Lines and Angles

Answer

Step 1: Identify that ∠AOC and ∠BOD are vertically opposite angles. Step 2: Apply Theorem - Vertically opposite angles are equal. Step 3: Therefore, ∠BOD = ∠AOC = 65°. Answer: ∠BOD = 65°

Answer

Step 1: Since ray OC stands on line AB, ∠AOC + ∠BOC = 180° (Linear Pair Axiom). Step 2: Let ∠AOC = 3x and ∠BOC = 2x. Step 3: 3x + 2x = 180° → 5x = 180° → x = 36°. Step 4: ∠AOC = 3(36°) = 108° and ∠BOC

Answer

Step 1: ∠POR + ∠ROQ = 180° (Linear pair). Step 2: Let ∠POR = 5x and ∠ROQ = 7x. Step 3: 5x + 7x = 180° → 12x = 180° → x = 15°. Step 4: ∠POR = 5(15°) = 75°, ∠ROQ = 7(15°) = 105°. Step 5: ∠POS = ∠ROQ = 1

Answer

Step 1: Identify that co-interior angles are interior angles on the same side of transversal. Step 2: Apply theorem - Sum of co-interior angles = 180° when lines are parallel. Step 3: Let co-interior

Answer

Step 1: Identify that ∠APQ and ∠PQD are corresponding angles. Step 2: Apply theorem - Corresponding angles are equal when lines are parallel. Step 3: Therefore, ∠PQD = ∠APQ. Step 4: ∠PQD = 65°. Answer

Answer

Step 1: Identify the alternate interior angles. Step 2: Apply theorem - Alternate interior angles are equal when lines are parallel. Step 3: If one alternate interior angle = 45°, then the other = 45°

Answer

Step 1: Adjacent angles forming linear pair sum to 180°. Step 2: (3x + 10)° + (2x + 20)° = 180°. Step 3: 3x + 10 + 2x + 20 = 180. Step 4: 5x + 30 = 180. Step 5: 5x = 150. Step 6: x = 30. Answer: x = 3

Answer

Use when: A ray stands on a line, creating two adjacent angles. Theorem: Sum of adjacent angles = 180°. Example: Ray OC on line AB creates ∠AOC and ∠BOC. ∠AOC + ∠BOC = 180°. If ∠AOC = 110°, then ∠BOC