Problem For $∠ABC$ and $∠DEF$, if the sides $AB$ and $DE$ are parallel in the same direction, and $BC$ and $EF$ are parallel in opposite direction, then $$∠ABC+∠DEF=2∠R.$$  $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem When the two sides $AB$ and $BC$ of $∠ABC$ are, respectively, parallel to the two sides $DE$ and $EF$ of $∠DEF$ in the same direction or in the opposite direction, then $$∠ABC=∠DEF.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem When there are two straight lines that intersect on the same plane, two straight lines that are parallel to each of them will always intersect. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem The two straight lines $EF$ and $GH$, which are perpendicular to the two parallel lines $AB$ and $CD$, respectively, are parallel to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…

Problem When a straight line $XY$ is perpendicular to $CD$ of the parallel lines $AB$ and $CD$, it is also perpendicular to the other $AB$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution…

Problem Two straight lines perpendicular to the same straight line are parallel to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Assume that $AB$ and $CD$ are both orthogonal to $XY$…

Problem When three straight lines $a, b$ and $c$ are on the same plane, and $a∥b$ and $a∥c$, then $$b∥c.$$ $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Assuming $b∦c$, two straight lines…

Problem When two parallel lines intersect with a straight line, the corresponding angles or alternate angles are equal, and the interior angles (or the exterior angles) on the same side are complementary to each other. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$…

Problem Two straight lines are parallel when they intersect with a straight line and alternate angles they form are equal. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Let the two straight lines…

Problem The point $P$ on the bisector of $∠BAC$ is equidistant from its two sides $AB$ and $AC$. $$ $$ $$ $$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ $$ Solution Drop a perpendicular line from the point…