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

Two lines cut by a transversal are parallel if the co-interior (same-side interior) angles add up to 180°. Here: 72° + 108° = 180° Since the sum is exactly 180°, the co-interior angles are supplementary, so the two lines ARE parallel.

When two lines intersect, they form two pairs of vertically opposite angles, and adjacent angles form linear pairs (summing to 180°). Let the given angle be 55°. Its vertically opposite angle is also 55° (vertically opposite angles are always equal). Each angle adjacent to it forms a linear pair: 180° − 55° = 125°. The four angles are: 55°, 125°, 55°, 125° (going around the intersection point).

The angle sum property of a triangle states that the three interior angles always add up to 180°. ∠A + ∠B + ∠C = 180° 50° + 65° + ∠C = 180° ∠C = 180° − 115° ∠C = 65° Verification: 50° + 65° + 65° = 180° ✓

The Exterior Angle Theorem states that an exterior angle of a triangle equals the sum of the two interior angles that are not adjacent to it. Let the unknown angle be x. 140° = 55° + x x = 140° − 55° x = 85° The other non-adjacent interior angle is 85°.

The positive x-axis and negative x-axis together form a straight line, so the angles a line makes with each of them are supplementary (they add up to 180°). Angle with negative x-axis = 180° − 130° = 50°.

Alternate interior angles formed when a transversal cuts two parallel lines are always equal. 3x + 10 = 4x − 15 10 + 15 = 4x − 3x 25 = x Substitute back: First angle = 3(25) + 10 = 75 + 10 = 85° Second angle = 4(25) − 15 = 100 − 15 = 85° Both angles equal 85°, confirming x = 25.

Angles that together form a straight line add up to 180°. (x + 15) + (2x + 5) + (3x − 20) = 180 6x + 0 = 180 6x = 180 x = 30 Substituting back: First angle = 30 + 15 = 45° Second angle = 2(30) + 5 = 65° Third angle = 3(30) − 20 = 70° Verification: 45° + 65° + 70° = 180° ✓

Given: A quadrilateral ABCD. To prove: ∠A + ∠B + ∠C + ∠D = 360° Proof: Step 1: Draw diagonal AC. This splits quadrilateral ABCD into two triangles: △ABC and △ACD. Step 2: By the angle sum property of a triangle, the angles of △ABC add up to 180°: ∠BAC + ∠B + ∠BCA = 180° Step 3: Similarly, the angles of △ACD add up to 180°: ∠DAC + ∠D + ∠DCA = 180° Step 4: Add both equations: (∠BAC + ∠DAC) + ∠B + (∠BCA + ∠DCA) + ∠D = 360° Step 5: Notice that ∠BAC + ∠DAC = ∠A (the full angle at vertex A), and ∠BCA + ∠DCA = ∠C (the full angle at vertex C). So: ∠A + ∠B + ∠C + ∠D = 360° Hence proved: the sum of the angles of any quadrilateral is 360°.