Question

angle angles within parallel lines no protractor ref: g421.3f1 a1 find the values of x and y a2 find the values of x and y a3 find the values of x and y a4 find the values of x and y b1 find the values of x and y b2 find the values of x and y b3 find the values of x, y and z b4 find the values of x, y and z c1 find the values of x and y c2 find the values of x and y c3 find the values of x, y and z c4 find the value of x

Explanation:

Step1: Recall angle - properties of parallel lines

When two parallel lines are cut by a transversal, corresponding angles are equal, alternate - interior angles are equal, and co - interior angles are supplementary (add up to 180°), and vertical angles are equal.

Step2: Solve for A1

Since the 52° angle and \(x\) are corresponding angles, \(x = 52^{\circ}\). The 52° angle and \(y\) are co - interior angles. So \(y=180 - 52=128^{\circ}\).

Step3: Solve for B1

The 118° angle and \(y\) are corresponding angles, so \(y = 118^{\circ}\). The 118° angle and \(x\) are co - interior angles. So \(x=180 - 118 = 62^{\circ}\).

Step4: Solve for C1

The 49° angle and \(y\) are alternate - interior angles, so \(y = 49^{\circ}\). The 49° angle and \(x\) are co - interior angles. So \(x=180 - 49=131^{\circ}\).

Step5: Solve for A2

The 132° angle and \(y\) are corresponding angles, so \(y = 132^{\circ}\). The 132° angle and \(x\) are co - interior angles. So \(x=180 - 132 = 48^{\circ}\).

Step6: Solve for B2

The 57° angle and \(x\) are corresponding angles, so \(x = 57^{\circ}\). The 57° angle and \(y\) are co - interior angles. So \(y=180 - 57=123^{\circ}\).

Step7: Solve for C2

The 127° angle and \(y\) are corresponding angles, so \(y = 127^{\circ}\). The 127° angle and \(x\) are co - interior angles. So \(x=180 - 127 = 53^{\circ}\).

Step8: Solve for A3

The 141° angle and \(y\) are corresponding angles, so \(y = 141^{\circ}\). The 141° angle and \(x\) are co - interior angles. So \(x=180 - 141=39^{\circ}\).

Step9: Solve for B3

The 126° angle and \(y\) are corresponding angles, so \(y = 126^{\circ}\). The 126° angle and \(x\) are co - interior angles. So \(x=180 - 126 = 54^{\circ}\). The vertical angle to \(x\) is also \(x\), and if we consider the straight - line property, \(z = 180 - x=126^{\circ}\).

Step10: Solve for C3

The 119° angle and \(y\) are corresponding angles, so \(y = 119^{\circ}\). The 119° angle and \(x\) are co - interior angles. So \(x=180 - 119 = 61^{\circ}\). The vertical angle to \(x\) is \(x\), and \(z = 180 - x = 119^{\circ}\).

Step11: Solve for A4

The 58° angle and \(x\) are corresponding angles, so \(x = 58^{\circ}\). The 58° angle and \(y\) are co - interior angles. So \(y=180 - 58=122^{\circ}\).

Step12: Solve for B4

The 63° angle and \(x\) are corresponding angles, so \(x = 63^{\circ}\). The 63° angle and \(y\) are co - interior angles. So \(y=180 - 63=117^{\circ}\). The vertical angle to \(x\) is \(x\), and \(z = 180 - x=117^{\circ}\).

Step13: Solve for C4

The 133° angle and \(x\) are co - interior angles. So \(x=180 - 133 = 47^{\circ}\).

Answer:

A1: \(x = 52^{\circ},y = 128^{\circ}\)
A2: \(x = 48^{\circ},y = 132^{\circ}\)
A3: \(x = 39^{\circ},y = 141^{\circ}\)
A4: \(x = 58^{\circ},y = 122^{\circ}\)
B1: \(x = 62^{\circ},y = 118^{\circ}\)
B2: \(x = 57^{\circ},y = 123^{\circ}\)
B3: \(x = 54^{\circ},y = 126^{\circ},z = 126^{\circ}\)
B4: \(x = 63^{\circ},y = 117^{\circ},z = 117^{\circ}\)
C1: \(x = 131^{\circ},y = 49^{\circ}\)
C2: \(x = 53^{\circ},y = 127^{\circ}\)
C3: \(x = 61^{\circ},y = 119^{\circ},z = 119^{\circ}\)
C4: \(x = 47^{\circ}\)