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(a) Any correct letter that names a 110° angle (e.g. vertically opposite, corresponding, or alternate with the marked angle).
(b) Valid reason matching the chosen letter, e.g. vertically opposite angles are equal; corresponding angles are equal (parallel lines); alternate angles are equal (parallel lines).
(a) 53°
(b) e.g. vertically opposite angles are equal (or appropriate reason for the marked configuration).
(c) 53°
(d) e.g. corresponding angles are equal (parallel lines), matching the relationship shown.
(a) 61°
(b) e.g. corresponding angles are equal (parallel lines), matching the marked configuration.
(c) 61°
(d) e.g. vertically opposite angles are equal (or appropriate reason for the marked configuration).
(a) 48°
(b) Valid angle fact for the diagram (e.g. vertically opposite / corresponding / alternate with parallels).
(c) 48°
(d) Valid angle fact consistent with the route used for y.
(a) 67°
(b) e.g. vertically opposite angles are equal; or alternate / corresponding angles with parallel lines.
(c) 67°
(d) Matching angle property for the second unknown.
(a) 72°
(b) e.g. corresponding angles are equal (parallel lines), as shown.
(c) 72°
(d) e.g. vertically opposite angles are equal (or appropriate reason for the marked configuration).
(a) 58°
(b) Any correct supporting fact for x.
(c) 58°
(d) Any correct supporting fact for y.
(a) 44°
(b) e.g. vertically opposite angles are equal (or appropriate reason for the marked configuration).
(c) 44°
(d) e.g. corresponding angles are equal (parallel lines), matching the relationship shown.
(a) 76°
(b) Angle facts as used (vertically opposite; corresponding / alternate with AB \parallel CD).
(c) 76°
(d) Angle facts as used, consistent with the diagram.
(a) 39°
(b) Short geometric justification (vertically opposite / corresponding / alternate as appropriate).
(c) 39°
(d) Short geometric justification consistent with the diagram.
Angle x = 46°.
Typical route: use angles on a straight line at each intersection together with AB \parallel CD (corresponding / alternate angles), then angle sum in the triangle formed by the segment of the upper line between the two transversals and the two sloping sides through P.
Angle y = 117°.
Use the given angles with angles on a straight line where needed, then AB \parallel CD (alternate / corresponding angles) to relate the upper and lower intersections, finishing with angles on a straight line at the lower intersection on the right.
Angle x = 55°.
Combine the angle at P with angles on a straight line / in the triangle formed by the two transversals and the upper parallel, then use AB \parallel CD to transfer angles to the lower line.
Angle z = 122°.
Use angles on a straight line at the lower intersections, then AB \parallel CD to obtain matching angles at the upper line, and finish at the left-hand upper intersection with angles on a straight line or in the small triangle containing P.
Angle w = 53°.
Link the two transversals via angles in the triangle at P (or corresponding angles with the parallels), using angles on a straight line at the lower line where needed.
Angle t = 64°.
Use the lower-line angle with AB \parallel CD to find a matching angle at the upper line, then angles on a straight line / in the triangle containing P to reach the angle between the transversals.
Angle x = 72°.
Use the angle at P with angles on a straight line at the upper line, then AB \parallel CD to transfer information to the right-hand transversal.
Angle u = 134°.
Transfer the lower-line information across the parallels, then use angles on a straight line or the triangle containing P to reach the angle at the left-hand upper intersection.
Angle v = 116°.
Use the angle at P together with angles on a straight line at the upper line, then AB \parallel CD to match angles at the lower line.
Angle r = 120°.
Relate the lower angle to the upper line using AB \parallel CD, then use angles on a straight line or the triangle through P to isolate the angle between the left transversal and the upper parallel.
∠BAE = 24° for the diagram as drawn. Typical reasoning: consecutive angles in a parallelogram sum to 180°, angles on a straight line at B, then angle sum in △ABE.
∠FGD = 58° for the diagram as drawn. Angles on a straight line at E, isosceles base angles, then alternate angles (AB \parallel CD).
x = 139° for the diagram as drawn. Use parallelogram angle facts and alternate angles with AB \parallel DC, then angle sum in a triangle through E.
∠ABE = 122° for the diagram as drawn. Typical route: use consecutive angles in the parallelogram and angles on a straight line at B, then angle sum in △ABE to reach the angle at B.
∠BAE = 26° for the diagram as drawn. Typical reasoning: consecutive angles in a parallelogram sum to 180°, angles on a straight line at B, then angle sum in △ABE.
∠ABE = 105° for the diagram as drawn. Typical route: use consecutive angles in the parallelogram and angles on a straight line at B, then angle sum in △ABE to reach the angle at B.
∠EFG = 62° for the diagram as drawn. Use angles on a straight line at E, isosceles base angles at F and G, then angle sum in △EFG together with AB \parallel CD where needed.
∠BAE = 26° for the diagram as drawn. Typical reasoning: consecutive angles in a parallelogram sum to 180°, angles on a straight line at B, then angle sum in △ABE.
∠ABE = 100° for the diagram as drawn. Typical route: use consecutive angles in the parallelogram and angles on a straight line at B, then angle sum in △ABE to reach the angle at B.
∠EGF = 55° for the diagram as drawn. Angles on a straight line at E, isosceles symmetry, then angle sum in △EFG and alternate angles with AB \parallel CD as appropriate.