Question

copying an angle

copy \\(\angle def\\) to the line so that \\(s\\) is the vertex.

this task will be complete when you have constructed an angle with vertex \\(s\\) that is congruent to \\(\angle def\\).

Explanation:

⚡ Using what you learned: Justifying Constructions and Properties

Step 1: Draw the reference arc on the original angle

Place the compass point on the vertex \(E\) of the original angle \(\angle DEF\). Draw an arc that intersects both rays of the angle. Let the intersection point on ray \(ED\) be \(P_1\) and the intersection point on ray \(EF\) be \(P_2\). This establishes a fixed radius \(r = EP_1 = EP_2\).

Step 2: Copy the reference arc to the new vertex

Without changing the compass width \(r\), place the compass point on the new vertex \(S\). Draw a similar arc that intersects the given line. Let this intersection point on the line be \(Q_1\).

Step 3: Measure the opening of the original angle

Place the compass point on the intersection point \(P_1\) on ray \(ED\). Adjust the compass width so that the pencil point rests exactly on the second intersection point \(P_2\) on ray \(EF\). This sets the compass to the distance \(d = P_1P_2\), which represents the "opening size" of the angle at radius \(r\).

Step 4: Mark the opening on the new construction

Without changing this new compass width \(d\), place the compass point on the intersection point \(Q_1\) on the new line. Draw an arc that intersects the first arc you drew around \(S\). Let this intersection point be \(Q_2\).

Step 5: Draw the final ray

Use a straightedge to draw a ray starting from the vertex \(S\) and passing through the intersection point \(Q_2\). The constructed angle with vertex \(S\) is now congruent to \(\angle DEF\).

Answer:

To copy \(\angle DEF\) to vertex \(S\):

1. Draw an arc centered at \(E\) intersecting rays \(ED\) and \(EF\).
2. Using the same radius, draw an arc centered at \(S\) intersecting the line at a point.
3. Measure the distance between the two intersection points on the original angle using your compass.
4. Place the compass on the intersection point on the new line, and draw an arc to intersect the first arc.
5. Draw a line from \(S\) through this new intersection point to complete the congruent angle.