Here are the step-by-step solutions for each problem on the worksheet.
1)
*
Reasoning: The angle labeled $x^\circ$ and the angle labeled $130^\circ$ are
consecutive interior angles (also called same-side interior angles). When lines are parallel, these angles add up to $180^\circ$.
*
Calculation:
$$x + 130 = 180$$
$$x = 180 - 130$$
$$x = 50$$
2)
*
Reasoning: The angle labeled $40^\circ$ and the angle labeled $(3x)^\circ$ are
corresponding angles. Corresponding angles are in the same position at each intersection and are equal when lines are parallel.
*
Calculation:
$$3x = 40$$
Divide both sides by 3:
$$x = \frac{40}{3}$$
$$x \approx 13.33$$
3)
*
Reasoning: The angle labeled $(2x + 10)^\circ$ and the angle labeled $70^\circ$ are
alternate exterior angles. These angles are on opposite sides of the transversal and outside the parallel lines. They are equal.
*
Calculation:
$$2x + 10 = 70$$
Subtract 10 from both sides:
$$2x = 60$$
Divide by 2:
$$x = 30$$
4)
*
Reasoning: The angle labeled $100^\circ$ and the angle labeled $(4x)^\circ$ are
consecutive interior angles (same-side interior). They add up to $180^\circ$.
*
Calculation:
$$4x + 100 = 180$$
Subtract 100 from both sides:
$$4x = 80$$
Divide by 4:
$$x = 20$$
5)
*
Reasoning: The angle labeled $(3x + 5)^\circ$ and the angle labeled $110^\circ$ are
alternate interior angles. Alternate interior angles are equal.
*
Calculation:
$$3x + 5 = 110$$
Subtract 5 from both sides:
$$3x = 105$$
Divide by 3:
$$x = 35$$
6)
*
Reasoning: The angle labeled $(x + 10)^\circ$ and the angle labeled $120^\circ$ are
corresponding angles. Corresponding angles are equal.
*
Calculation:
$$x + 10 = 120$$
Subtract 10 from both sides:
$$x = 110$$
7)
*
Reasoning: The angle labeled $70^\circ$ and the angle labeled $(2x - 10)^\circ$ are
alternate exterior angles. They are equal.
*
Calculation:
$$2x - 10 = 70$$
Add 10 to both sides:
$$2x = 80$$
Divide by 2:
$$x = 40$$
8)
*
Reasoning: The angle labeled $110^\circ$ and the angle labeled $(3x + 20)^\circ$ are
consecutive interior angles (same-side interior). They add up to $180^\circ$.
*
Calculation:
$$3x + 20 + 110 = 180$$
Combine the numbers:
$$3x + 130 = 180$$
Subtract 130 from both sides:
$$3x = 50$$
Divide by 3:
$$x = \frac{50}{3}$$
$$x \approx 16.67$$
Final Answer:
1) x = 50
2) x = 13.33 (or 40/3)
3) x = 30
4) x = 20
5) x = 35
6) x = 110
7) x = 40
8) x = 16.67 (or 50/3)
Parent Tip: Review the logic above to help your child master the concept of transversal angles worksheet middle school.
