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Constructing Lines, Segments, and Angles

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Constructing Lines, Segments, and Angles

MCC9-12.G.CO.12

- Two basic instruments used in geometry are the straightedge and the compass.
- A straightedge is a bar or strip of wood, plastic, or metal that has at least one long edge of reliable straightness, similar to a ruler, but without any measurement markings.
- A compass is an instrument for creating circles or transferring measurements. It consists of two pointed branches joined at the top by a pivot.
- It is believed that during early geometry, all geometric figures were created using just a straightedge and a compass.

- Though technology and computers abound today to help us make sense of geometry problems, the straightedge and compass are still widely used to construct figures, or create precise geometric representations.
- Constructions allow you to draw accurate segments and angles, segment and angle bisectors, and parallel and perpendicular lines.
- A geometric figure precisely created using only a straightedge and compass is called a construction.
- A straightedge can be used with patty paper (tracing paper) or a reflecting device to create precise representations.
- Constructions are different from drawings or sketches.

- A drawingisa precise representation of a figure, created with measurement tools such as a protractor and a ruler.
- A sketch is a quickly done representation of a figure or a rough approximation of a figure.
- When constructing figures, it is very important not to erase your markings.
- Markings show that your figure was constructed and not measured and drawn.
- An endpoint is either of two points that mark the ends of a line, or the point that marks the end of a ray.

- A line segmentis a part of a line that is noted by two endpoints.
- An angle is formed when two rays or line segments share a common endpoint.
- A constructed figure and the original figure are congruent; they have the same shape, size, or angle.
- Follow the steps outlined on the next few slides to copy a segment and an angle.

- Use the given line segment to construct a new line segment with length 2AB.

- Use your straightedge to draw a long ray. Label the endpoint C.
- Put the sharp point of your compass on endpoint A of the original segment. Open the compass until the pencil end touches B.
- Without changing your compass setting, put the sharp point of your compass on C and make a large arc that intersects your ray, as shown on the next slide.

Mark the point of intersection as point D.

- Without changing your compass setting, put the sharp point of your compass on D and make a large arc that intersects your ray.

- Mark the point of intersection as point E.

Do not erase any of your markings.

CE = 2AB

- Use the given angle to construct a new angle equal to ∠A +∠A.

- Follow the steps from Example 1 to copy ∠A. Label the vertex of the copied angle G.
- Put the sharp point of the compass on vertex A of the original angle. Set the compass to any width that will cross both sides of the original angle.
- Draw an arc across both sides of ∠A. Label where the arc intersects the angle as points B and C.

- Without changing the compass setting, put the sharp point of the compass on G. Draw a large arc that intersects one side of your newly constructed angle. Label the point of intersection H, as shown on the next slide.

- Put the sharp point of the compass on C of the original angle and set the width of the compass so it touches B.
- Without changing the compass setting, put the sharp point of the compass on point H and make an arc that intersects the arc created in step 4. Label the point of intersection as J, as shown on the next slide.

- Draw a ray from point G to point J.
- Do not erase any of your markings ∠G = ∠A + ∠A

- Segments and angles are often described with measurements.
- Segments have lengths that can be measured with a ruler.
- Angles have measures that can be determined by a protractor.
- It is possible to determine the midpoint of a segment.
- The midpointis a point on the segment that divides it into two equal parts.
- When drawing the midpoint, you can measure the length of the segment and divide the length in half.
- When constructing the midpoint, you cannot use a ruler, but you can use a compass and a straightedge (or patty paper and a straightedge) to determine the midpoint of the segment. This procedure is called bisecting a segment.

- To bisect means to cut in half. It is also possible to bisect an angle, or cut an angle in half using the same construction tools. A midsegment is created when two midpoints of a figure are connected. A triangle has three midsegments.
- Bisecting a Segment
- A segment bisector cuts a segment in half.
- Each half of the segment measures exactly the same length.
- A point, line, ray, or segment can bisect a segment.
- A point on the bisector is equidistant, or is the same distance, from either endpoint of the segment.
- The point where the segment is bisected is called the midpoint of the segment.

- Construct a segment whose measure is the length of

- Copy the segment and label it PQ.
- Make a large arc intersecting PQ .Put the sharp point of your compass on endpoint P. Open the compass wider than half the distance of PQ
- Draw the arc, as shown.

- Make a second large arc.
- Without changing your compass setting, put the sharp point of the compass on endpoint Q, then make the second arc, as shown on the next slide.
- It is important that the arcs intersect each other in two places.

- Connect the points of intersection of the arcs.
- Use your straightedge to connect the points of intersection. Label the midpoint of the segment M, as shown on the next slide.

- Make a second large arc.
- Without changing your compass setting, put the sharp point of the compass on endpoint M, and then draw the second arc.
- It is important that the arcs intersect each other in two places, as shown on the next slide.

- Connect the points of intersection of the arcs.
- Use your straightedge to connect the points of intersection.
- Label the midpoint of the smaller segment N, as shown on the next slide.

- Construct an angle whose measure is 3/4 the measure of ∠S.

- Copy the angle and label the vertex S.

- Make a large arc intersecting the sides of ∠S.
- Put the sharp point of the compass on the vertex of the angle and swing the compass so that it passes through each side of the angle.
- Label where the arc intersects the angle as points T and U, as shown on the next slide.

- Find a point that is equidistant from both sides of ∠S.
- Put the sharp point of the compass on point T.
- Open the compass wider than half the distance from T to U.
- Make an arc beyond the arc you made for points T and U, as shown on the next slide.

- Without changing the compass setting, put the sharp point of the compass on U.
- Make a second arc that crosses the arc you just made. It is important that the arcs intersect each other.
- Label the point of intersection W, as shown on the next slide.

- ∠TSW is congruent to ∠WSU.
- The measure of ∠TSW is 1/2 the measure of ∠S.
- The measure of ∠WSU is 1/2 the measure of ∠S.
- Find a point that is equidistant from both sides of ∠WSU.
- Make a large arc intersecting the sides of ∠WSU.
- Put the sharp point of the compass on the vertex of ∠S and swing the compass so that it passes through each side of ∠WSU.
- Label where the arc intersects the angle as points X and Y.

- Put the sharp point of the compass on point X.
- Open the compass wider than half the distance from X to Y.
- Make an arc, as shown on the next slide.

- Without changing the compass setting, put the sharp point of the compass on Y.
- Make a second arc. It is important that the arcs intersect each other.
- Label the point of intersection Z, as shown on the next slide.

- Draw the angle bisector.
- Use your straightedge to create a ray connecting point Z with the vertex of the original angle, S.

- Do not erase any of your markings.
- ∠XSZ is congruent to ∠ZSY.
- ∠TSZ is 3/4 the measure of ∠TSU

- Geometry construction tools can also be used to create perpendicular and parallel lines.
- While performing each construction, it is important to remember that the only tools you are allowed to use are a compass and a straightedge, a reflective device and a straightedge, or patty paper and a straightedge.
- You may be tempted to measure angles or lengths, but in constructions this is not allowed. You can adjust the opening of your compass to verify that lengths are equal.

- Perpendicular lines are two lines that intersect at a right angle (90˚).
- A perpendicular line can be constructed through the midpoint of a segment. This line is called the perpendicular bisector of the line segment.
- It is impossible to create a perpendicular bisector of a line, since a line goes on infinitely in both directions, but similar methods can be used to construct a line perpendicular to a given line.
- It is possible to construct a perpendicular line through a point on the given line as well as through a point not on a given line.

- Parallel lines are lines that either do not share any points and never intersect, or share all points.
- Any two points on one parallel line are equidistant from the other line.
- There are many ways to construct parallel lines.
- One method is to construct two lines that are both perpendicular to the same given line.

- Use a compass and a straightedge to construct a line perpendicular to line through point B that is not on the line.
- Draw line with point B not on the line.

- Make a large arc that intersects line .Put the sharp point of your compass on point B.
- Open the compass until it extends farther than line .
- Make a large arc that intersects the given line in exactly two places.
- Label the points of intersection F and G, as shown on the next slide.

- Make a set of arcs above line .

- Without changing your compass setting, put the sharp point of the compass on point F. Make a second arc above the given line.

- Without changing your compass setting, put the sharp point of the compass on point G. Make a third arc above the given line. The third arc must intersect the second arc.
- Label the point of intersection H, as shown on the next slide.

- Draw the perpendicular line.
- Use your straightedge to connect points B and H.
- Label the new line , as shown on the next slide.

- Do not erase any of your markings.
- Line is perpendicular to line .

- Use a compass and a straightedge to construct a line parallel to line through point C that is not on the line.
- Draw line with point C not on the line.

- Construct a line perpendicular to line through point C.
- Make a large arc that intersects line .
- Put the sharp point of your compass on point C.
- Open the compass until it extends farther than line .
- Make a large arc that intersects the given line in exactly two places.
- Label the points of intersection J and K, as shown on the next slide.

- Make a set of arcs below line .
- Without changing your compass setting, put the sharp point of the compass on point J. Make a second arc below the given line.

- Without changing your compass setting, put the sharp point of the compass on point K. Make a third arc below the given line.
- Label the point of intersection R.

- Draw the perpendicular line.
- Use your straightedge to connect points C and R.
- Label the new line .
- Do not erase any of your markings.
- Line is perpendicular to line .

- Construct a second line perpendicular to line .

- Put the sharp point of your compass on point C.
- Make a large arc that intersects line on either side of point C.
- Label the points of intersection X and Y, as shown on the next slide.

- Make a set of arcs to the right of line .
- Put the sharp point of your compass on point X.
- Open the compass so that it extends beyond point C.
- Make an arc to the right of line , as shown on the next slide.

- Without changing your compass setting, put the sharp point of the compass on point Y. Make another arc to the right of line .
- Label the point of intersection S, as shown on the next slide.

Draw the perpendicular line.

- Use your straightedge to connect points C and S.
- Label the new line , as shown on the next slide.

Line is perpendicular

to line .

Line is parallel to line .