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.
TL;DR: Free step-by-step Karnataka (SSLC) Class 9 Mathematics solutions for "Lines and Angles" — important questions with detailed answers, download PDF for…
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Q1: If a transversal intersects two lines and makes angles of 65° and 115° on the same side of the transversal, determine if the lines are parallel.
Given: Transversal intersects two lines
Angles on the same side of transversal = 65° and 115°
Determine: Are the lines parallel?
Step 1: Recall the co-interior angles property
Co-interior angles (also called consecutive interior angles or allied angles):
- Located on the same side of the transversal
- Between the two lines (interior)
- Property: If two lines are parallel, co-interior angles are supplementary (sum to 180°)
Step 2: Check if the angles are supplementary
Sum of angles = 65° + 115°…
Q2: Two lines intersect. If one angle formed is 48°, find all four angles.
Given: Two lines intersect
One angle = 48°
Find: All four angles
Step 1: Understand vertical angles
When two lines intersect, they form four angles.
Vertical angles (opposite angles) are equal.
Step 2: Identify the angles
Let's label the angles as ∠1, ∠2, ∠3, ∠4 going around.
Given: ∠1 = 48°
Step 3: Find the vertically opposite angle
Vertical angle to ∠1 is ∠3.
∠3 = ∠1 = 48° (vertically opposite angles are equal)
Step 4: Find adjacent angles using linear pair property
Adjacent angles are sup…
Q3: In triangle ABC, if ∠A = 60°, ∠B = 70°, find ∠C and verify that the sum equals 180°.
Given: Triangle ABC with ∠A = 60°, ∠B = 70°
Find: ∠C
Step 1: Use the angle sum property of a triangle
The sum of all interior angles in a triangle = 180°
∠A + ∠B + ∠C = 180°
Step 2: Substitute known values
60° + 70° + ∠C = 180°
130° + ∠C = 180°
∠C = 180° - 130°
∠C = 50°
Step 3: Verify the sum
∠A + ∠B + ∠C = 60° + 70° + 50° = 180° ✓
Step 4: Geometric interpretation
The three angles are 60°, 70°, and 50°.
Since all angles are less than 90°, this is an acute-angled triangle.
Answer: ∠C = 50°, …
Q4: An exterior angle of a triangle is 125°. One of the non-adjacent interior angles is 65°. Find the other non-adjacent interior angle.
Given: Exterior angle = 125°
One non-adjacent interior angle = 65°
Find: The other non-adjacent interior angle
Step 1: Recall the exterior angle property
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
Exterior angle = ∠1 + ∠2 (where ∠1 and ∠2 are non-adjacent interior angles)
Step 2: Apply the property
125° = 65° + (other non-adjacent angle)
Other non-adjacent angle = 125° - 65° = 60°
Step 3: Verify using triangle angle sum
Let the interior angle adjac…
Q5: A line makes an angle of 120° with the positive x-axis. Find the angle it makes with the negative x-axis.
Given: Line makes 120° with positive x-axis
Find: Angle with negative x-axis
Step 1: Visualize the problem
The positive x-axis and negative x-axis form a straight line.
They are collinear and opposite in direction.
Step 2: Understand the angle relationship
If a line makes angle θ with the positive x-axis (measured counterclockwise),
then it makes angle (180° - θ) with the negative x-axis.
Step 3: Apply the formula
Angle with positive x-axis = 120°
Angle with negative x-axis = 180° - 120° = 60…
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