
Use only your compass and straight edge when drawing a construction. No freehand drawing! 

The constructions associated with copying a segment and copying an angle require that you have a "place" to begin your copy. It is customary to draw a straight line upon which you then produce your copy. Such a line is called a "reference line".
Given: (a line segment)
Construct: a line segment congruent to .
(make a copy of the segment). 

STEPS:
1. Using a straightedge, draw a reference line, if one is not provided.
2. Draw a dot on the reference line to mark your starting point for the construction.
3. Place the point of the compass on point A on the given figure.
4. Stretch the compass so that the pencil is exactly on B. Make a small arc through B.
(This small arc will show that you measured the length of the segment with your compass.)
5. Without changing the span of the compass, place the compass point on the starting point (dot) on the reference line and swing the pencil to create an arc crossing the reference line.
6. Label your copy.
Proof of Construction: The compass was used as a measuring tool to obtain (and copy) the length of the given segment. Since the given segment and the copy are the same length, the segments are congruent.
Note: You may also think of the length from A to B as being the radius of a circle with center at A. By copying the circle at A', the radii segments will be congruent.
Given: ∠ABC
Construct: an angle congruent to ∠ABC.
(make a copy of the angle) 

STEPS:
1. Using a straightedge, draw a reference line, if one is not provided.
2. Place a dot (starting point) on the reference line.
3. Place the point of the compass on the vertex of the given angle, ∠ABC (vertex at point B).
4. Stretch the compass to any length that will stay "on" the angle.
5. Swing an arc so the pencil will cross BOTH sides (rays) of the angle.
6. Without changing the size of the compass, place the compass point on the starting point (dot) on the reference line and swing an arc that will intersect the reference line and go above the reference line.
7. Go back to the given angle ∠ABC and measure the span (width) of the arc from where it crosses one side of the angle to where it crosses the other side of the angle. (Place a small arc to show you measured this distance.)
8. Using this width, place the compass point on the reference line where the previous arc crosses the reference line and mark off this new width on your new arc.
9. Connect this new intersection point to the starting point (dot) on your reference line.
10. Label your copy.
Proof of Construction: When your construction is finished, draw line segments connecting where the first arcs cross the sides of the angles. We know these lengths are the same in both drawings since they represent the measured spans of the same arcs. These addition of these segments will create two triangles that have 3 sets of congruent (equal) sides. Since the triangles are congruent by SSS and any leftover corresponding parts will also be congruent (by CPCTC), we know that ∠ABC is congruent to ∠A'B'C'.
(Note: These triangles are not necessarily ΔABC and ΔA'B'C', since the first arc in the construction need not pass through A and/or C on the given angle.)
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