Here are the step-by-step solutions for each diagram.
(1)
*
Identify the relationship: The angle labeled $81^\circ$ and angle $x$ are on opposite sides of the transversal line and between the two parallel lines. These are called
alternate interior angles.
*
Rule: Alternate interior angles are equal.
*
Calculation: Since they are equal, $x = 81$.
(2)
*
Identify the relationship: The angle labeled $76^\circ$ and angle $x$ are on the same side of the transversal line and between the two parallel lines. These are called
consecutive interior angles (or co-interior angles).
*
Rule: Consecutive interior angles add up to $180^\circ$.
*
Calculation: $x + 76 = 180$.
Subtract 76 from 180: $180 - 76 = 104$.
So, $x = 104$.
(3)
*
Identify the relationship: Angle $x$ and the angle labeled $99^\circ$ are in the same relative position at each intersection (bottom-left corner). These are called
corresponding angles.
*
Rule: Corresponding angles are equal.
*
Calculation: Since they are equal, $x = 99$.
(4)
*
Identify the relationship: First, look at the angle labeled $83^\circ$. The angle directly opposite it (vertically opposite) is also $83^\circ$. This new $83^\circ$ angle and angle $x$ are in the same relative position (top-right corner). These are
corresponding angles.
*
Rule: Corresponding angles are equal.
*
Calculation: Therefore, $x = 83$.
(5)
*
Identify the relationship: The angle labeled $112^\circ$ and angle $x$ are on opposite sides of the vertical transversal line and between the two parallel lines. These are
alternate interior angles.
*
Rule: Alternate interior angles are equal.
*
Calculation: Since they are equal, $x = 112$.
(6)
*
Identify the relationship: Look at the angle labeled $154^\circ$. The angle vertically opposite to it (directly across the intersection) is also $154^\circ$. Now, compare this new $154^\circ$ angle with angle $x$. They are on the same side of the transversal and between the parallel lines. These are
consecutive interior angles.
*
Rule: Consecutive interior angles add up to $180^\circ$.
*
Calculation: $x + 154 = 180$.
Subtract 154 from 180: $180 - 154 = 26$.
So, $x = 26$.
Final Answer:
(1) x = 81
(2) x = 104
(3) x = 99
(4) x = 83
(5) x = 112
(6) x = 26
Parent Tip: Review the logic above to help your child master the concept of alternate angles worksheet.
