Answer:
- Angle bisector
- Perpendicular bisector
- Congruent angles
- Equilateral triangle
Step-by-step explanation:
You might find it enlightening to actually carry out these constructions using a real physical compass and straightedge. It can help impress on your memory what the steps are and what the end result is.
<u>Angle bisector</u>
Starting with an angle (only picture 1 has this), you draw two arcs at the same distance from the vertex, then you draw two intersecting arcs at the same distance from where the first arcs crossed the legs of the angle. This point of intersection is equidistant from the legs of the angle, so is a point on the angle bisector. The construction is completed by using the straightedge to draw a line through the point of intersection and the angle vertex.
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<u>Perpendicular bisector</u>
The construction is very similar to that for an angle bisector. Here, you draw the intersecting arcs at the same distance from the endpoints of the segment you want to bisect. As before, these points of intersection are equidistant from the endpoints of the segment, so are on the bisector of the segment.
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<u>Congruent angles</u>
As with the angle bisector, you start by drawing arcs at the same distance from the vertex of the reference angle, and similar arcs at that distance from the vertex of the angle you're copying to. Then you adjust the compass to the distance between the points on the angle legs were the arcs intersect, and you copy that distance to the other angle. This makes the two isosceles triangles connecting the vertex and points where the arcs cross the legs congruent. All the angles in the congruent triangles will be congruent, but you're only interested in the one at the original vertex location. You complete the construction by drawing the missing side of that angle using the straightedge.
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<u>Equilateral triangle</u>
The construction starts a lot like the ones for bisectors, but you start by setting the compass to the segment length. Then the point of intersection of the crossing arcs is the same distance from the segment ends as the length of the original segment. The meaning of "equilateral" is "same-length sides". The segments joining the original endpoints with the intersecting arcs will all be the same length, hence form an equilateral triangle.
