Question

using the compass and ruler, construct a congruent angle with vertex a. (to change the width of your compass, click on the compass button.)

Explanation:

Step1: Draw a ray from \( A' \)

First, use the ruler to draw a ray with vertex \( A' \). Let's call this ray \( A'B' \). This will be one side of the congruent angle.

Step2: Set compass on original angle

Place the compass tip on the vertex of the original angle (let's call it \( A \)). Draw an arc that intersects both sides of the original angle. Let the intersection points be \( C \) (on one side) and \( D \) (on the other side).

Step3: Transfer arc to \( A' \)

Without changing the compass width, place the compass tip on \( A' \) and draw an arc that intersects the ray \( A'B' \) at point \( C' \).

Step4: Measure arc length \( CD \)

Set the compass width to the distance between \( C \) and \( D \) (the length of the arc on the original angle).

Step5: Mark point \( D' \)

With the compass tip on \( C' \), draw an arc that intersects the previously drawn arc (from Step 3) at point \( D' \).

Step6: Draw ray \( A'D' \)

Use the ruler to draw a ray from \( A' \) through \( D' \). The angle \( \angle B'A'D' \) is congruent to the original angle at \( A \).

Answer:

The congruent angle with vertex \( A' \) is constructed by following the steps of angle - construction using a compass and ruler as described above. The final constructed angle has vertex \( A' \), one side \( A'B' \) and the other side \( A'D' \), and \( \angle B'A'D' \cong \) the original angle at \( A \).