Question

(01.02 mc)
dan uses a compass to draw an arc from q, as shown. he wants to construct a line segment through r that makes the same angle with \\( \overline { q r } \\) as \\( \overline { p q } \\).
which figure shows the next step to construct a congruent angle at r?
\\( \bigcirc \\) a
\\( \bigcirc \\) b
\\( \bigcirc \\) c
\\( \bigcirc \\) d

Explanation:

Step1: Recall the steps of angle - congruence construction

When constructing a congruent angle, after drawing an arc from the vertex of the given angle (here \(Q\)), the next step is to draw an arc with the same compass - width from the new vertex (here \(R\)).

Step2: Analyze each option

• Option a: It shows drawing an arc from \(R\) with the same radius as the arc from \(Q\) (used to measure \(\angle PQS\)). This is the correct next step.
• Option b: Using a straight - edge (ruler) in this step is incorrect. The compass is used for arc - drawing in the congruent - angle construction at this stage.
• Option c: The incorrect use of a tool (a non - compass tool in an improper way for this construction step) makes it wrong.
• Option d: Repeating the arc - drawing from \(Q\) is not the next step. The focus should be on \(R\) now.

Answer:

A. Option a