Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. The two arcs above \(XY\) must use the same radius from \(X\) and \(Y\) so that their intersection is equidistant from \(X\) and \(Y\).
2. The two arcs below \(XY\) must also use the same radius from \(X\) and \(Y\) for the same reason.
3. The radius used below need not be equal to the radius used above.
4. Both intersection points lie on the perpendicular bisector of \(XY\), so joining them gives the same perpendicular bisector.

No. The upper pair and lower pair may have different radii, but within each pair the radius from \(X\) and \(Y\) must be the same.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. A pair of equal-radius arcs from \(X\) and \(Y\) gives one point on the perpendicular bisector.
2. A second pair of equal-radius arcs, using a different radius, can give another point on the same side of \(XY\).
3. Two distinct points determine a line.
4. Since both points are equidistant from \(X\) and \(Y\), the line through them is the perpendicular bisector.

Yes. Both pairs can be on the same side if they give two distinct intersection points on the perpendicular bisector.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. The construction needs an intersection point whose distance from \(X\) equals its distance from \(Y\).
2. If the radius from \(X\) is different from the radius from \(Y\), the intersection point is generally not equidistant from the two endpoints.
3. Then the point need not lie on the perpendicular bisector.
4. So equal radii are required within one pair of arcs.

Yes. For one pair of arcs, the radius from \(X\) and the radius from \(Y\) must be equal.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Draw a base segment \(XY\).
2. Construct its perpendicular bisector by drawing equal-radius arcs from \(X\) and \(Y\).
3. Use the arc intersections as centres for symmetric arcs.
4. Repeat equal-radius arcs around the support lines and finally trace only the boundary of the design.

The design can be recreated by first constructing the perpendicular bisector of the base segment and then tracing equal-radius symmetric arcs.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. The rope has loops fixed at \(X\) and \(Y\) and its midpoint is pulled to \(A\).
2. Because \(A\) is the midpoint of the rope, the stretched lengths \(AX\) and \(AY\) are equal.
3. Similarly, pulling the same midpoint below gives \(BX=BY\).
4. Any point equidistant from \(X\) and \(Y\) lies on the perpendicular bisector of \(XY\).
5. Both \(A\) and \(B\) lie on this perpendicular bisector, so line \(AB\) is the perpendicular bisector.

\(AB\) is the perpendicular bisector because both \(A\) and \(B\) are equidistant from \(X\) and \(Y\).

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Mark equal distances \(OX\) and \(OY\) on the given line using the rope.
2. Use equal rope lengths from \(X\) and \(Y\) to locate a point \(A\) off the line.
3. Join \(O\) to \(A\).
4. Since \(A\) is equidistant from \(X\) and \(Y\), \(AO\) lies on the perpendicular bisector of \(XY\) and therefore makes a \(90^\circ\) angle with the line.

One method is to mark equal points \(X\) and \(Y\) around \(O\), locate an equidistant point \(A\) with the rope, and join \(OA\).

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Draw any angle \(XOY\).
2. With centre \(O\), mark equal points \(A\) and \(B\) on the two arms.
3. With centres \(A\) and \(B\) and the same radius, draw arcs meeting at \(C\) inside the angle.
4. Draw \(OC\). By SSS congruence, \(OC\) bisects the angle.
5. Repeat the same steps for four differently opened angles.

Each angle is bisected by drawing equal points on its arms, intersecting equal arcs from those points, and joining the vertex to the arc intersection.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Draw a point \(O\) and construct a \(90^\circ\) angle.
2. Bisect the \(90^\circ\) angle to get a \(45^\circ\) ray.
3. Repeat around \(O\) so that the full \(360^\circ\) angle is divided into eight equal \(45^\circ\) parts.
4. Use equal compass radii on adjacent rays to draw matching petal arcs.

The support rays are eight \(45^\circ\) rays around one centre; equal arcs between adjacent rays form the eight petals.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. The new point \(C\) is still at equal distance from the two marked points \(A\) and \(B\).
2. Also \(OA=OB\) by construction.
3. Thus triangles formed with \(O\), \(A\), \(B\) and \(C\) are congruent by SSS.
4. So \(OC\) still makes equal angles with the two arms, though it may bisect the vertically opposite or external angle depending on where \(C\) lies.

Yes. Equal arcs on the other side still give a line through \(O\) that bisects the corresponding angle.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. From a straight angle, right angle and \(60^\circ\) angle, we can construct many angles.
2. Bisection gives \(90^\circ,45^\circ,22.5^\circ,11.25^\circ\) and also \(60^\circ,30^\circ,15^\circ,7.5^\circ\).
3. Adding and subtracting these gives more angles such as \(75^\circ,105^\circ,135^\circ\).
4. A \(65.5^\circ\) angle is not obtained by the basic ruler-compass bisections introduced here because it needs an exact half-degree construction not generated by these starting angles.

Many angles such as \(30^\circ,45^\circ,60^\circ,75^\circ\) and \(22.5^\circ\) can be made. With the methods here, \(65.5^\circ\) is not constructed exactly.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Use the rope to mark equal lengths \(OA\) and \(OB\) on the two arms of the angle.
2. With the same rope length from \(A\) and \(B\), locate a point \(C\) inside the angle.
3. Stretch the rope straight from \(O\) to \(C\).
4. Because \(OA=OB\) and \(AC=BC\), triangles \(OAC\) and \(OBC\) are congruent.
5. Therefore \(OC\) bisects the angle.

Mark equal points on the arms, find an equal-distance point from them using the rope, and join it to the vertex.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Draw the square and its diagonals or midlines as support lines.
2. Use angle bisection to place equal directions for the petals.
3. The largest petal radius is limited by the side of the square and neighbouring petals.
4. Choose centres on the support lines so that each arc just touches the square boundary or adjacent arc without crossing it.

Use the square as a boundary, construct symmetric support rays, and take the largest equal radius that keeps every petal inside the square.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. For each original angle, draw an arc cutting its arms at \(B\) and \(C\).
2. At the new point \(X\), draw an arc with the same radius to cut one arm at \(Z\).
3. Measure \(BC\) with the compass.
4. Transfer that length on the new arc to get \(Y\).
5. Join \(XY\). The copied angle equals the original by SSS congruence.

Each angle is copied by transferring the same arc radius and the same chord length between the arc-cut points.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Identify one repeating unit: two arms with a fixed angle between them.
2. Construct the first unit using ruler and compass.
3. Copy the angle wherever the same orientation is needed.
4. Use the compass to keep all corresponding arm lengths equal.
5. Repeat the copied unit to complete the whole pattern.

Construct one unit, copy its angle, transfer equal arm lengths, and repeat the unit in the required orientations.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Draw a line \(m\) and a transversal \(l\) meeting it at \(A\).
2. Choose a point \(B\) on the transversal.
3. Copy the angle made by \(m\) and \(l\) at \(A\) to point \(B\).
4. Draw the new line through the copied angle.
5. Equal corresponding angles make the two lines parallel.

Each parallel pair is made by copying a corresponding angle from the first line to a chosen point on a transversal.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Start with one side of the figure and mark its endpoints.
2. Copy the needed angles at the next vertices using the compass method.
3. Transfer equal or intended side lengths with the compass.
4. Construct opposite sides by copying corresponding angles so that they are parallel.
5. Continue around the figure until all vertices are joined.

The figure can be built by repeated angle copying and compass transfer of side lengths, keeping opposite support lines parallel.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Draw two equal support segments and mark their midpoints.
2. Use the endpoints and midpoints as centres for arcs.
3. Choose a radius that lets the upper arcs meet cleanly at the top.
4. Trace the outer boundary to form the pointed arch.
5. Changing the radius changes the sharpness and width of the arch.

A pointed arch is constructed from equal support segments and equal arcs meeting at the top; changing the radius gives different arch shapes.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Choose a base line and mark symmetric support points.
2. Construct equal angles or equal lengths where symmetry is needed.
3. Draw arcs from selected centres with compass radii that meet neatly.
4. Trace the final arch boundary and erase or ignore support lines.

Own designs should use symmetric support points, equal arcs, and a clean traced boundary.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Draw a segment \(AB=5\) cm.
2. With radius \(5\) cm, draw arcs from \(A\) and \(B\) to locate the centre \(O\) of an equilateral triangle on \(AB\).
3. Draw a circle with centre \(O\) and radius \(5\) cm.
4. Step the same radius around the circle to mark six points.
5. Join consecutive points. Each side is \(5\) cm and the angles are equal.

A regular hexagon of side \(5\) cm is made by stepping a \(5\) cm compass radius six times around a circle.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Construct a \(60^\circ\) angle using an equilateral-triangle construction.
2. Bisect the \(60^\circ\) angle to get \(30^\circ\).
3. Bisect the \(30^\circ\) angle to get \(15^\circ\).

\(30^\circ\) is half of \(60^\circ\), and \(15^\circ\) is half of \(30^\circ\).

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Construct a regular hexagon.
2. Extend or construct outward triangles on each side or on the indicated support segments.
3. The central angle pattern gives \(60^\circ\) directions.
4. If each point triangle has all three angles \(60^\circ\) and equal side construction, then it is equilateral.

Yes, the six point triangles are equilateral when they are constructed on the regular-hexagon support with \(60^\circ\) angles.

Strategic angle. First identify what must stay equal. Then decide whether the construction needs a perpendicular bisector, an angle bisector, a copied angle, or a repeated equal length.

1. Draw the basic support segment or polygon shown in the figure.
2. Mark equal distances with the compass.
3. Use the marked points as centres for the required arcs.
4. Trace only the intended final arcs after the support construction is complete.

Both figures are constructed by first building equal support lengths and then drawing the required arcs from those support points.