertically opposite angles are equal, and angles in a linear pair sum to \( 180^\circ \). Use these properties to find unknown angles.

Question 2. Give an example for vertically opposite angles in your surroundings.
Answer: Vertically opposite angles can be seen in many places. For example, when two roads cross, the angles formed across from each other are vertically opposite. Another instance is the "X" shape created by the scissor blades when open; the angles at the pivot point facing each other are vertically opposite. The intersection of railway tracks also creates vertically opposite angles.
In simple words: When two lines cross, the angles that are directly opposite each other are vertically opposite angles. Think of an 'X' shape.

Exam Tip: Look for any 'X' formation in objects or structures. The angles directly across from each other at the intersection point will be vertically opposite.

Think, Discuss and Write (Page 104)

Question 1. In the figure, AC and BE intersect at P. AC and BC intersect at C, AC and EC intersect at C. Try to find another ten pairs of intersecting line segments.
Answer: Various pairs of intersecting lines are:
(i) Intersecting at B are: [CB and EB], [PB and CB], [AB and EB], [AB and CB], [AB and PB]
(ii) Intersecting at C are: [AC and BC], [PC and BC], [EC and BC], [AC and EC], [PC and EC]
(iii) Intersecting at E are: [BE and CE], [PE and CE]
(iv) Intersecting at P are: [BP and CP], [EP and CP]
(v) [AP and BP] and [AP and EP]
In simple words: Look for any two lines that cross each other at a single point. Each point where lines meet creates a set of intersecting line segments.

Exam Tip: When listing intersecting line segments, specify the point of intersection. Be methodical by focusing on one intersection point at a time to ensure all pairs are identified.

Question 2. Can two lines intersect in more than one point? Think about it.
Answer: No, two distinct lines cannot intersect in more than one point. If they did, they would effectively be the same line, as two points define a unique line.
In simple words: No, two different straight lines can only cross each other at one single spot. If they cross at more than one spot, they are actually the same line.

Exam Tip: This is a fundamental postulate in geometry: two distinct lines intersect at most at one point. Understanding this helps prevent logical errors.

Try These (Page 104)

Question 1. Find examples from your surroundings where lines intersect at right angles.
Answer: Lines intersecting at right angles are very common. Consider the corner of a room, where two walls meet the floor or ceiling. The edges form right angles. The cross-sections of windowpanes, the intersection of horizontal and vertical lines on a graph paper, or the corners of a book or a table all demonstrate lines intersecting at \( 90^\circ \).
In simple words: Look for corners or cross shapes in your daily life, like where walls meet, or the edges of a window. These often make perfect 90-degree angles.

Exam Tip: When providing real-world examples, pick clear and undeniable instances of perpendicular lines or right angles to demonstrate understanding.

Question 2. Find the measures of the angles made by the intersecting lines at the vertices of an equilateral triangle.
Answer: For an equilateral triangle, all its interior angles are equal and each measures \( 60^\circ \). Therefore, the angles made by the intersecting lines at the vertices A, B, and C are:
Measure of \( \angle A = 60^\circ \)
Measure of \( \angle B = 60^\circ \)
Measure of \( \angle C = 60^\circ \)
In simple words: In an equilateral triangle, all three angles are the same, so each angle is 60 degrees.

Exam Tip: Remember that an equilateral triangle has three equal sides and three equal angles, each measuring \( 60^\circ \).

Question 3. Find the measures of angles at the four vertices made by the intersecting lines of a rectangle.
Answer: A rectangle is a quadrilateral where all four interior angles are right angles, meaning each measures \( 90^\circ \). Therefore, the measures of the angles made by the intersecting sides at the four vertices A, B, C, and D are:
Measure of \( \angle A = 90^\circ \)
Measure of \( \angle B = 90^\circ \)
Measure of \( \angle C = 90^\circ \)
Measure of \( \angle D = 90^\circ \)
In simple words: In a rectangle, all four corners are perfect square corners, which means each angle is 90 degrees.

Exam Tip: Rectangles and squares are defined by their right-angle corners. Always recall that each interior angle in these shapes is \( 90^\circ \).

Question 4. If two lines intersect, do they always intersect at right angles?
Answer: No. Two lines can intersect at any angle. They only intersect at right angles if they are perpendicular to each other. For example, roads often cross at various angles, not just right angles.
In simple words: No, lines can cross at any angle. They only make 90-degree angles if they are perpendicular.

Exam Tip: Differentiate between intersecting lines (which can cross at any angle) and perpendicular lines (which specifically intersect at \( 90^\circ \)).

Try These (Page 105)

Question 1. Suppose two lines are given. How many transversals can you draw for these lines?
Answer: We can draw an infinite number of transversals to two given lines. A transversal is simply a line that crosses two or more other lines, and there are countless ways to draw such a line.
In simple words: You can draw endless lines that cut across two other lines.

Exam Tip: A transversal can be drawn at any angle and position across two lines. Therefore, there are limitless possibilities.

Question 2. If a line is a transversal to three lines, how many points of intersections are there?
Answer: As shown in the adjoining figure, there are 3 distinct points of intersection. Each time the transversal crosses one of the three lines, it creates a new point where they meet.
In simple words: If one line cuts across three other lines, it will create three separate crossing points.

Exam Tip: Each intersection of the transversal with one of the other lines counts as one distinct point of intersection. So, for 'n' lines, there will be 'n' intersection points if they are distinct.

Question 3. Try to identify a few transversals in your surroundings.
Answer: You can find transversals everywhere. Imagine railway tracks (two parallel lines) crossed by a road (the transversal). Or Venetian blinds (parallel slats) crossed by the pull cords (transversals). Even lines on ruled paper (parallel lines) intersected by a diagonal pencil mark (a transversal). Another example could be the support beams (transversals) that cut across the horizontal planks (parallel lines) of a fence.
In simple words: Look for examples where one line crosses two or more other lines, like a road crossing railway tracks or strings on a guitar.

Exam Tip: Focus on identifying a line that cuts across other lines, regardless of whether those other lines are parallel or not.

Try These (Page 106)

Question 1. Name the pairs of angles in each figure:
(i) Figure showing \( \angle 1 \) and \( \angle 2 \)
(ii) Figure showing \( \angle 3 \) and \( \angle 4 \)
(iii) Figure showing \( \angle 5 \) and \( \angle 6 \)
(iv) Figure showing \( \angle 7 \) and \( \angle 8 \)
(v) Figure showing \( \angle 9 \) and \( \angle 10 \)
(vi) Figure showing \( \angle 11 \) and \( \angle 12 \)
Answer:
(i) \( \angle 1 \) and \( \angle 2 \) are a pair of corresponding angles.
(ii) \( \angle 3 \) and \( \angle 4 \) are a pair of alternate interior angles.
(iii) \( \angle 5 \) and \( \angle 6 \) are a pair of interior angles on the same side of the transversal.
(iv) \( \angle 7 \) and \( \angle 8 \) are a pair of corresponding angles.
(v) \( \angle 9 \) and \( \angle 10 \) are a pair of alternate interior angles.
(vi) \( \angle 11 \) and \( \angle 12 \) are linear pair of angles.
In simple words: When a line cuts two other lines, special pairs of angles are formed like corresponding (same position), alternate interior (inside, opposite sides), same-side interior (inside, same side), or linear pair (next to each other on a straight line).

Exam Tip: Familiarize yourself with all angle pairs formed by a transversal: corresponding, alternate interior, alternate exterior, interior on the same side, and vertically opposite. Practice identifying them quickly.

Try These (Page 109)

Question 1.
(i) Lines \( l \parallel m \); t is a transversal \( \angle x = ? \)
(ii) Lines \( a \parallel b \); c is a transversal \( \angle y = ? \)
(iii) \( l_1 \), \( l_2 \) be two lines t is a transversal Is \( \angle 1 = \angle 2 \)?
(iv) Lines \( l \parallel m \); t is a transversal \( \angle z = ? \)
(v) Lines \( l \parallel m \); t is a transversal \( \angle x = ? \)
(vi) Lines \( l \parallel m \); t is a transversal find a, b, c, d.

Answer:
(i) \( x = 60^\circ \). This is because \( \angle x \) and \( 60^\circ \) are alternate interior angles, and these angles are always equal when lines are parallel.

Exam Tip: Remember that alternate interior angles are equal only when the lines intersected by the transversal are parallel. Always check for the parallel symbol.

Answer:
(ii) \( y = 55^\circ \). This happens because \( \angle y \) and \( 55^\circ \) are alternate interior angles, and alternate interior angles are equal when lines are parallel.

Exam Tip: Alternate interior angles lie between the two lines on opposite sides of the transversal. Identifying their positions is important for solving these problems.

Answer:
(iii) No, \( \angle 1 \) and \( \angle 2 \) are not equal. This is because lines \( l_1 \) and \( l_2 \) are not parallel. When lines are not parallel, pairs of angles like corresponding or alternate interior are generally not equal.

Exam Tip: The equality of angles formed by a transversal (like corresponding, alternate interior, alternate exterior) is a property specific to parallel lines. Without parallel lines, these angle pairs are not necessarily equal.

Answer:
(iv) \( 60^\circ + z = 180^\circ \)
\( \implies z = 180^\circ - 60^\circ \)
\( \implies z = 120^\circ \). This happens because \( \angle z \) and \( 60^\circ \) are interior angles on the same side of the transversal, which are supplementary when the lines are parallel.

Exam Tip: Interior angles on the same side of the transversal are also known as consecutive interior angles or co-interior angles. Their sum is always \( 180^\circ \) for parallel lines.

Answer:
(v) \( x = 120^\circ \). This is because \( \angle x \) and \( 120^\circ \) are corresponding angles, and corresponding angles are equal when lines are parallel.

Exam Tip: Corresponding angles are found in the same relative position at each intersection where a transversal crosses two lines. Look for the 'F' shape to easily identify them.

Answer:
(vi) We are given a \( 60^\circ \) angle. Angles are a, b, c, d.
\( a + 60^\circ = 180^\circ \)
\( \implies a = 180^\circ - 60^\circ \)
\( \implies a = 120^\circ \) (Interior angles on the same side of the transversal are supplementary).
Since \( a \) and \( d \) are alternate exterior angles, \( a = d = 120^\circ \).
Since \( b \) and \( d \) form a linear pair, \( b + d = 180^\circ \)
\( \implies b + 120^\circ = 180^\circ \)
\( \implies b = 180^\circ - 120^\circ \)
\( \implies b = 60^\circ \).
Since \( c \) and \( b \) are vertically opposite angles, \( c = b = 60^\circ \).
So, \( a = 120^\circ \), \( b = 60^\circ \), \( c = 60^\circ \), and \( d = 120^\circ \).

Exam Tip: When multiple angles are involved, systematically use one angle property at a time (e.g., linear pairs, vertically opposite, corresponding, alternate interior/exterior, co-interior) to find the unknown angles.

Try These (Page 110)

Question 1.
(i) Is \( l \parallel m \)? Why?
(ii) Is \( l \parallel m \)? Why?
(iii) If \( l \parallel m \), What is \( \angle x \)?

Answer:
(i) Yes, \( l \parallel m \). If a transversal line crosses two given lines such that the alternate interior angles are equal (both are \( 50^\circ \)), then the given lines are indeed parallel.

Exam Tip: Remember the converse: if alternate interior angles are equal, the lines are parallel. This is a fundamental condition for proving lines parallel.

Answer:
(ii) Yes, \( l \parallel m \). The pair of interior angles on the same side of the transversal are supplementary. Here, \( x = 130^\circ \) because it is vertically opposite to the given \( 130^\circ \) angle. We then check if \( 50^\circ + 130^\circ = 180^\circ \). Since their sum is \( 180^\circ \), the lines \( l \) and \( m \) are parallel.

Exam Tip: When using co-interior angles to prove parallelism, always calculate their sum. If the sum is \( 180^\circ \), the lines are parallel.

Answer:
(iii) Given \( l \parallel m \). The angle \( x \) and the \( 70^\circ \) angle are consecutive interior angles. Therefore, their sum must be \( 180^\circ \).
\( x + 70^\circ = 180^\circ \)
\( \implies x = 180^\circ - 70^\circ \)
\( \implies x = 110^\circ \).

Exam Tip: Always identify the relationship between the given angles (e.g., alternate, corresponding, co-interior) before applying the rules for parallel lines.