Lines and Angles

125°

Step 1: When a transversal intersects two parallel lines, corresponding angles are formed. Step 2: By the corresponding angles axiom, corresponding angles are equal when lines are parallel. Step 3: Since one corresponding angle is 125°, the other corresponding angle is also 125°. Step 4: This property holds for all pairs of corresponding angles formed by parallel lines and a transversal.

75°

Step 1: Apply the exterior angle theorem: An exterior angle equals the sum of the two interior opposite angles. Step 2: Let the unknown interior opposite angle be x. Step 3: Set up the equation: 120° = 45° + x. Step 4: Solve for x: x = 120° - 45° = 75°. Step 5: Therefore, the other interior opposite angle is 75°.

20

Step 1: When parallel lines are cut by a transversal, alternate interior angles are equal. Step 2: Set up the equation: 3x + 10 = 5x - 30. Step 3: Rearrange: 10 + 30 = 5x - 3x, so 40 = 2x. Step 4: Solve for x: x = 20. Step 5: Check: 3(20) + 10 = 70° and 5(20) - 30 = 70°, so both angles are equal.

Obtuse triangle

Step 1: Find the third angle using angle sum property: ∠R = 180° - 40° - 80° = 60°. Step 2: Identify the angles: ∠P = 40°, ∠Q = 80°, ∠R = 60°. Step 3: Since one angle (∠Q = 80°) is greater than 90°, this is an obtuse triangle. Step 4: Wait, let me recalculate: ∠R = 180° - 40° - 80° = 60°. All angles are less than 90°, so this is actually an acute triangle. Step 5: Correction: This is an acute triangle since all angles are less than 90°.