Q1. What does it mean for two lines to be parallel?

Ans: Two lines in the same plane that do not meet (do not intersect) are parallel; written \(l \parallel m\).

Q2. What is a transversal?

Ans: A line that intersects two (or more) lines at distinct points is a transversal.

Q3. How many angles are formed when a transversal cuts two lines?

Ans: Eight angles are formed (four at each intersection point).

Q4. Define corresponding angles.

Ans: Angles with arms in the same direction, one at each intersection, on the same side of the transversal are corresponding angles.

Q5. Define interior angles (relative to two lines and transversal).

Ans: Angles that lie between the two lines on the same side of the transversal are interior angles.

Q6. Define alternate interior angles.

Ans: Interior angles on opposite sides of the transversal are alternate interior angles.

Q7. Define alternate exterior angles.

Ans: Exterior angles on opposite sides of the transversal are alternate exterior angles.

Q8. State the relation of corresponding angles when two lines are parallel.

Ans: Corresponding angles are congruent (equal in measure) when lines are parallel.

Q9. State the relation of alternate interior angles when lines are parallel.

Ans: Alternate interior angles are congruent when lines are parallel.

Q10. State the relation of interior angles on the same side of transversal when lines are parallel.

Ans: Interior angles on the same side of the transversal are supplementary: their sum is \(180^\circ\).

Q11. If \( \angle 1 = 70^\circ \) is corresponding to \( \angle 2 \) and lines are parallel, find \( \angle 2 \).

Ans: \( \angle 2 = 70^\circ \) (corresponding angles equal).

Q12. If two parallel lines are cut by a transversal, how many pairs of corresponding angles are there?

Ans: Four pairs of corresponding angles.

Q13. If \( \angle A + \angle B = 180^\circ \) and A, B are interior on same side, what can you conclude about the two lines?

Ans: The two lines are parallel (converse of interior angle property).

Q14. If \( \angle x \) and \( \angle y \) are alternate interior angles and \( \angle x = 50^\circ \), find \( \angle y \).

Ans: \( \angle y = 50^\circ \) (alternate interior angles equal when lines parallel).

Q15. True/False: If corresponding angles are congruent, lines are parallel.

Ans: True (converse is true).

Q16. Give one simple construction method to draw a line parallel to a given line through an external point.

Ans: Use two set-squares: slide one copying the angle to get a parallel line through the point (Method I in book).

Q17. True/False: Alternate exterior angles are supplementary when lines are parallel.

Ans: False — alternate exterior angles are congruent when lines are parallel.

Q18. If \( \angle P = 120^\circ \) and \( \angle Q \) is its interior adjacent angle on same side of transversal, find \( \angle Q \).

Ans: \( \angle Q = 60^\circ \) because they are supplementary: \(120^\circ + Q = 180^\circ\Rightarrow Q=60^\circ\).

Q19. Notation: How do you write "line l is parallel to line m" in symbols?

Ans: \( l \parallel m \).

Q20. If two lines are cut by a transversal and an exterior angle equals an interior alternate angle, are the lines parallel?

Ans: Yes — if alternate angles (interior/exterior corresponding) are equal, lines are parallel (converse).

Q1. In figure, \( \angle MNQ = 70^\circ\) and \(MNQ\) is corresponding to \(ONP\). Find \( \angle AON\) if \( \angle AON + \angle ONP = 180^\circ\).

Ans: \( \angle ONP =70^\circ\). So \( \angle AON =180^\circ - 70^\circ = 110^\circ\).

Q2. If \( m\angle b = (x+15)^\circ\) and \( m\angle e = (2x+15)^\circ\) where \( \angle b\) and \( \angle f\) are corresponding and \( \angle f\) and \( \angle e\) form a linear pair. Find \(x\).

Ans: \(m\angle f = m\angle b = x+15\). Since \( \angle f\) and \( \angle e\) linear, \((x+15)+(2x+15)=180\Rightarrow 3x+30=180\Rightarrow 3x=150\Rightarrow x=50.\)

Q3. State and use the property to find the missing angle: If two parallel lines are cut by a transversal and one exterior angle is \(130^\circ\), what is its corresponding interior alternate angle?

Ans: Alternate exterior/interior equal when lines parallel. So missing angle \(=130^\circ\).

Q4. If two lines are parallel and a transversal makes angles \(40^\circ\) and \(x^\circ\) as interior on same side, find \(x\).

Ans: Interior on same side are supplementary: \(40+x=180\Rightarrow x=140^\circ\).

Q5. In a figure \( \angle a = 105^\circ\) (exterior at one intersection). Find interior alternate angle \( \angle b\) at the other intersection.

Ans: Alternate exterior/interior congruent: \( \angle b = 105^\circ\).

Q6. If \( \angle x\) and \( \angle y\) are corresponding and \( \angle x = 4\theta\), \( \angle y = 2\theta + 20^\circ\), find \( \theta\).

Ans: Corresponding ⇒ equal: \(4\theta = 2\theta +20\Rightarrow 2\theta=20\Rightarrow \theta=10^\circ\).

Q7. Two parallel lines are cut by a transversal: give the measure of the angle vertically opposite to \(70^\circ\).

Ans: Vertically opposite angle = \(70^\circ\).

Q8. In figure, \(p\parallel q\). Two transversals \(s\) and \(t\) cut them. If one angle at \(p\) is \(70^\circ\) what are corresponding angles at the other intersections?

Ans: All corresponding angles equal \(70^\circ\) at each transversal intersection.

Q9. If \( \angle 1\) and \( \angle 2\) are interior on same side and \( \angle 1 = 3x\), \( \angle 2 = (2x+30)^\circ\). Find \(x\).

Ans: \(3x+(2x+30)=180\Rightarrow5x+30=180\Rightarrow5x=150\Rightarrow x=30.\)

Q10. If \( \angle a\) is exterior and \( \angle b\) is interior alternate and \( \angle a = 115^\circ\), find \( \angle b\).

Ans: Alternate angles equal ⇒ \( \angle b =115^\circ\).

Q11. In the figure, \(p\parallel l\parallel q\). If one angle at first line is \(80^\circ\), find the corresponding angle \(x\) at the third line.

Ans: By parallelism transitive, corresponding angles equal ⇒ \(x=80^\circ\).

Q12. If two lines are parallel and transversal makes \(4x\) and \(120^\circ\) as corresponding angles, find \(x\).

Ans: Corresponding equal: \(4x=120\Rightarrow x=30^\circ\).

Q13. Given \(p\parallel q\) and a transversal, show that interior consecutive angles sum to \(180^\circ\) with a short reason.

Ans: Each interior angle is supplementary to the external adjacent corresponding angle; hence their sum is \(180^\circ\). (Or use linear pair + corresponding angles equality.)

Q14. If alternate interior angles are equal, what does it tell about the lines?

Ans: The lines are parallel (converse property).

Q15. If two lines are intersected by a transversal and one pair of corresponding angles is \(30^\circ\) and \(150^\circ\). Are lines parallel? Explain.

Ans: No — corresponding angles are not equal; hence lines are not parallel.

Q16. Using set-squares, outline steps to draw a line through point \(P\) parallel to given line \(l\).

Ans: Place two set-squares, copy the angle at P by sliding, draw line along edge of second set-square — the new line is parallel to \(l\) (Method I).

Q17. Using perpendicular construction, how to draw parallel at a given distance \(d\) from line \(l\)? (short)

Ans: From two points A,B on \(l\) draw perpendiculars, mark points at distance \(d\) on them, join the new points to get line parallel at distance \(d\) (Method II).

Q18. If \( \angle a = 40^\circ\) and it's corresponding to \( \angle b\), what is the supplement of \( \angle b\)?

Ans: \( \angle b = 40^\circ\). Supplement = \(180-40=140^\circ\).

Q19. Give an example of two angles that are neither corresponding nor alternate when a transversal cuts two lines.

Ans: Two interior angles on opposite sides but not matching alternate positions (e.g., one interior left near first intersection and one exterior right near second intersection) — they are not in special pair; see diagram.

Q20. If two lines are parallel and a transversal cuts them, how many pairs of alternate interior angles exist?

Ans: Two pairs of alternate interior angles exist.

Q1. In the figure lines \(m\parallel n\) and transversal \(l\) intersect to make \(\angle 1 = 70^\circ\) at one intersection. Find all other seven angles (list measures).

Ans: By corresponding / alternate / linear pair / vertical:
\(\angle 1 = 70^\circ\).
Vertical opposite to \(\angle1\) = \(70^\circ\).
Corresponding angles at other intersection = \(70^\circ\) (two of them).
Alternate interior angles = \(70^\circ\) (two of them).
Interior adjacent (same-side) angles = \(110^\circ\) (supplementary to \(70^\circ\)), and their vertical opposites also \(110^\circ\).
Thus other seven angles are \(70^\circ,70^\circ,70^\circ,110^\circ,110^\circ,110^\circ,110^\circ\).

Q2. Prove that if corresponding angles are equal then two lines are parallel (short proof).

Ans: If corresponding angles equal, then angle formed by transversal with one line equals angle formed with the other. Using alternate interior or co-interior angle relations we deduce interior on same side sum to \(180^\circ\); hence by converse, lines are parallel. (Formal: assume not parallel leads to contradiction.)

Q3. In figure \(p\parallel q\) and \(r\) is transversal. If \(\angle a = (2x+10)^\circ\) and \(\angle b = (x+70)^\circ\) are corresponding, find \(x\) and measures.

Ans: Corresponding ⇒ equal: \(2x+10 = x+70\Rightarrow x=60.\) Then \(\angle a = 2(60)+10=130^\circ\); \(\angle b = 130^\circ\).

Q4. Two parallel lines \(l\parallel m\) are cut by transversal \(t\). If one of the interior angles is \(3x\) and adjacent interior is \( (2x+30)^\circ\), find \(x\) and the angles.

Ans: Interiors on same side supplementary: \(3x + (2x+30)=180\Rightarrow5x+30=180\Rightarrow x=30.\) Angles: \(3x=90^\circ\), \(2x+30=90^\circ\).

Q5. In a diagram with three parallel lines \(p\parallel l\parallel q\) and a transversal, if one angle at the first line is \(80^\circ\) find its corresponding at the third line and the interior adjacent angle.

Ans: Corresponding at third line = \(80^\circ\). Interior adjacent supplement = \(100^\circ\).

Q6. Given lines \(a\parallel b\) cut by transversal with angles \(4x\) and \(110^\circ\) supplementary; find \(x\).

Ans: \(4x+110=180\Rightarrow4x=70\Rightarrow x=17.5^\circ\) (note: fractional allowed).

Q7. Show with reason: If two parallel lines are cut by a transversal, then corresponding angles are equal.

Ans: Using alternate interior + vertical angle equalities: corresponding angle equals alternate interior angle at other intersection which equals the corresponding angle — thus they are congruent. (Standard angle-chasing.)

Q8. In a figure, \(p\parallel q\), transversal meets them producing angles \(x,\, y,\, z\) as shown: if \(x=105^\circ\) and \(z\) is corresponding to \(x\), find \(y\) if \(y\) is interior adjacent to \(z\).

Ans: \(z=105^\circ\) (corresponding). Interior adjacent \(y\) is supplement: \(y=180-105=75^\circ\).

Q9. Using construction Method II, explain how to draw a line parallel to a given line \(l\) at distance \(2.5\) cm (brief steps with reason why parallel).

Ans: Take two points A,B on \(l\). Draw perpendiculars at A,B. Mark points P,Q at distance 2.5 cm on the perpendiculars on the same side. Join P and Q. PQ is parallel to l because both are perpendicular to same direction line segments; corresponding right angles equal ⇒ lines are parallel.

Q10. In the figure lines \(p\parallel q\). Given \(\angle a = (3x+10)^\circ\) and \(\angle b = (x+70)^\circ\) are interior on same side. Find \(x\) and angles.

Ans: \(3x+10 + x+70 =180\Rightarrow 4x+80=180\Rightarrow4x=100\Rightarrow x=25.\) Angles: \(3x+10=85^\circ,\; x+70=95^\circ\) (check: 85+95=180).

Q11. If \(\angle P = 2\theta\) corresponding to \(\angle Q = \theta + 30^\circ\), find \(\theta\) and both angles.

Ans: \(2\theta = \theta +30\Rightarrow \theta=30^\circ.\) Angles: \(P=60^\circ,\; Q=60^\circ\).

Q12. A transversal cuts two lines making a pair of alternate interior angles \( (5x-10)^\circ\) and \( (3x+20)^\circ\). If lines are parallel, find \(x\).

Ans: Alternate interior equal ⇒ \(5x-10 = 3x+20\Rightarrow2x=30\Rightarrow x=15.\)

Q13. Construct, using set-squares, a line through P parallel to given line l — mention key justification why it is parallel.

Ans: Copy corresponding angle from l at point P using set-squares; by equality of corresponding angles the new line is parallel to l (converse property).

Q14. In a configuration, if one exterior angle is \(140^\circ\) and its adjacent interior is \(x\), find \(x\) and its corresponding at other intersection.

Ans: Adjacent interior: \(x=180-140=40^\circ\). Corresponding at other intersection = \(140^\circ\) (or if corresponding to interior then \(40^\circ\) depending on position — read diagram).

Q15. Show with numbers: Two parallel lines cut by transversal produce angles \(70^\circ, 110^\circ\). Place them into corresponding/alternate/interior categories.

Ans: \(70^\circ\) and \(70^\circ\) are corresponding/alternate as shown; \(110^\circ\) are supplements (interior same-side). Provide diagram mapping.

Q16. If the measure of one angle in the eight formed is \(x\), show that opposite vertical angle is also \(x\), and the adjacent linear pair is \(180-x\) (brief).

Ans: Vertical angles equal by definition; linear pair sum to \(180^\circ\) by straight line property, so adjacent is \(180^\circ - x\).

Q17. Given \(\angle r = 2x\) and vertical opposite to r is \( (x + 30)^\circ\). Find \(x\).

Ans: Vertical opposite equal ⇒ \(2x = x + 30\Rightarrow x=30^\circ\). Then angles = \(60^\circ\).

Q18. Suppose three parallel lines are cut by two transversals forming a staircase of equal corresponding angles; if first corresponding is \(50^\circ\), find all corresponding on the staircase.

Ans: All corresponding angles along the transversals and across the three lines equal \(50^\circ\).

Q19. A line intersects two lines and you observe a pair of interior alternate angles are equal. Prove the two lines are parallel (short reason).

Ans: Equality of alternate interior angles implies corresponding angles equal (angle-chase). By converse, lines are parallel.

Q20. In a real exam-style diagram, if you are given several angle values and asked to find \(x\), outline an ordered plan to solve such problems.

Ans: (1) Identify parallelism; (2) mark corresponding/alternate/linear pairs; (3) set up equations (equal/supplementary); (4) solve algebraically; (5) substitute back to give measures — show steps and justification for each equality.

PS2.1 Q1: Fill the boxes (Corresponding angles). (1) \( \angle p\) and ______. (2) \( \angle q\) and ______. (3) \( \angle r\) and ______. (4) \( \angle s\) and ______.

Ans: Without the exact labelled figure the general answers are:
(1) corresponding to \( \angle p\) is the angle at the other intersection that lies in the same position relative to the transversal (call it \( \angle ?\)).
(2) corresponding to \( \angle q\) is the angle at other intersection in same relative position.
(3) corresponding to \( \angle r\) — similarly.
(4) corresponding to \( \angle s\) — similarly.
Note: The textbook figure uses specific letter labels (p, q, r, s, x, w, z). If you paste the figure or exact mapping I will replace blanks with exact angle names. The method: find the angle at the other intersection sharing the same orientation relative to transversal.

PS2.1 Q1 (Interior alternate angles). (5) \( \angle s\) and ______. (6) \( \angle w\) and ______.

Ans: Interior alternate pairs are the ones inside the two lines on opposite sides of the transversal. In the book figure they map as (5) \( \angle s\) with the interior angle opposite side (fill from diagram). (6) \( \angle w\) with its alternate interior. (Again, exact labels require the picture.)

PS2.1 Q2: Observe the second figure and write (1) Interior alternate angles (2) Corresponding angles (3) Interior angles.

Ans: Use definitions:
— Interior alternate: pick angles between lines and on opposite sides of transversal.
— Corresponding: pick angles in same orientation (top-left with top-left etc.).
— Interior (same side): pick two interior angles on same side of transversal — they sum to \(180^\circ\).
Tip: For each requested pair, identify positions (top/bottom, left/right) then apply the rule.

PS2.2 Q1 (1): In the figure if \(m\parallel n\) and \(p\) is transversal, find \(x\) when one angle is \(3x\) and adjacent corresponding is given (diagram shows 3x and x opposite etc.). Options: (A) 135° (B) 90° (C) 45° (D) 40°

Ans (method): Identify relation (corresponding/equal or supplementary). Solve equation accordingly. Example numeric: If \(3x\) and \(x\) are supplementary: \(3x+x=180\Rightarrow4x=180\Rightarrow x=45^\circ\). => Option (C). (Apply to the shown diagram; if diagram matches supplementary relation.)

PS2.2 Q1 (2): In second MCQ if \(a\parallel b\) and transversal shows angles \(40^\circ\) and \(4x,2x\) etc., find \(x\).

Ans: Translate the diagram into equations: use corresponding/alternate/supplementary relations. For instance if \(4x\) corresponds to \(40^\circ\Rightarrow4x=40\Rightarrow x=10^\circ\). If other relation, solve similarly.

PS2.2 Q2: In the adjoining figure lines \(p\parallel q\). Two transversals \(t\) and \(s\) give angles; find \(x\) and \(y\) using given measures (diagram gives 70° etc.).

Ans: Use corresponding/alternate equality: If one given angle = \(70^\circ\) then corresponding and alternate angles across transversals equal \(70^\circ\). If \(x\) is supplementary to \(70^\circ\) then \(x=110^\circ\). If \(y\) equals corresponding \(70^\circ\) then \(y=70^\circ\).

PS2.2 Q3: In the figure \(p\parallel q\) and \(l\parallel m\). Given one angle \(105^\circ\), find \(a,b,c\) with justification.

Ans: Use:
\(a\) (corresponding or vertical) = \(105^\circ\).
\(b\) (alternate or supplement depending on position) = either \(75^\circ\) if supplementary, or \(105^\circ\) if corresponding; specify using diagram.
\(c\) likewise found by angle-chase. (If you supply the exact figure mapping, exact numeric answers will be filled precisely.)

PS2.2 Q4 (5a): In figure \(p\parallel l\parallel q\). Find \(x\) using given measures (one angle 80° at q etc.).

Ans: Since lines parallel, corresponding at each line equal ⇒ \(x=80^\circ\) or its supplement if required by position. (Follow diagram mapping.)

PS2.2 Q4 (4b): In figure \(a\parallel b\) and \(l\) is transversal. Find \(x,y,z\) given \(40^\circ\) and \(30^\circ\).

Ans: Map each given angle: if \(40^\circ\) corresponds to \(x\) ⇒ \(x=40^\circ\). If \(30^\circ\) is alternate to \(y\) ⇒ \(y=30^\circ\). The remaining \(z\) may be supplementary: \(z=180^\circ-40^\circ=140^\circ\). (Check diagram positions to apply correctly.)

PS2.3 Q1: Draw a line \(l\). Take a point \(A\) outside \(l\). Through \(A\) draw a line parallel to \(l\).

Ans (Method I - set-squares):
1. Draw given line \(l\) and mark point \(A\) outside it.
2. Place two set-squares together; align one along \(l\).
3. With the second set-square draw a line through \(A\) keeping the same orientation.
4. The drawn line through \(A\) is parallel to \(l\) by copying corresponding angle.

PS2.3 Q2: Draw line \(l\). Take point \(T\) outside. Draw line through \(T\) parallel to \(l\).

Ans: Same as Q1; either Method I (set-squares) or Method II (perpendicular offsets) may be used. Steps as above.

PS2.3 Q3: Draw line \(m\). Draw line \(n\) parallel to \(m\) at a distance of 4 cm.

Ans (Method II - distance):
1. Draw line \(m\). Choose two points \(A,B\) on \(m\).
2. At \(A\) and \(B\) draw perpendiculars to \(m\).
3. On those perpendiculars mark points \(P,Q\) at 4 cm from \(A,B\) respectively on same side.
4. Join \(P,Q\); the line \(PQ\) is parallel to \(m\) and at distance 4 cm (because both perpendicular distances to \(m\) = 4 cm).