Question

In the straightedge and compass construction of the congruent angle below, which of the following reasons can you use to prove that ∠def ≅ ∠ljk?

1) draw a line through points d and f and another line through points l and k. then Δdef ≅ Δljk by sss, and ∠def ≅ ∠ljk by cpctc.
2) draw a line through points d and f and another line through points l and k. then Δdef ≅ Δljk by ssa, and ∠def ≅ ∠ljk by cpctc.
3) draw a line through points d and f and another line through points l and k. then Δdef ≅ Δljk by aas, and ∠def ≅ ∠ljk by cpctc.
4) draw a line through points d and f and another line through points l and k. then Δdef ≅ Δljk by asa, and ∠def ≅ ∠ljk by cpc

Answer 1

The correct reason to prove that ∠def ≅ ∠ljk is option 4, which states, "Draw a line through points d and f and another line through points l and k. Then Δdef ≅ Δljk by ASA, and ∠def ≅ ∠ljk by CPCTC."

Step-by-step explanation:

In a straightedge and compass construction, the congruence of angles is established through a sequence of steps. The crucial aspect is choosing the correct congruence criterion. Option 4 correctly utilizes the Angle-Side-Angle (ASA) criterion. It suggests drawing lines through points d and f and another through points l and k, forming two triangles, Δdef and Δljk. By ASA, if two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. Therefore, ∠def ≅ ∠ljk.

This method adheres to the principles of geometric constructions and congruence, ensuring a valid and accurate proof.

To delve deeper into geometric proofs and congruence criteria, explore the principles of angle and side relationships in triangles. Understanding the nuances of different congruence criteria, such as ASA, provides a solid foundation in geometric reasoning and proof construction.