Find angles formed when parallel lines are cut by a transversal, including co-interior (same-side) angles, as in a parallelogram.
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When a line (the transversal) crosses two parallel lines, it makes only two angle sizes: a small one and a big one, and they add to \(180^\circ\).
So if one angle is \(70^\circ\), every other angle in the picture is either \(70^\circ\) or \(180 - 70 = 110^\circ\). Ask yourself: is the angle I want the small one or the big one?
Common traps: treating same-side interior angles as equal (they are supplementary), and treating corresponding or alternate angles as supplementary. The same rule shows up inside parallelograms: two angles next to each other add to \(180^\circ\).
Alternate interior angles formed by a transversal across parallel lines are equal, so the angle also measures \(53^\circ\). (Treating the angles as supplementary gives \(180 - 53 = 127^\circ\). Subtracting from \(90^\circ\) gives \(90 - 53 = 37^\circ\). Subtracting from \(360^\circ\) gives \(360 - 53 = 307^\circ\).)
Same-side (co-interior) interior angles formed by a transversal across parallel lines are supplementary, so the angle measures \(180 - 57 = 123^\circ\). (Treating the angles as equal gives \(57^\circ\). Subtracting from \(90^\circ\) gives \(90 - 57 = 33^\circ\). Subtracting from \(360^\circ\) gives \(360 - 57 = 303^\circ\).)
Co-interior (same-side interior) angles formed by a transversal cutting two parallel lines are supplementary, so angle \(QRS = 180 - 72 = 108^\circ\). (Treating co-interior angles as equal gives \(72^\circ\). Subtracting from \(90^\circ\) gives \(90 - 72 = 18^\circ\). Subtracting from \(360^\circ\) gives \(360 - 72 = 288^\circ\).)