Lines and Angles — Karnataka (SSLC) Class 9 Mathematics Solutions (Free)
Free step-by-step Karnataka (SSLC) Class 9 Mathematics solutions for "Lines and Angles" — important questions with detailed answers, download PDF for board exam preparation.
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Lines and Angles — Textbook Questions Answered (8)
Q1: A transversal intersects two lines, making angles of 80° and 100° on the same side of the transversal. Are the two lines parallel?
Two lines cut by a transversal are parallel if the co-interior (same-side interior) angles are supplementary, adding up to 180°.
80° + 100° = 180°
Since the sum is exactly 180°, the lines ARE parallel.
Q2: Two straight lines intersect, and one of the four angles formed is 62°. Find the other three angles.
When two lines intersect, vertically opposite angles are equal, and each pair of adjacent angles forms a linear pair summing to 180°.
Given angle: 62°. Its vertically opposite angle is also 62°.
Each adjacent angle: 180° − 62° = 118°.
The four angles, going around the point of intersection, are: 62°, 118°, 62°, 118°.
Q3: In triangle ABC, ∠A = 55° and ∠B = 75°. Find ∠C and verify the angle sum property.
Angle sum property: ∠A + ∠B + ∠C = 180°.
55° + 75° + ∠C = 180°
∠C = 180° − 130°
∠C = 50°
Verification: 55° + 75° + 50° = 180° ✓
Q4: An exterior angle of a triangle is 130°, and one non-adjacent interior angle is 70°. Find the other non-adjacent interior angle.
By the Exterior Angle Theorem, an exterior angle equals the sum of the two non-adjacent interior angles.
130° = 70° + x
x = 130° − 70°
x = 60°
The other non-adjacent interior angle is 60°.
Q5: A line makes an angle of 145° with the positive x-axis. What angle does it make with the negative x-axis?
The positive and negative x-axis together form a straight line, so the two angles the line makes with them are supplementary.
Angle with negative x-axis = 180° − 145° = 35°.
Q6: Two parallel lines are cut by a transversal. One alternate interior angle is (5x − 10)° and the other is (4x + 15)°. Find x and both angles.
Alternate interior angles formed by a transversal cutting two parallel lines are equal.
5x − 10 = 4x + 15
5x − 4x = 15 + 10
x = 25
Substitute back:
First angle = 5(25) − 10 = 125 − 10 = 115°
Second angle = 4(25) + 15 = 100 + 15 = 115°
Both angles equal 115°, confirming x = 25.
Q7: Three angles (x + 5)°, (2x + 10)°, and (3x − 15)° lie along a straight line. Find all three angles.
Angles forming a straight line add up to 180°.
(x + 5) + (2x + 10) + (3x − 15) = 180
6x = 180
x = 30
Substituting back:
First angle = 30 + 5 = 35°
Second angle = 2(30) + 10 = 70°
Third angle = 3(30) − 15 = 75°
Verification: 35° + 70° + 75° = 180° ✓
Q8: Prove that the sum of the interior angles of a quadrilateral is 360°.
Given: A quadrilateral PQRS.
To prove: ∠P + ∠Q + ∠R + ∠S = 360°
Proof:
Step 1: Draw diagonal PR, splitting the quadrilateral into two triangles: △PQR and △PRS.
Step 2: In △PQR, the angles sum to 180°: ∠QPR + ∠Q + ∠QRP = 180°.
Step 3: In △PRS, the angles sum to 180°: ∠SPR + ∠S + ∠SRP = 180°.
Step 4: Adding these: (∠QPR + ∠SPR) + ∠Q + (∠QRP + ∠SRP) + ∠S = 360°.
Step 5: Since ∠QPR + ∠SPR = ∠P (the full angle at P) and ∠QRP + ∠SRP = ∠R (the full angle at R):
∠P + ∠Q + ∠R + ∠S = 360°
Hence proved.
Related
- NCERT Solutions: Lines and Angles — the CBSE version of this same topic.
- Constructions — more Karnataka (SSLC) Class 9 Mathematics.
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