To construct a $75^\circ$ angle, you will combine a $60^\circ$ angle with a $15^\circ$ angle using only a compass and straightedge.

Step 1: Construct a $60^\circ$ Angle

1. Draw a Straight Line: Begin by drawing a straight line and marking a point on it, which will serve as the vertex of the angle.

2. Create an Arc: With a compass, place its point on the vertex and draw an arc that intersects the line at two points. Let’s label these intersection points as $A$ and $B$.

3. Draw Intersecting Arcs: Without adjusting the width of the compass, place the compass on point $A$ and draw an arc above the line. Repeat this from point $B$ to create two arcs that intersect each other above the line. Label the intersection of these arcs as point $C$.

4. Draw the Angle: Draw a straight line from the vertex through point $C$. This line forms a $60^\circ$ angle with the original line.

Step 2: Construct a $15^\circ$ Angle

To create a $15^\circ$ angle, you will need to bisect a $30^\circ$ angle. First, you will construct a $30^\circ$ angle by bisecting the $60^\circ$ angle you just created.

1. Bisect the $60^\circ$ Angle: Place the compass point on the vertex of the $60^\circ$ angle and draw an arc that intersects both arms of the angle. Let’s label the points of intersection as $D$ and $E$.

2. Draw New Arcs: Without changing the compass width, place the compass on point $D$ and draw an arc inside the angle. Then, do the same from point $E$. The two arcs will intersect at a new point, which we can label as point $F$.

3. Draw the Bisector: Draw a straight line from the vertex through point $F$. This line bisects the $60^\circ$ angle, creating two $30^\circ$ angles.

Step 3: Bisect the $30^\circ$ Angle to Create a $15^\circ$ Angle

1. Bisect One of the $30^\circ$ Angles: Select one of the $30^\circ$ angles to bisect. Place the compass point at the vertex of this $30^\circ$ angle and draw an arc that intersects both arms of the angle. Label the intersection points as $G$ and $H$.

2. Draw New Arcs: Again, without changing the compass width, draw arcs from points $G$ and $H$ to create a new intersection point inside the angle. Label this intersection point as $I$.

3. Draw the Final Bisector: Draw a straight line from the vertex through point $I$. This line bisects the $30^\circ$ angle, resulting in two $15^\circ$ angles.

Step 4: Combine the $60^\circ$ and $15^\circ$ Angles

Finally, to form a $75^\circ$ angle, align the vertices of the $60^\circ$ angle and the $15^\circ$ angle. Draw a straight line through the combined vertex to complete the construction of the $75^\circ$ angle.

By following these steps, you have successfully constructed a $75^\circ$ angle using a $60^\circ$ angle and a $15^\circ$ angle.