Angles, Parallel Lines and Polygons

Two angles are complementary if they add to $90^{\circ }$; the complement of $38^{\circ }$ is $90^{\circ }-38^{\circ }=52^{\circ }$. Two angles are supplementary if they add to $180^{\circ }$; the supplement of $38^{\circ }$ is $180^{\circ }-38^{\circ }=142^{\circ }$. Angles that sit side by side along a straight line are supplementary, because together they make the straight angle. When two straight lines cross, they make four angles. The angles directly across from each other are called vertically opposite angles, and they are always equal. Here is why: call the four angles $p,q,r,s$ going around. Then $p+q=180^{\circ }$ (along one line) and $q+r=180^{\circ }$ (along the other line). Both sums equal $180^{\circ }$, so $p+q=q+r$, and subtracting $q$ from both sides gives $p=r$.

Two lines are parallel if they are always the same distance apart and never meet. Arrowheads on the lines (like > on each) show that they are parallel. A transversal is a line that crosses two or more other lines. Where a transversal crosses two lines it makes eight angles, four at each crossing. If the two lines are parallel, the transversal meets them both at the same slant, so the four angles at the upper crossing are an exact copy of the four angles at the lower crossing. That one idea gives three named relationships.

Angle Relationships (Grade 9 Destreamed, MTH1W)

A polygon is a closed shape made of straight sides. You already know that the three angles of a triangle add to $180^{\circ }$. To find the total for any polygon, split it into triangles. In a convex polygon (one with no dents, so every interior angle is less than $180^{\circ }$), pick one corner and draw a diagonal to every corner it is not already joined to. A polygon with $n$ sides splits into $n-2$ triangles this way: a pentagon into 3, a hexagon into 4. The angles of those triangles fill the polygon's corners exactly, so the interior angle sum is $180^{\circ }$ times the number of triangles. The same formula holds for a polygon with a dent (a concave polygon), as long as its sides do not cross each other.

An exterior angle is made by extending one side past a corner; it sits along a straight line with that corner's interior angle, so interior $+$ exterior $=180^{\circ }$. Imagine walking all the way around a convex polygon: at each corner you turn through its exterior angle, and by the time you are back where you started you have turned exactly once around, $360^{\circ }$. So the exterior angles of a convex polygon, one at each corner, always add to $360^{\circ }$, whether it has 3 sides or 30. In a regular polygon (all sides equal and all angles equal), each exterior angle is $\frac{360^{\circ }}{n}$.

Sum of interior angles of a polygon | Angles and intersecting lines | Geometry | Khan Academy