Geometry 2

In a compass-and-straightedge construction, what job does each tool do?

The straightedge draws lines, rays, and segments. The compass draws arcs and transfers an exact distance. Do not measure with a ruler or protractor unless the teacher allows it.

What does “use the same compass opening” mean?

Do not change the distance between the compass point and pencil. Move the compass to the new center while keeping its width locked.

What do the geometry symbols ≅, ⟂, and ∥ mean?

≅ means congruent, or the same size or measure. ⟂ means perpendicular, meeting at 90°. ∥ means parallel, staying the same distance apart.

How is a perpendicular bisector different from an angle bisector?

A perpendicular bisector crosses a segment at its midpoint and at 90°. An angle bisector divides one angle into two congruent angles.

Copy segment AB: What is the setup step?

Use the straightedge to draw a new segment longer than AB. Label one endpoint R.

Copy segment AB: How do you transfer its exact length?

Put the compass point at A and pencil at B. Without changing that opening, put the compass point at R and draw an arc across the new segment. Label the intersection S.

Copy segment AB: How do you finish and check the construction?

Erase the excess beyond S. The finished statement is RS ≅ AB. Memory sequence: set A-to-B, move to R, mark S.

Copy ∠BAC: What is the setup step?

Draw a new ray and label its endpoint D. Point D will be the vertex of the copied angle.

Copy ∠BAC: How do you copy the first arc?

Center the compass at A and draw an arc across both sides of the original angle. Label the crossings B and C. Keep the same opening, center at D, and draw an arc across the new ray. Label that crossing E.

Copy ∠BAC: What second distance must the compass copy?

Set the compass width to the distance BC. This copies the chord between the two places where the first arc crossed the original angle.

Copy ∠BAC: How do you finish the new angle?

With compass width BC, center at E and mark F where the new arc intersects the arc centered at D. Draw ray DF. Then ∠EDF ≅ ∠BAC.

Construct the perpendicular bisector of AB: How do you begin?

Choose a compass opening greater than half of AB but less than AB. Center at A and draw arcs above and below the segment.

Construct the perpendicular bisector of AB: How do you make the second pair of arcs?

Do not change the compass opening. Center at B and draw arcs above and below AB so they intersect the arcs from A.

Construct the perpendicular bisector of AB: What line do you draw, and what two facts are true?

Draw the line through the two arc-intersection points. It crosses AB at its midpoint and is perpendicular to AB.

What is the memory sequence for a perpendicular-bisector construction?

Wide same-width arcs from A, wide same-width arcs from B, then connect the two arc intersections. Think: endpoint, endpoint, connect the lens tips.

Bisect ∠A: How do you create points B and C?

Center the compass at vertex A and draw one arc across both sides of the angle. Label the two crossings B and C.

Bisect ∠A: How do you create the interior point D?

Choose a compass opening large enough for the arcs to meet. Draw an interior arc centered at B. With the same opening, draw an interior arc centered at C. Label their intersection D.

Bisect ∠A: How do you finish and check the construction?

Draw ray AD. It is the angle bisector, so ∠BAD ≅ ∠CAD.

What is the memory sequence for an angle-bisector construction?

A makes B and C. B and C make D. Connect A to D.

Construct a line through X parallel to line m: What is the setup?

Choose any point A on line m. Draw line XA. This line acts as the transversal used to copy an angle.

Construct a parallel through X: How do you make the original reference arc?

Center the compass at A and draw an arc that crosses line m at B and line XA at Q.

Construct a parallel through X: How do you copy the reference arc?

Keep the same compass opening. Center at X and draw an arc of the same size across line XA. Label its intersection with XA as T.

Construct a parallel through X: How do you copy the chord BQ?

Set the compass width to BQ. Without changing it, center at T and draw a small arc intersecting the arc centered at X. Label the intersection R.

Construct a parallel through X: How do you finish, and why does it work?

Draw line XR. Then XR ∥ m because you copied the corresponding angle made by transversal XA. Memory sequence: A-arc to X-arc, then BQ to TR.

Construct a perpendicular through point A on line m: How do you begin?

Center the compass at A and draw arcs that cross line m on both sides of A. Label the crossings C and D.

Perpendicular through A on line m: How do you create point X?

Use a compass opening greater than half of CD. Draw an arc centered at C. With the same opening, draw an arc centered at D so the arcs intersect. Label the intersection X.

Perpendicular through A on line m: How do you finish and remember it?

Draw line AX. Then AX ⟂ m at A. Memory sequence: A finds C and D; C and D find X; connect A to X.

Construct a perpendicular from point P not on line m: How do you begin?

Center the compass at P and draw an arc that crosses line m in two places. Label the crossings A and B.

Perpendicular from P not on line m: How do you create point X?

Use a compass opening greater than half of AB. Draw an arc centered at A on the side of m opposite P. With the same opening, draw an arc centered at B so the arcs intersect. Label the intersection X.

Perpendicular from P not on line m: How do you finish and remember it?

Draw line PX. Then PX ⟂ m. Memory sequence: P finds A and B; A and B find X; connect P to X.