1.05 Angles and constructions

The angles in the diagram are marked as congruent. That means they have equal measure.

x=152

Complementary angles have a sum of 90\degree. Use this to write an equation that includes the two angles in the diagram knowing that the angles are complementary. Then we want to solve for x.

If the angles were supplementary, then the sum of the angles would equal 180 \degree instead of 90 \degree.

To construct a copy of an angle, we will follow the series of steps detailed in the concept summary above, using technology in the form of GeoGebra.

We start by drawing a ray nearby that will form one leg of the angle. We can do this by using the Vector tool as shown.

Next, we want to create an arc on the original angle that intersects both legs, and then duplicate an arc with the same radius centered on the new ray.

To do so, we can make use of the Circle: Center & Radius tool, to ensure that both arcs have the same radius.

We can then use the Point tool to create points at all of the intersections between an arc/circle and a ray.

We now want to get the radius of the circle centered at a point of intersection on the original angle (point F in this case) that passes through the other point of intersection (point G).

Once again, we can make use of the Circle: Center & Radius tool, by clicking on F (to be the center) and then dragging to G before releasing. Once we have created the circle, we can check its radius in the algebra tab.

We can then duplicate this circle across to the new ray, centering it on the point of intersection (point H) and giving it the same radius using the Circle: Center & Radius tool.

Finally, we can mark the point of intersection of the two circles we have created, and then use the Vector tool to create the other ray of our copied angle.

Note that the new copy of the angle doesn't have to be drawn in the same orientation as the original angle - it can be rotated, as shown here.

Also note that because we have used a circle tool to draw the arcs, there are two possible points of intersection to use at the last step. Using either of these will create an angle of the correct measure.

The ray which bisects \angle{AOB} will intersect the circle at the midpoint of \overset{\large\frown}{AB}. So we can construct an angle bisector and then mark the point of intersection with the circle.

We will do so by making use of technology in the form of GeoGebra.

We start by drawing an arc centered at O that intersects both radii. The easiest way to do this is to use one of the Circle tools.

We then mark both of the points of intersection, using the point tool.

Next, we draw an arc centered at one of these points of intersection that passes halfway between the interior of \angle{AOB}.

It is easiest to do this using the Circle: Center & Radius tool, since the next step will be to create another arc (i.e. circle) of the same radius.

We now repeat this, creating another circle of the same radius centered at the other point of intersection, so that it crosses the circle we just drew.

We then mark the newly formed point of intersection.

We now construct the bisector of \angle{AOB} by drawing a line through this latest point of intersection and O.

Finally, we can label point C as the intersection of the bisector and \overset{\large\frown}{AB}.

We can also find the midpoint of minor arc \overset{\large\frown}{AB} using string.

First, take a piece of string and tie it to the end of a pencil. Use a push pin to secure the other end of the string on point A and draw an arc.

Then, using the same string, place the push pin to secure the string on point B and draw another arc, intersecting the first.

Finally, draw a line from point O through the intersection of the two drawn arcs to the circle's edge to find the midpoint of minor arc \overset{\large\frown}{AB}.