Question

in the figure, two parallel lines are cut by a transversal. two of the angles are labeled. find the measures of the other angles and label them. 135° 45° 135° 115° 45° 25°

Explanation:

Step1: Use vertical angles property

Vertical angles are equal. So the angle vertical to \(135^{\circ}\) is \(135^{\circ}\), and the angle vertical to \(45^{\circ}\) is \(45^{\circ}\).

Step2: Use linear - pair property

Angles in a linear pair sum to \(180^{\circ}\). For the angle adjacent to \(135^{\circ}\), it is \(180 - 135=45^{\circ}\). For the angle adjacent to \(45^{\circ}\), it is \(180 - 45 = 135^{\circ}\).

Step3: Use corresponding angles property (since lines are parallel)

Corresponding angles are equal. The angle corresponding to \(135^{\circ}\) is \(135^{\circ}\), and the angle corresponding to \(45^{\circ}\) is \(45^{\circ}\).

Step4: Use alternate - interior angles property (since lines are parallel)

Alternate - interior angles are equal. The angle with measure \(135^{\circ}\) has an alternate - interior angle of \(135^{\circ}\), and the angle with measure \(45^{\circ}\) has an alternate - interior angle of \(45^{\circ}\).

Answer:

The angles are \(45^{\circ},135^{\circ},45^{\circ},135^{\circ},45^{\circ},135^{\circ}\) (in the order of the blanks from top - left, top - middle, middle - left, middle - middle, bottom - left, bottom - middle)