Step 1: Understand the angle relationships: When a transversal crosses parallel lines, it creates:

• Corresponding angles (equal)
• Alternate interior angles (equal)
• Alternate exterior angles (equal)
• Consecutive interior angles (supplementary - add to 180°)

Step 2: Identify the given angle: Let's say angle 1 = 65°

Step 3: Find angles equal to 65°: All corresponding angles = 65° All alternate interior angles = 65° All alternate exterior angles = 65° There are 4 angles that measure 65°

Step 4: Find the supplementary angles: The other 4 angles are supplementary to 65° 180° - 65° = 115°

Step 5: Summary of all eight angles: Four angles measure 65° Four angles measure 115°

Step 6: Verify: 65° + 115° = 180° ✓ (linear pairs)

Answer: Four angles are 65° and four angles are 115°

Corresponding angles are congruent when lines are parallel.

Answer: $65°$

Step 1: Recall corresponding angles: When parallel lines are cut by a transversal, corresponding angles are congruent (equal)

Step 2: Identify corresponding angles: Corresponding angles are in the same relative position at each intersection point

Step 3: Apply the property: If angle 3 = 112° Then its corresponding angle = 112°

Step 4: Verify the concept: Corresponding angles are on the same side of the transversal and in the same position (both above or both below the parallel lines)

Answer: The corresponding angle measures 112°

Consecutive interior angles are supplementary.

$(2x + 10) + (3x - 15) = 180$

$5x - 5 = 180$

$5x = 185$

$x = 37$

Answer: $x = 37$

Step 1: Recall alternate interior angles: When parallel lines are cut by a transversal, alternate interior angles are congruent

Step 2: Set up the equation: 3x + 20 = 5x - 40

Step 3: Solve for x: 20 + 40 = 5x - 3x 60 = 2x x = 30

Step 4: Find the angle measures: First angle: 3x + 20 = 3(30) + 20 = 90 + 20 = 110° Second angle: 5x - 40 = 5(30) - 40 = 150 - 40 = 110°

Step 5: Verify: Both angles equal 110° ✓ (alternate interior angles are equal)

Answer: x = 30, both angles measure 110°

Step 1: Recall consecutive interior angles: Also called co-interior or same-side interior angles When lines are parallel, consecutive interior angles are supplementary (sum to 180°)

Step 2: Set up the equation: (2x + 15) + (3x + 25) = 180

Step 3: Simplify and solve: 2x + 15 + 3x + 25 = 180 5x + 40 = 180 5x = 140 x = 28

Step 4: Find both angle measures: First angle: 2x + 15 = 2(28) + 15 = 56 + 15 = 71° Second angle: 3x + 25 = 3(28) + 25 = 84 + 25 = 109°

Step 5: Verify: 71° + 109° = 180° ✓ (consecutive interior angles are supplementary)

Answer: x = 28, angles are 71° and 109°

For the lines to be parallel, alternate interior angles must be congruent.

Set them equal: $5x - 20 = 3x + 40$

$2x = 60$

$x = 30$

When $x = 30$:

• First angle: $5(30) - 20 = 130°$
• Second angle: $3(30) + 40 = 130°$

Since the angles are equal, the lines are parallel.

Answer: Yes, the lines are parallel

Step 1: Identify the angle relationship: Angles AEF and EFC appear to be corresponding angles (both on the same side of the transversal)

Step 2: Determine the condition for parallel lines: If AB ∥ CD, then corresponding angles must be equal So we need: 4x - 10 = 2x + 50

Step 3: Test if this equation is consistent: 4x - 10 = 2x + 50 4x - 2x = 50 + 10 2x = 60 x = 30

Step 4: Find the angle measures: Angle AEF = 4x - 10 = 4(30) - 10 = 120 - 10 = 110° Angle EFC = 2x + 50 = 2(30) + 50 = 60 + 50 = 110°

Step 5: Verify: Both angles equal 110° ✓ Since corresponding angles are equal, the lines ARE parallel

Alternative check - if these were consecutive interior angles: 110° + 110° = 220° ≠ 180° So they cannot be consecutive interior angles

Step 6: Conclusion: Since we can find a consistent value of x that makes the corresponding angles equal, the lines are parallel

Answer: Yes, the lines are parallel when x = 30. Both angles measure 110°.