in exercises 9 and 10, use a compass and straightedge to construct a line through point p that is parallel to line m.
9.
p
m
10.
p
m
proving a theorem in exercises 11 and 12, prove the theorem. (see example 2.)
11. alternate exterior angles converse (theorem 3.7)
12. consecutive interior angles converse (theorem 3.8)

Question

in exercises 9 and 10, use a compass and straightedge to construct a line through point p that is parallel to line m.
9.
  p
m
10.
  p
m
proving a theorem in exercises 11 and 12, prove the theorem. (see example 2.)
11. alternate exterior angles converse (theorem 3.7)
12. consecutive interior angles converse (theorem 3.8)

Accepted Answer

# Explanation:
## Step1: Identify the construction method
To construct a line through $ P $ parallel to $ m $, we use the "corresponding angles" or "copying an angle" method with compass and straightedge. First, draw a transversal (a line) through $ P $ and intersecting $ m $ at a point, say $ Q $.
## Step2: Copy the angle
Using the compass, measure the angle formed between the transversal and $ m $ at $ Q $. Then, with $ P $ as the vertex, construct an angle equal to that measured angle on the transversal through $ P $. The side of this new angle will be the line parallel to $ m $ through $ P $.
## Step3: Verify (optional for construction)
Visually check that the constructed line has the same slope (direction) as $ m $ and does not intersect $ m $ (since they are parallel).

For example, in problem 9:
- Draw a transversal from $ P $ to intersect $ m $ at a point $ Q $.
- Measure $ \angle PQM $ (where $ M $ is a point on $ m $) with the compass.
- At $ P $, construct $ \angle QPR $ equal to $ \angle PQM $, so line $ PR $ is parallel to $ m $.

(Note: The actual construction involves precise compass and straightedge steps, but the key is using angle - copying to ensure parallelism via equal corresponding angles, which is based on the converse of the corresponding angles theorem.)

# Answer:
The construction involves drawing a transversal through $ P $ and $ m $, copying the angle formed by the transversal and $ m $ at the intersection point to $ P $, and the resulting line through $ P $ is parallel to $ m $. (The final constructed line is parallel to $ m $ through $ P $, and its exact drawing follows the compass - straightedge angle - copying method.)

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the construction method

Step2: Copy the angle

Step3: Verify (optional for construction)

Answer: