Explain how you can use a straight edge and a compass to construct an angle that is both congruent an adjacent to a given angle.
If you have angle AOB and want to construct BOC such that AOB and BOC are congruent,
with the compass draw an arc subtended by AOB and extending beyond OB. Let the arc intersect OA at D and OB at E.
Place the compass at E and set to the distance ED. Then draw a circle with radius ED which intersects the extended arc DE at C.
angle BOC is congruent to AOB because it subtends an equal arc and has the same radius
To construct an angle that is both congruent and adjacent to a given angle, you can follow these steps using a straight edge (ruler) and a compass:
Step 1: Draw the given angle:
- Start by drawing a line segment using a straight edge. Label one endpoint as point A, which will act as the vertex of the given angle.
- Place the compass pointer at point A and open it to any reasonable radius (let's say r).
- Draw an arc on both sides of the line, intersecting it at points B and C.
Step 2: Construct a congruent angle:
- Without changing the compass width, place the compass pointer at point B and draw an arc intersecting the previously drawn arc (centered at A) at point D.
- Place the straight edge on points A and D and draw a line segment (AD).
Step 3: Construct an adjacent angle:
- Place the compass pointer at point D, which is the new vertex of the constructed angle, and open it to any reasonable radius.
- Draw an arc that intersects the line segment AD at point E.
Step 4: Complete the construction:
- Place the straight edge on points D and E and draw a line segment (DE).
- Label point F, which is the intersection of the line segment DE with the arc (centered at point D).
- The angle ∠DEF is now both congruent and adjacent to the given angle, ∠BAC.
Remember, the key is to use the compass to create congruent arcs and the straight edge to connect the points accordingly.