Angular Constructions Help

Drawing Angles

The following paragraphs describe how to reproduce (copy) an angle, and also how to bisect an angle.

Reproducing An Angle

Figure 5-13 illustrates the process for reproducing an angle. First, suppose two rays intersect at a point P , as shown in drawing A. Set down the non-marking tip of the compass on point P , and construct an arc from one ray to the other. Let R and Q be the two points where the arc intersects the rays (drawing B). Call the angle in question ∠ RPQ , where points R and Q are equidistant from point P .

Fig. 5-13 . Reproducing an angle.

Now, place a new point S somewhere on the page a good distance away from point P , and construct a ray emanating outward from point S , as shown in illustration C. (This ray can be in any direction, but it's easiest if you make it go in approximately the same direction as ray PQ ). Make the new ray at least as long as ray PQ . Without changing the compass span from its previous setting, place its non-marking tip down on point S and construct a sweeping arc that is larger than arc QR . (You can do this by estimation, as shown in drawing D. You can make a full circle if you want.) Let point T represent the intersection of the new arc and the new ray.

Now return to the original arc, place the non-marking tip of the compass down on point Q , and construct a small arc through point R so the compass spans the distance QR , as shown in drawing E. Then, without changing the span of the compass, place its non-marking tip on point T , and construct an arc that intersects the arc centered on point S . Call this intersection point U . Finally, construct ray SU , as shown in drawing F. You now have a new angle with the same measure as the original angle. That is, ∠ UST ≅ ∠ RPQ .

Bisecting An Angle

Figure 5-14 illustrates one method that can be used to bisect an angle, that is, to divide it in half. First, suppose two rays intersect at a point P , as shown in drawing A. Set down the non-marking tip of the compass on point P , and construct an arc from one ray to the other. Call the two points where the arc intersects the rays point R and point Q (drawing B). We can now call the angle in question ∠ RPQ , where points R and Q are equidistant from point P .

Fig. 5-14 . Bisection of an angle.

Now, place the non-marking tip of the compass on point Q , increase its span somewhat from the setting used to generate arc QR , and construct a new arc. Next, without changing the span of the compass, set its non-marking tip down on point R and construct an arc that intersects the arc centered on point Q . (If the arc centered on point Q isn't long enough, go back and make it longer. You can make it a full circle if you want.) Let S be the point at which the two arcs intersect (drawing C). Finally, construct ray PS , as shown at D. This ray bisects ∠ RPQ . This means that ∠ RPS ∠ SPQ , and also that the sum of the measures of ∠ RPS and ∠ SPQ is equal to the measure of ∠ RPQ .