Question

select the correct answer.
what needs to be corrected in the following construction for copying \\( \angle a b c \\) with point \\( d \\) as the vertex?
\\( \bigcirc \\) a. the second arc should be drawn centered at \\( k \\) through \\( a \\).
\\( \bigcirc \\) b. the second arc should be drawn centered at \\( j \\) through \\( a \\).
\\( \bigcirc \\) c. the third arc should cross the second arc.
\\( \bigcirc \\) d. the third arc should pass through \\( d \\).

Explanation:

Step1: Recall the steps of copying an angle

When copying an angle \(\angle ABC\) to \(\angle DEF\) (in this case with vertex \(D\)), we first draw a ray \(DE\). Then, we draw an arc centered at \(B\) intersecting the sides of \(\angle ABC\) (say at \(J\) and \(K\)). Next, we draw an arc of the same radius centered at \(D\) (the second - arc). Then, we measure the distance between \(J\) and \(K\) with a compass and draw the third - arc (centered at the intersection of the second - arc and the ray \(DE\)) such that it intersects the second - arc.

Step2: Analyze each option

• Option A: The second arc is centered at \(D\) (not \(K\)) when copying \(\angle ABC\) to a new angle with vertex \(D\).
• Option B: The second arc is centered at \(D\) (not \(J\)) when copying \(\angle ABC\) to a new angle with vertex \(D\).
• Option C: The third arc (centered at \(L\)) should cross the second arc (centered at \(D\)). This is a key step in the angle - copying construction. If the third arc does not cross the second arc, we cannot complete the construction of the copied angle.
• Option D: The third arc is centered at \(L\) (a point on the ray with vertex \(D\)) and is used to find the intersection point to complete the angle - copying, not to pass through \(D\).

Answer:

C. The third arc should cross the second arc.