-- There had to be instructions that came along with this picture. Somewhere in those instructions, it said that the top line and bottom line are parallel ... the ones with the little arrowheads across them, they're parallel.
-- One factoid about parallel lines is: When parallel lines are cut by a transversal, the alternate interior angles are equal.
Both of the sloping lines are transversals that cut the parallel lines. Look at the longer one. Angle-m and 46° are the alternate interior angles, so they're equal.
-- Angle-m = 46° . Now the whole thing falls apart.
-- Look at the triangle standing up on one vertex. One angle in the triangle is 92°. Another angle in the triangle is 46° (angle-m) . The three angles in the triangle have to add up to 180° .
-- Angle-k is what's left when you start with 180°, take away 92°, and then take away another 46° . I get 42° . ___________________________________
Another way to do it, using another fact about parallel lines:
-- When parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary. (I haven't used this rule in 61 years. I hope they haven't changed it.)
-- Look at the shorter sloping line. It's a transversal that cuts the parallel lines.
-- The interior angles on the same side of the transversal are the 92° and the big one at the bottom ... the (k + 46°).
-- Those angles are supplementary ... they add up to 180° .
92° + (k + 46°) = 180°
-- AGAIN ... 'k' is what's left when you start with 180°, take away 92°, then take away 46° more. I get 42° again.
