Question
An illustration shows two angles O E F and P C D. An arc intersects ray O E and O F. An arc intersects rays P C and P D and is itself intersected by a smaller arc at the intersection of ray P C and the larger arc. Ray P D is part of a larger line A P D B.
Omar constructed ∠CPD
as a copy of ∠EOF
. Which set of steps did he follow in order to construct a copy of the angle seen in the image?
Omar constructed ∠CPD
as a copy of ∠EOF
. Which set of steps did he follow in order to construct a copy of the angle seen in the image?
Answers
GPT-4o mini
To construct an angle \( \angle CPD \) that is a copy of the angle \( \angle EOF \), Omar could follow these geometric steps:
1. **Draw Ray CP**: Begin by drawing a ray \( CP \) where you want the vertex of the new angle, ensuring that it has a similar orientation to ray \( OE \).
2. **Draw an Arc**: Place the compass point at point \( E \) and draw an arc that intersects both rays \( OE \) and \( OF \). Let the points where the arc intersects rays \( OE \) and \( OF \) be labeled as \( A \) and \( B \).
3. **Transfer the Length**: Without changing the width of the compass, place the compass point at point \( P \) and draw a similar arc that intersects ray \( CP \).
4. **Mark Intersections**: Label the intersection of the arc with ray \( CP \) as point \( X \).
5. **Copy the Arc**: Now, with the same compass width used previously, place the compass point at point \( A \) and draw an arc that marks a point on ray \( CD \). Label this point as \( Y \).
6. **Draw the Second Ray**: Finally, place the compass point at point \( B \) and set the compass width to the same measurement, marking the intersection on ray \( PD \) accordingly.
7. **Complete the Angle**: Draw the ray \( PD \) through point \( Y \). This gives you \( \angle CPD \) such that \( \angle CPD \) is congruent to \( \angle EOF \).
These steps utilize the principles of angle construction using arcs to ensure that angles are congruent.
1. **Draw Ray CP**: Begin by drawing a ray \( CP \) where you want the vertex of the new angle, ensuring that it has a similar orientation to ray \( OE \).
2. **Draw an Arc**: Place the compass point at point \( E \) and draw an arc that intersects both rays \( OE \) and \( OF \). Let the points where the arc intersects rays \( OE \) and \( OF \) be labeled as \( A \) and \( B \).
3. **Transfer the Length**: Without changing the width of the compass, place the compass point at point \( P \) and draw a similar arc that intersects ray \( CP \).
4. **Mark Intersections**: Label the intersection of the arc with ray \( CP \) as point \( X \).
5. **Copy the Arc**: Now, with the same compass width used previously, place the compass point at point \( A \) and draw an arc that marks a point on ray \( CD \). Label this point as \( Y \).
6. **Draw the Second Ray**: Finally, place the compass point at point \( B \) and set the compass width to the same measurement, marking the intersection on ray \( PD \) accordingly.
7. **Complete the Angle**: Draw the ray \( PD \) through point \( Y \). This gives you \( \angle CPD \) such that \( \angle CPD \) is congruent to \( \angle EOF \).
These steps utilize the principles of angle construction using arcs to ensure that angles are congruent.