Match each step with the correct ordered description for how to construct a segment bisector.

Given: (a line segment) Construction: bisect .

STEPS:
1. Place your compass point on A and stretch the compass MORE THAN half way to point B (you may also stretch to point B).
2. With this length, swing a large arc that will go above and below

You may also see this construction done where only small portions of the arcs are shown both above and below the segment. The larger arcs, as shown here, are often easier to remember (since they form a "crayfish" looking creature) and they visually reinforce the circle concept needed for the proof of this construction.

The congruent triangle proof of this simple construction is quite lengthy. Here we go!

Proof of Construction: Label the points of intersection of the arcs with the letters D and E. Draw segments

First Triangles: With we know ΔDAE and ΔDBE are congruent by SSS (they share side ). Now, ∠BDE ∠ADE since corresponding parts of Δs are (CPCTC).

Second Triangles: With ∠BDE ∠ADE, and the shared side , ΔADC and ΔBDC are congruent by SAS. Now, by CPCTC. We now have bisecting since two congruent segments were formed.

Perpendicular: We also have ∠DCA ∠DCB by CPCTC. These ∠s form a linear pair making them supplementary. m∠DCA + m∠DCB = 180 (definition of supplementary). By substitution, m∠DCA + m∠DCB = 180, making m∠DCA = 90. ∠DCA is a right angle by definition, making since ⊥lines form rt.∠s.

Notice that after this construction, the sides of quadrilateral ADBE are congruent making ADBE a rhombus. We know that the diagonals of a rhombus bisect each other and are also perpendicular, supporting our construction.

Also, keep in mind that all points on the perpendicular bisector of a segment are equidistant from the endpoints of the segment, which can be seen in this construction.

Given: ∠ABC Construction: bisect ∠ABC.

STEPS:
1. Place compass point on the vertex of the angle (point B).
2. Stretch the compass to any length that will stay ON the angle.
3. Swing an arc so the pencil crosses both sides (rays) of the given angle. You should now have two intersection points with the sides (rays) of the angle.
4. Place the compass point on one of these new intersection points on the sides of the angle. If needed, stretch the compass to a sufficient length to place your pencil well into the interior of the angle. Stay between the sides (rays) of the angle. Place an arc in this interior (it is not necessary to cross the sides of the angle).

5. Without changing the span on the compass, place the point of the compass on the other intersection point on the side of the angle and make a similar arc. The two small arcs in the interior of the angle should be intersecting.

Proof of Construction: Label the points where the first arc intersects with the sides (rays) of the angle as E and F. The intersection of the two small arcs will be labeled D. Draw