Geometry reasoning chains

PQRS is a parallelogram, with one side produced to form an exterior angle. The exterior angle at P is 61°. Calculate ∠R, giving a reason for each step.

In quadrilateral WXYZ, one side is produced to form an exterior angle. ∠$Z=100^\circ$, ∠$W=86^\circ$ and the exterior angle at Y is 109°. Calculate ∠X, giving a reason for each step.

△PQR is similar to △XYZ (the triangles are equiangular). $PQ=6$ cm, $QR=10$ cm and $XY=6$ cm. Calculate the length of YZ.

PQRS is a parallelogram, with one side produced to form an exterior angle. The exterior angle at P is 97°. Calculate ∠S, giving a reason for each step.

In quadrilateral KLMN, one side is produced to form an exterior angle. ∠$L=57^\circ$, ∠$M=101^\circ$ and the exterior angle at N is 73°. Calculate ∠K, giving a reason for each step.

ABCD is a parallelogram, with one side produced to form an exterior angle. The exterior angle at A is 67°. Calculate ∠C, giving a reason for each step.

PQRS is a parallelogram, with one side produced to form an exterior angle. The exterior angle at P is 54°. Calculate ∠R, giving a reason for each step.

△ABC is similar to △XYZ (the triangles are equiangular). $AB=27$ cm, $BC=12$ cm and $XY=27$ cm. Calculate the length of YZ.

In quadrilateral KLMN, one side is produced to form an exterior angle. ∠$N=82^\circ$, ∠$L=75^\circ$ and the exterior angle at K is 103°. Calculate ∠M, giving a reason for each step.

△RST is similar to △PQR (the triangles are equiangular). $RS=8$ cm, $ST=7$ cm and $PQ=32$ cm. Calculate the length of QR.

△DEF is similar to △XYZ (the triangles are equiangular). $DE=5$ cm, $EF=6$ cm and $XY=15$ cm. Calculate the length of YZ.

KLMN is a parallelogram, with one side produced to form an exterior angle. The exterior angle at K is 96°. Calculate ∠M, giving a reason for each step.