We need to prove that segments AB and CD are equal, given that arcs AB and CD are equal.
The picture shows 2 triangles that can be formed.
It can be seen from the pictures that segments OA, OB, OC and OD are all equal since they go from the center to the perimeter of the circumference. All those segments are equal to the radius of the circumference.
Arcs AB and CD are equal, then, correspondent central angles 1 and 2 are equal.
Whe have two triangles with a pair of correspondent sides being congruent (OA = OC and OB = OD) and the angle formed by those pair of sides are also equal (Angles 1 and 2). Then, by Side-Angle-Side (SAS) criterion, we can say both triangles (OAB and OCD) are congruent. Both triangles are congruent, then by CPCTE (Corresponding parts of congruent triangles are equal) we can say AB and CD are equal, since both sides are correspondent sides, and both triangles are congruent.
Now we can build the table.
1) First we drew auxiliary lines to identify triangles and sides
Statement: Draw OA, OB, OC and OD
Reason: Auxiliary lines.
2) Sides OA, OB, OC and OD are equal because all of them are a radius line of the circumference:
Statement: OA = OC, OB = OD
Reason: Radii of the same circle are:
3) Next we stated angles 1 and 2 are equal because they are correspondent central angles of two congruent arcs (arc AB and arc CD)
Statement: m∠1 = m∠2
Reason: If arcs are =, then central ∠s are equal.
4) Next, we stated that triangles AOB and COD are equal following the Side-Angle-Side criterian for congruency of triangles:
Statement: ΔAOB = ΔCOD
Reason: SAS
5) Finally we said segments AB and CD are equal because they are corresponding parts of congruent angles:
Statement: AB = CD
Reason: CPCTE