Practice Set 2.2 Geometry 9th Std Maths Part 2 Answers Chapter 2 Parallel Lines
Question 1
Maharashtra Board Solution
In the given figure, y = 108° and x = 71°. Are the lines m and n parallel? Justify?
Solution & Step-by-Step Answer:
y = 108°, x = 71° …[Given] x + y = 71° + 108° = 179° ∴ x + y = 180° ∴ The angles x andy are not supplementary. ∴ The angles do not satisfy the interior angles test for parallel lines ∴ line m and line n are not parallel lines.
Question 2
Maharashtra Board Solution
In the given figure, if ∠a = ∠b then prove that line l || line m. Given: ∠a ≅ ∠b To prove: line l| line m
Solution & Step-by-Step Answer:
Proof: consider ∠c as shown in the figure ∠a ≅ ∠c …….. (i) [Vertically opposite angles] But, ∠a ≅ ∠b I (ii) [Given] ∴ ∠b ≅ ∠c [From (i) and (ii)] But, ∠b and ∠c are corresponding angles on lines l and m when line n is the transversal. ∴ line l || line m. [Corresponding angles test]
Question 3
Maharashtra Board Solution
In the given figure, if ∠a ≅ ∠b and ∠x ≅ ∠y, then prove that line l | line n. Given: ∠a ≅ ∠b and ∠x ≅ ∠y To prove: line l | line n
Solution & Step-by-Step Answer:
Proof: ∠a = ∠b [Given] But, ∠a and ∠b are corresponding angles on lines l and m when line k is the transversal. ∴ line l || line m ….(i) [Corresponding angles test] ∠x ≅ ∠y [Given] But, ∠x and ∠y are alternate angles on lines m and n when seg PQ is the transversal, ∴ line m || line n ……(ii) [Alternate angles test] ∴ From (i) and (ii), line l || line m || line n i.e., line l || line n
Question 4
Maharashtra Board Solution
In the given figure, if ray BA || ray DE, ∠C = 50° and ∠D = 100°. Find the measure of ∠ABC. (Hint: Draw a line passing through point C and parallel to line AB.)
Solution & Step-by-Step Answer:
Draw a line FG passing through point C and parallel to line AB line FG || ray BA …….(i) [Construction] Ray BA || ray DE ….(ii) [Given] line FG || ray BA || ray DE …(iii) [From (i) and (ii)] line FG||rayDE [From (iii)] and seg DC is their transvensal ∴ ∠ DCF = ∠ EDC [Alternate angles] ∴ ∠ DCF = 100° [∵ ∠D = 100°] Now, ∠ DCF = ∠ BCF + ∠ BCD [Angle addition property] ∴ 100° = ∠BCF + 50° ∴ 100° – 50° = ∠BCF ∴ ∠BCF = 50° ….(iv) Now, line FG || ray BA and seg BC is their transversal. ∴ ∠ABC + ∠BCF = 180° [Interior angles] ∴ ∠ABC + 50° = 180° [From (iv)] ∴ ∠ABC = 180°- 50° ∴ ∠ABC = 130°
Question 5
Maharashtra Board Solution
In the given figure, ray AE || ray BD, ray AF is the bisector of ∠EAB and ray BC is the bisector of ∠ABD. Prove that line AF || line BC. Given: Ray AE || ray BD, and ray AF and ray BC are the bisectors of ∠EAB and ∠ABD respectively. To prove: line AF || line BC
Solution & Step-by-Step Answer:
Proof: Ray AE || ray BD and seg AB is their transversal. ∴ ∠EAB = ∠ABD ….(i) [Alternate angles] ∠FAB = ∠EAB [Ray AF bisects ∠EAB] ∴ 2∠FAB = ∠EAB …..(ii) ∠CBA = ∠ABD [Ray BC bisects ∠ABD] ∴ 2∠CBA = ∠ABD …(iii) ∴ 2∠FAB = 2∠CBA [From (i), (ii) and (iii)] ∴ ∠FAB = ∠CBA But, ∠FAB and ∠ABC are alternate angles on lines AF and BC when seg AB is the transversal. ∴ line AF || line BC [Alternate angles test]
Question 6
Maharashtra Board Solution
A transversal EF of line AB and line CD intersects the lines at points P and Q respectively. Ray PR and ray QS are parallel and bisectors of ∠BPQ and ∠PQC respectively. Prove that line AB || line CD. Given: Ray PR || ray QS Ray PR and ray QS are the bisectors of ∠BPQ and ∠PQC respectively. To prove: line AB || line CD
Solution & Step-by-Step Answer:
Proof: Ray PR || ray QS and seg PQ is their transversal. ∠RPQ = ∠SQP ….(i) [Alternate angles] ∠RPQ = ∠BPQ …. (ii) [Ray PR bisects ∠BPQ] ∠SQP = ∠PQC [Ray QS bisects ∠PQC] ∴ ∠BPQ = ∠PQC ∴ ∠BPQ = ∠PQC But, ∠BPQ and ∠PQC are alternate angles on lines AB and CD when line EF is the transversal. ∴ line AB || line CD [Alternate angles test]
Maharashtra Board Class 9 Maths Chapter 2 Parallel Lines Practice Set 2.2 Intext Questions and Activities
Question 1
Maharashtra Board Solution
In the given figure, how will you decide whether line ¡ and line m are parallel or not? (Textbook pg. no. 19)
Solution & Step-by-Step Answer:
In the figure, we observe that line I and line m are coplanar and do not intersect each other. ∴ Line l and line m are parallel lines.