Question

Proof: BABC is given. ∠2 is congruent to ∠2 by the definition of an angle bisector. BX is congruent to BX by the reflexive property of congruence. Therefore, ΔABX is congruent to ΔCBX by the postulate. Finally, ∠A is congruent to ∠C because corresponding parts of congruent triangles are congruent. Which of the following correctly completes the proof of the isosceles triangle theorem? a. SSS b. AAA c. SAS d. ASA

Answer 1

The correct choice to complete the proof of the isosceles triangle theorem is the side-angle-side (SAS) congruence postulate, as it justifies ΔABX being congruent to ΔCBX.

Step-by-step explanation:

The proof is confirming the principles of the isosceles triangle theorem, which states that if two sides of a triangle are congruent, then the angles opposite to those sides are also congruent. As stated in the proof, angle 2 is congruent to angle 2 by the definition of an angle bisector, and segment BX is congruent to itself by the reflexive property. To complete the proof, we observe that AB is congruent to CB by the given that BABC is isosceles, with AB and CB being the congruent sides. Thus, the congruent parts are: angle 2 is congruent (angle-angle), and BX is congruent to itself (side). These leave us with the side-angle-side (SAS) congruence postulate, justifying that ΔABX is congruent to ΔCBX. Hence, corresponding parts of congruent triangles are congruent, which means ∠A is congruent to ∠C.

Answer 2

The isosceles triangle theorem is proved using the ASA (Angle-Side-Angle) postulate, which is identified as option d. This is due to the congruency of one side and the two angles surrounding it.

Step-by-step explanation:

The correct option to complete the proof of the isosceles triangle theorem is d. ASA (Angle-Side-Angle). The proof states that ΔABX is congruent to ΔCBX by a postulate, which is the ASA postulate since two angles and the included side are known to be congruent. Specifically, Δ2 is congruent to Δ2 by the definition of an angle bisector, BX is congruent to BX by the reflexive property, and one side of each triangle (AB or BC) is congruent to itself by the reflexive property as well. It follows that ΔA is congruent to ΔC, proving the isosceles triangle theorem.