Question

Line m and point P are shown below. Part A: Using a compass and straightedge, construct line n parallel to line m and passing through point P. Leave all construction marks. Part B: Explain the process that you used to construct line n.

Answer 1

To draw a line parallel to a given line through a certain point, use a compass to create two equi-distant points on the given line. Based on these, form a similar set up at the given point and draw the parallel line.

Step-by-step explanation:

To construct line n parallel to line m through point P using a compass and straightedge, follow these steps:

1. Place the point of the compass at point P, and draw an arc cutting line m at any point, let's say Q.
2. Without changing the compass width, move the compass point to Q, and make another arc above the line m to create a point, say R.
3. Now, without changing the compass width, draw another arc from point P, creating a similar point S on the other side of point P.
4. Draw a line from point P through point S. This is line n, which is parallel to line m.

This method is based on the geometric principle that if a line is drawn crossing two parallel lines, the alternate angles are equal. Hence, by constructing equal arcs, we ensure that line n and line m are parallel as they form equal corresponding angles.

Learn more about Line Construction

Answer 2

see the attached

Step-by-step explanation:

In the attachment, we will refer to the circles, bottom-to-top, as circles 1, 2 and 3. The black points of intersection, bottom-to-top, will be referred to by the letters A, B, C, D. The transversal line through the white point (W) and pink point (P) will be line q.

Step 1. Draw line q through point P so it intersects line m at some convenient point. Label that point W.

Step 2. Choose an arbitrary radius for your compass. Here, we have chosen it to be the length WB. It happens to be less than half the length of WP, but that is not a requirement.

Step 3. Draw an arc of the chosen radius centered at W and intersecting line q and line m. Label the intersection points A (on line m) and B (on line q). These intersection points are on circle 1.

Step 4. Draw an arc of the same radius centered at P. It should be a long enough arc that it would intersect the proposed line parallel to m. Label the intersection point on line q with label C. This intersection point is on circle 3.

Step 5. Adjust the compass width (radius) to the distance from A to B. This is the radius of circle 2.

Step 6. Draw an arc centered at C so that it intersects the arc of Step 4. This is circle 2, and you want it to intersect circle 3. Label that point of intersection D.

Step 7. Draw line PD parallel to m.

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The point of the construction is to create congruent alternate interior angles AWB and CPD, so that lines AW and PD are parallel.