Question

order the steps from 1-5 for constructing a copy of ∠a.
image of ∠a
place the compass point at a. draw an arc that intersects both rays of ∠a. label the points of intersection b and c.
place the compass point at c. open the compass to the distance between b and c.
place the compass point at y and draw an arc. label the point z where the two arcs intersect. use a straightedge to draw $overrightarrow{xy}$.
use a straightedge to draw a ray with endpoint x.
place the compass point at x and draw an arc intersecting the ray. mark the point y at the intersection.

Explanation:

Step1: Start with the original angle

First, we need to create an arc on the original angle \(\angle A\) to get the length of the angle's "opening". So the first step is: Place the compass point at \(A\). Draw an arc that intersects both rays of \(\angle A\). Label the points of intersection \(B\) and \(C\). (Step 1)

Step2: Draw the initial ray for the copy

Next, we need a starting ray for the copied angle. So: Use a straightedge to draw a ray with endpoint \(X\). (Step 2)

Step3: Transfer the arc to the new ray

Now, we draw an arc on the new ray (with endpoint \(X\)) to match the arc on \(\angle A\). So: Place the compass point at \(X\) and draw an arc intersecting the ray. Mark the point \(Y\) at the intersection. (Step 3)

Step4: Set the compass to the angle's width

We need to set the compass to the distance between \(B\) and \(C\) (the width of the angle at the arc) to transfer the angle. So: Place the compass point at \(C\). Open the compass to the distance between \(B\) and \(C\). (Step 4)

Step5: Complete the copied angle

Finally, we draw the second ray of the copied angle by intersecting the arcs. So: Place the compass point at \(Y\) and draw an arc. Label the point \(Z\) where the two arcs intersect. Use a straightedge to draw \(\overrightarrow{XZ}\). (Step 5) Wait, the original step says \(\overrightarrow{XY}\)? Wait, no, the last step should be drawing the ray from \(X\) through \(Z\) to form the angle. But following the given steps, the order is:

1. Place the compass point at \(A\). Draw an arc that intersects both rays of \(\angle A\). Label the points of intersection \(B\) and \(C\).
2. Use a straightedge to draw a ray with endpoint \(X\).
3. Place the compass point at \(X\) and draw an arc intersecting the ray. Mark the point \(Y\) at the intersection.
4. Place the compass point at \(C\). Open the compass to the distance between \(B\) and \(C\).
5. Place the compass point at \(Y\) and draw an arc. Label the point \(Z\) where the two arcs intersect. Use a straightedge to draw \(\overrightarrow{XZ}\) (but the given step says \(\overrightarrow{XY}\), maybe a typo, but following the steps as given)

So the order of the steps (numbered 1 - 5) is:

1. First step (as above) - number 1
2. Second step - number 2
3. Third step - number 3
4. Fourth step - number 4
5. Fifth step - number 5

So the order of the boxes (from top to bottom) should be:

1. 1
2. 4
3. 5
4. 2
5. 3

Wait, let's re - order properly:

The correct steps for copying an angle are:

1. Draw a ray with endpoint \(X\) (this is the base ray of the copied angle).
2. On \(\angle A\), draw an arc intersecting both sides at \(B\) and \(C\).
3. On the ray \(X\), draw an arc intersecting the ray at \(Y\) (using the same compass width as in step 2? No, step 2 is the original angle, step 3 is the new ray. Wait, the correct sequence is:

- Step 1: Draw a ray with endpoint \(X\) (let's call this ray \( \overrightarrow{X}\) (wait, no, a ray with endpoint \(X\), say \( \overrightarrow{XY}\) but initially, it's just a ray from \(X\)).

- Step 2: On \(\angle A\), place compass at \(A\), draw arc intersecting both sides at \(B\) and \(C\).

- Step 3: On the ray from \(X\), place compass at \(X\), draw arc intersecting the ray at \(Y\) (using the same radius as in step 2? No, the radius for step 3 is the same as step 2? Wait, no, the correct method:

1. Draw a ray \( \overrightarrow{XZ}\) (wait, the standard construction:

1. Given \(\angle A\), draw a ray \( \overrightarrow{XY}\) (this will be one side of the copied angle).

2. With compass at \(A\), draw an arc that intersects both sides of \(\angle A\) at \(B\) and \(C\).

3. With compass at \(X\), draw an arc (same radius as step 2) that intersects \( \overrightarrow{XY}\) at \(Y\).

4. With compass at \(C\), set the width to \(BC\) (the distance between \(B\) and \(C\)).

5. With compass at \(Y\), draw an arc (same radius as step 4) that intersects the arc from step 3 at \(Z\). Then draw \( \overrightarrow{XZ}\) to form the copied angle.

So the steps in order:

1. Draw a ray with endpoint \(X\) (step: Use a straightedge to draw a ray with endpoint \(X\)) - let's call this step 2 (wait, no, let's list the given steps:

Given steps (in the table, from top to bottom):

1. Place the compass point at \(A\). Draw an arc that intersects both rays of \(\angle A\). Label the points of intersection \(B\) and \(C\).

2. Place the compass point at \(C\). Open the compass to the distance between \(B\) and \(C\).

3. Place the compass point at \(Y\) and draw an arc. Label the point \(Z\) where the two arcs intersect. Use a straightedge to draw \(\overrightarrow{XY}\). (Wait, this has a typo, should be \(\overrightarrow{XZ}\))

4. Use a straightedge to draw a ray with endpoint \(X\).

5. Place the compass point at \(X\) and draw an arc intersecting the ray. Mark the point \(Y\) at the intersection.

Now, let's order them correctly:

- First, we need a ray to start the copy: Step 4 (Use a straightedge to draw a ray with endpoint \(X\)) - order 1.

- Then, on the original angle, draw the arc: Step 1 (Place the compass point at \(A\)... ) - order 2.

- Then, on the new ray, draw the arc: Step 5 (Place the compass point at \(X\)... ) - order 3.

- Then, set the compass to the angle's width: Step 2 (Place the compass point at \(C\)... ) - order 4.

- Then, draw the second ray: Step 3 (Place the compass point at \(Y\)... ) - order 5.

So the order of the boxes (from top to bottom) is:

1. 2 (the first step in the table is step 2 of the construction)

2. 4 (the second step in the table is step 4 of the construction)

3. 5 (the third step in the table is step 5 of the construction)

4. 1 (the fourth step in the table is step 1 of the construction)

5. 3 (the fifth step in the table is step 3 of the construction)

Wait, no, let's list the construction steps in order and map to the given steps:

Construction Step 1: Draw a ray with endpoint \(X\). (Given step: "Use a straightedge to draw a ray with endpoint \(X\)" - this is the 4th row in the table)

Construction Step 2: Place the compass point at \(A\). Draw an arc that intersects both rays of \(\angle A\). Label the points of intersection \(B\) and \(C\). (Given step: 1st row)

Construction Step 3: Place the compass point at \(X\) and draw an arc intersecting the ray. Mark the point \(Y\) at the intersection. (Given step: 5th row)

Construction Step 4: Place the compass point at \(C\). Open the compass to the distance between \(B\) and \(C\). (Given step: 2nd row)

Construction Step 5: Place the compass point at \(Y\) and draw an arc. Label the point \(Z\) where the two arcs intersect. Use a straightedge to draw \(\overrightarrow{XZ}\) (Given step: 3rd row, with a typo in the ray name)

So the order of the given steps (from top to bottom) should be numbered as:

1st row: 2

2nd row: 4

3rd row: 5

4th row: 1

5th row: 3

So the numbers to put in the boxes (from top to bottom) are 2, 4, 5, 1, 3.

Answer:

The order of the steps (from top to bottom) is: 2, 4, 5, 1, 3.