### Step-by-Step Solution:
1. **Draw Angle ABC:**
- Start by drawing a horizontal line segment BC.
- Choose a point B on this line and draw a semicircle with center B that intersects the line at two points.
- Using a compass, set the width to a suitable measure (for example, 6 cm) and draw an arc from point B that intersects the semicircle.
- Label the intersection point as A.
- The angle ABC is formed by the line segments AB and BC.
**Hint:** Make sure to keep the compass width consistent when drawing arcs to ensure accuracy.
2. **Draw BP, the bisector of angle ABC:**
- With the compass still set to the same width, place the compass point at point A and draw an arc that intersects both lines AB and BC.
- Label the intersection points as D and E.
- Now, without changing the compass width, place the compass point at D and draw an arc.
- Repeat this from point E, ensuring the two arcs intersect.
- Label the intersection of the arcs as P.
- Draw a straight line from B through P. This line BP is the bisector of angle ABC.
**Hint:** The intersection of the arcs from points D and E gives you the point P, which is crucial for accurately bisecting the angle.
3. **Draw BR, the bisector of angle PBC:**
- Now, we need to bisect angle PBC.
- Place the compass point at point P and draw an arc that intersects lines PB and BC.
- Label the intersection points as F and G.
- Again, without changing the compass width, place the compass point at F and draw an arc.
- Repeat this from point G, ensuring the two arcs intersect.
- Label the intersection of these arcs as R.
- Draw a straight line from B through R. This line BR is the bisector of angle PBC.
**Hint:** Ensure that the arcs from points F and G intersect clearly to find point R accurately.
4. **Draw BQ, the bisector of angle ABP:**
- To bisect angle ABP, place the compass point at point A and draw an arc that intersects lines AB and BP.
- Label the intersection points as H and I.
- Place the compass point at H and draw an arc.
- Repeat this from point I, ensuring the two arcs intersect.
- Label the intersection of these arcs as Q.
- Draw a straight line from B through Q. This line BQ is the bisector of angle ABP.
**Hint:** The angle bisector helps in dividing the angle into two equal parts, so ensure that the arcs intersect properly.
5. **Check if angles ABQ, QBP, PBR, and RBC are equal:**
- Measure angles ABQ, QBP, PBR, and RBC using a protractor or by reasoning through the construction.
- Since BP, BR, and BQ are angle bisectors, angles ABQ, QBP, PBR, and RBC will all be equal.
**Hint:** You can use the property of angle bisectors to confirm that the angles are equal.
6. **Check if angles ABR and QBC are equal:**
- Measure angles ABR and QBC.
- Since ABR is composed of angles ABP and PBR, and QBC is composed of angles QBP and RBC, both should equal 45 degrees.
**Hint:** Use the sum of angles in a triangle or the properties of angle bisectors to verify the equality.
