Here are the solutions for each problem on the worksheet.
Problem 1 (Top Left)
*
Identify the relationship: The angles $6x$ and $70^\circ$ are on the same side of the transversal and between the parallel lines. These are
consecutive interior angles, which means they add up to $180^\circ$.
*
Equation: $6x + 70 = 180$
*
Solve:
* Subtract 70 from both sides: $6x = 110$
* Divide by 6: $x = \frac{110}{6}$
* Simplify the fraction: $x = \frac{55}{3}$ or $18\frac{1}{3}$
Problem 2 (Top Right)
*
Identify the relationship: The angle labeled $q^\circ$ and the angle labeled $110^\circ$ are in matching corners at each intersection. These are
corresponding angles, which means they are equal.
*
Equation: $q = 110$
*
Solve: Since they are equal, $q$ is simply 110.
Problem 3 (Middle Left)
*
Identify the relationship: The angles $4x$ and $90^\circ$ are on opposite sides of the transversal and between the parallel lines. These are
alternate interior angles, which means they are equal.
*
Equation: $4x = 90$
*
Solve:
* Divide by 4: $x = \frac{90}{4}$
* Simplify: $x = 22.5$
Problem 4 (Middle Right)
*
Identify the relationship: The angles $120^\circ$ and $(2y - 10)^\circ$ are on the same side of the transversal and between the parallel lines. These are
consecutive interior angles, so they add up to $180^\circ$.
*
Equation: $120 + (2y - 10) = 180$
*
Solve:
* Combine numbers on the left: $2y + 110 = 180$
* Subtract 110 from both sides: $2y = 70$
* Divide by 2: $y = 35$
Problem 5 (Bottom Left)
*
Identify the relationship: The angle labeled $r^\circ$ and the angle labeled $105^\circ$ form a straight line together. Angles on a straight line add up to $180^\circ$ (they are a linear pair).
*
Equation: $r + 105 = 180$
*
Solve:
* Subtract 105 from both sides: $r = 75$
Problem 6 (Bottom Right)
*
Identify the relationship: The angles $(3t + 15)^\circ$ and $135^\circ$ are on opposite sides of the transversal and outside the parallel lines. These are
alternate exterior angles, which means they are equal.
*
Equation: $3t + 15 = 135$
*
Solve:
* Subtract 15 from both sides: $3t = 120$
* Divide by 3: $t = 40$
Final Answer:
1. Equation: $6x + 70 = 180$, $x = \frac{55}{3}$ (or $18.33$)
2. Equation: $q = 110$, $q = 110$
3. Equation: $4x = 90$, $x = 22.5$
4. Equation: $120 + 2y - 10 = 180$, $y = 35$
5. Equation: $r + 105 = 180$, $r = 75$
6. Equation: $3t + 15 = 135$, $t = 40$
Parent Tip: Review the logic above to help your child master the concept of parallel lines worksheet answers.
