Lines and Angles — Andhra Pradesh (SSC) Class 9 Mathematics Solutions (Free)

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° = 180° Step 3: Apply the converse of the co-interior angle theorem If co-interior angles are supplementary, then the lines are parallel. Since 65° + 115° = 180°, the lines ARE parallel. Step 4: Conclusion The two lines are parallel to each other. Answer: Yes, the lines are parallel because the co-interior angles sum to 180°.

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 supplementary (sum to 180°). ∠2 = 180° - ∠1 = 180° - 48° = 132° Step 5: Find the fourth angle ∠4 = 180° - ∠3 = 180° - 48° = 132° Or, ∠4 = ∠2 = 132° (vertically opposite angles) Step 6: Verify ∠1 + ∠2 + ∠3 + ∠4 = 48° + 132° + 48° + 132° = 360° ✓ Answer: The four angles are 48°, 132°, 48°, and 132°

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°, and the sum of angles = 180°

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 adjacent to the exterior angle = x Then: 65° + 60° + x = 180° (angle sum property) x = 180° - 65° - 60° = 55° Also verify exterior angle = interior adjacent angle Exterior angle = 180° - x = 180° - 55° = 125° ✓ Answer: The other non-adjacent interior angle is 60°

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° Step 4: Verify using geometry The positive x-axis and negative x-axis are supplementary (form a straight line). If the line makes 120° on one side, it makes 180° - 120° = 60° on the other side. Alternative verification: Angle from positive x-axis (counterclockwise) = 120° Angle from negative x-axis (counterclockwise) = 120° - 180° = -60° (or 60° clockwise) Magnitude = 60° Answer: The line makes an angle of 60° with the negative x-axis

Given: Two parallel lines cut by transversal Alternate interior angles: (2x + 15)° and (3x - 5)° Find: Value of x and both angles Step 1: Apply the alternate interior angles property When two parallel lines are cut by a transversal, alternate interior angles are equal. 2x + 15 = 3x - 5 Step 2: Solve for x 2x + 15 = 3x - 5 15 + 5 = 3x - 2x 20 = x x = 20 Step 3: Find the first angle First angle = 2x + 15 = 2(20) + 15 = 40 + 15 = 55° Step 4: Find the second angle Second angle = 3x - 5 = 3(20) - 5 = 60 - 5 = 55° Step 5: Verify Both angles = 55° ✓ (they are equal as expected for alternate interior angles) Answer: x = 20, and both angles = 55°

Given: Three angles on a straight line Measures: (x + 10)°, (2x - 30)°, (3x - 40)° Find: All three angles Step 1: Apply the property of angles on a straight line Angles on a straight line sum to 180°. (x + 10) + (2x - 30) + (3x - 40) = 180 Step 2: Combine like terms x + 2x + 3x + 10 - 30 - 40 = 180 6x - 60 = 180 Step 3: Solve for x 6x = 180 + 60 6x = 240 x = 40 Step 4: Find all three angles First angle = x + 10 = 40 + 10 = 50° Second angle = 2x - 30 = 2(40) - 30 = 80 - 30 = 50° Third angle = 3x - 40 = 3(40) - 40 = 120 - 40 = 80° Step 5: Verify Sum = 50° + 50° + 80° = 180° ✓ Answer: The three angles are 50°, 50°, and 80°

Statement: The sum of interior angles of a quadrilateral = 360° Proof: Step 1: Draw a quadrilateral ABCD Label the interior angles as ∠A, ∠B, ∠C, ∠D. Step 2: Divide the quadrilateral into two triangles Draw diagonal AC from vertex A to vertex C. This divides quadrilateral ABCD into two triangles: △ABC and △ACD. Step 3: Apply the angle sum property to triangle ABC In △ABC: ∠CAB + ∠ABC + ∠BCA = 180° ... (1) Step 4: Apply the angle sum property to triangle ACD In △ACD: ∠CAD + ∠ACD + ∠ADC = 180° ... (2) Step 5: Add equations (1) and (2) (∠CAB + ∠ABC + ∠BCA) + (∠CAD + ∠ACD + ∠ADC) = 180° + 180° Step 6: Rearrange terms (∠CAB + ∠CAD) + ∠ABC + (∠BCA + ∠ACD) + ∠ADC = 360° Step 7: Recognize that combined angles form quadrilateral angles ∠A = ∠CAB + ∠CAD ∠B = ∠ABC ∠C = ∠BCA + ∠ACD ∠D = ∠ADC Step 8: Conclusion ∠A + ∠B + ∠C + ∠D = 360° Therefore, the sum of interior angles of a quadrilateral is 360°.