CONSTRUCTING LINES, SEGMENTS, AND ANGLES MCC9-12.G.CO.12 Slide 2
COPYING SEGMENTS AND ANGLES INTRODUCTION 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. Slide 3 INTRODUCTION 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. Slide 4 INTRODUCTON A drawing is a 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. Slide 5 INTRODUCTION A
line segment is 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. Slide 6 Slide 7 Slide 8 Slide 9 EXAMPLE Use the given line
segment to construct a new line segment with length 2AB. Slide 10
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. Slide 11 4.Mark the point of intersection as point D.
Slide 12 Without changing your compass setting, put the sharp point
of your compass on D and make a large arc that intersects your ray.
Slide 13 Mark the point of intersection as point E. Do not erase
any of your markings. CE = 2AB Slide 14 EXAMPLE Use the given angle
to construct a new angle equal to A + A. Slide 15 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. Slide 16 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. Slide 17 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. Slide 18 Draw a ray from point G to point J. Do not
erase any of your markings G = A + A Slide 19 BISECTING SEGMENTS
AND ANGLES INTRODUCTION 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 midpoint is 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. Slide 20 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. Slide 21 Slide 22 Slide 23 EXAMPLE
Construct a segment whose measure is the length of Slide 24 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. Slide 25
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. Slide 26 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. Slide 27 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. Slide 28 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.
Slide 29 EXAMPLE Construct an angle whose measure is 3/4 the
measure of S. Slide 30 Copy the angle and label the vertex S. Slide
31 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. Slide 32 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. Slide 33 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.
Slide 34 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. Slide 35 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. Slide 36 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.
Slide 37 Draw the angle bisector. Use your straightedge to create a
ray connecting point Z with the vertex of the original angle, S.
Slide 38 Do not erase any of your markings. XSZ is congruent to
ZSY. TSZ is 3/4 the measure of TSU Slide 39 CONSTRUCTING PARALLEL
AND PERPENDICULAR LINES INTRODUCTION 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. Slide 40
PERPENDICULAR LINES AND BISECTORS 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. Slide 41 Slide 42
Slide 43 Slide 44 Slide 45 Slide 46 Slide 47 PARALLEL LINES
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. Slide 48
Slide 49 Slide 50 Slide 51 Slide 52 EXAMPLE 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. Slide 53 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. Slide 54 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. Slide
55 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. Slide 56 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. Slide 57 EXAMPLE
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. Slide 58 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. Slide 59 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. Slide 60 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. Slide 61 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. Slide 62 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. Slide 63 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. Slide 64 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. Slide 65 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.
