Question

find the values of the missing angles in the diagram below. 147° 69° z° y° 99° w° x° identify the relationship between the 147° angle and angle z.

Explanation:

Step1: Find the value of \( z \)

Since the \( 147^{\circ} \) angle and \( z \) are alternate - exterior angles (parallel lines cut by a transversal), \( z = 147^{\circ} \) (alternate - exterior angles are equal).

Step2: Find the value of \( y \)

\( y\) and \( z \) are supplementary to the \( 69^{\circ} \) angle. Using the angle - sum property around a point (\( 180^{\circ} \) for a linear pair). \( y+z + 69^{\circ}=180^{\circ}\). Substitute \( z = 147^{\circ}\), then \( y+147^{\circ}+69^{\circ}=180^{\circ}\), \( y=180^{\circ}-(147^{\circ} + 69^{\circ})=180^{\circ}-216^{\circ}\) (wrong, actually \( z\) and \( 69^{\circ}\) are vertical angles with angles in a different relation. Correctly, \( z\) and the angle adjacent to \( 69^{\circ}\) in the linear - pair with \( y\): \( z\) and \( 69^{\circ}\) are vertical angles? No. Wait, parallel lines: the line cutting them. Let's use the property that \( z\) (alternate - exterior to \( 147^{\circ}\)) and then for \( y\): \( y + 69^{\circ}=z\) (corresponding angles or using the triangle - like part. Wait, better: since \( z = 147^{\circ}\) (alternate - exterior), and \( y+69^{\circ}=z\) (exterior angle of a triangle - like figure formed by the transversals and parallel lines). So \( y=z - 69^{\circ}=147^{\circ}-69^{\circ}=78^{\circ}\)

Step3: Find the value of \( w \)

Since the \( 99^{\circ}\) angle and \( w\) are alternate - interior angles (parallel lines cut by a transversal), \( w = 99^{\circ}\) (alternate - interior angles are equal).

Step4: Find the value of \( x \)

Using the angle - sum property of a triangle (sum of angles in a triangle is \( 180^{\circ}\)). Let's assume a triangle - like figure. If we consider the angles \( x\), \( y\), \( w\) (adjusting for the parallel - line and transversal properties). \( x + y+w=180^{\circ}\) (using the fact that the sum of angles in a triangle - like region formed by the transversals and parallel lines). Substitute \( y = 78^{\circ}\) and \( w = 99^{\circ}\). Then \( x=180^{\circ}-(78^{\circ}+99^{\circ})=180^{\circ}-177^{\circ}=3^{\circ}\)

Answer:

\( x = 3^{\circ}\), \( y = 78^{\circ}\), \( w = 99^{\circ}\), \( z = 147^{\circ}\)