Question

which of the following are reasons used in the proof that the angle - bisector construction can be used to bisect any angle? check all that apply.
a. corresponding parts of congruent triangles are congruent.
b. any line segment can be extended indefinitely.
c. every segment is congruent to itself.
d. all of the radii of a circle are congruent.

Explanation:

• Option A: In the angle - bisector construction proof, when we prove two triangles are congruent (using SSS - Side - Side - Side congruence criterion, for example), we use the fact that corresponding parts of congruent triangles are congruent to show that the angles formed by the bisector are equal.
• Option C: The reflexive property (every segment is congruent to itself) is used when we consider a common side in the triangles formed during the angle - bisector construction. For example, if we have two triangles \(\triangle ABD\) and \(\triangle ACD\), the side \(AD\) is common to both triangles, and by the reflexive property \(AD\cong AD\).
• Option D: When we construct arcs with the same radius (centered at the vertex of the angle), the radii of these arcs (which are parts of the sides of the triangles formed in the construction) are congruent. This is crucial for establishing the SSS congruence of the triangles used in the angle - bisector proof.

Option B (Any line segment can be extended indefinitely) is not relevant to the proof of the angle - bisector construction. The angle - bisector construction mainly focuses on triangle congruence and properties of circles (radii) and segments (reflexive property), not on the extension of line segments.

Answer:

A. Corresponding parts of congruent triangles are congruent.
C. Every segment is congruent to itself.
D. All of the radii of a circle are congruent.