Here are the step-by-step solutions for each problem.
1.
*
Angle 1: The angle labeled $60^\circ$ and $\angle 1$ form a straight line (a linear pair). Angles on a straight line add up to $180^\circ$.
$$180^\circ - 60^\circ = 120^\circ$$
So, $m\angle 1 = 120^\circ$.
*
Angle 2: $\angle 1$ and $\angle 2$ are corresponding angles (they are in the same position at each intersection). Since the lines are parallel, corresponding angles are equal.
So, $m\angle 2 = 120^\circ$.
2.
*
Angle 2: The angle labeled $40^\circ$ and $\angle 2$ form a straight line.
$$180^\circ - 40^\circ = 140^\circ$$
So, $m\angle 2 = 140^\circ$.
*
Angle 1: $\angle 1$ and $\angle 2$ are alternate exterior angles (outside the parallel lines on opposite sides of the transversal). These angles are equal.
So, $m\angle 1 = 140^\circ$.
3.
*
Angle 2: The angle labeled $45^\circ$ and $\angle 2$ are alternate interior angles. These angles are equal.
So, $m\angle 2 = 45^\circ$.
*
Angle 1: $\angle 1$ and $\angle 2$ form a straight line.
$$180^\circ - 45^\circ = 135^\circ$$
So, $m\angle 1 = 135^\circ$.
4.
*
Angle 2: The angle labeled $75^\circ$ and $\angle 2$ form a straight line.
$$180^\circ - 75^\circ = 105^\circ$$
So, $m\angle 2 = 105^\circ$.
*
Angle 1: $\angle 1$ and $\angle 2$ are alternate exterior angles. These angles are equal.
So, $m\angle 1 = 105^\circ$.
5.
*
Value of x: The angle labeled $x^\circ$ and the angle labeled $80^\circ$ are vertical angles (opposite each other where lines cross). Vertical angles are always equal.
So, $x = 80$.
*
Value of y: The angle labeled $y^\circ$ and the angle labeled $80^\circ$ are consecutive interior angles (inside the parallel lines on the same side). They add up to $180^\circ$.
$$y + 80 = 180$$
$$y = 100$$
6.
*
Value of y: The square symbol indicates a right angle ($90^\circ$). Because the vertical line is perpendicular to the bottom horizontal line, it is also perpendicular to the top parallel line. Therefore, $y = 90$.
*
Value of x: Angle $x$ and angle $y$ form a straight line along the vertical axis. Since $y = 90$, $x$ must also be $90$ to make $180^\circ$.
So, $x = 90$.
7.
* The two angles shown ($80^\circ$ and $2x^\circ$) are consecutive interior angles. They add up to $180^\circ$.
$$80 + 2x = 180$$
Subtract 80 from both sides:
$$2x = 100$$
Divide by 2:
$$x = 50$$
8.
* The angle labeled $2x^\circ$ and the angle labeled $110^\circ$ are alternate exterior angles. They are equal.
$$2x = 110$$
Divide by 2:
$$x = 55$$
9.
* The angle labeled $100^\circ$ and the angle labeled $(x - 10)^\circ$ are alternate exterior angles. They are equal.
$$x - 10 = 100$$
Add 10 to both sides:
$$x = 110$$
10.
* The angle labeled $60^\circ$ and the angle labeled $(3x + 15)^\circ$ are consecutive interior angles. They add up to $180^\circ$.
$$60 + (3x + 15) = 180$$
Combine like terms ($60 + 15$):
$$3x + 75 = 180$$
Subtract 75 from both sides:
$$3x = 105$$
Divide by 3:
$$x = 35$$
Final Answer:
1. m∠1 = 120°, m∠2 = 120°
2. m∠1 = 140°, m∠2 = 140°
3. m∠1 = 135°, m∠2 = 45°
4. m∠1 = 105°, m∠2 = 105°
5. x = 80, y = 100
6. x = 90, y = 90
7. x = 50
8. x = 55
9. x = 110
10. x = 35
Parent Tip: Review the logic above to help your child master the concept of parallel line and transversal worksheet.
