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CONSTRUCTING LINES, SEGMENTS, AND ANGLES MCC9-12.G.CO.12

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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.

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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.

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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.

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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.

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

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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.

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4.Mark the point of intersection as point D.

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

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Mark the point of intersection as point E. Do not erase any of your markings. CE = 2AB

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

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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.

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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.

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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.

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

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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.

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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.

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EXAMPLE Construct a segment whose measure is the length of

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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.

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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.

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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.

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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.

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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.

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

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Copy the angle and label the vertex S.

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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.

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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.

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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.

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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.

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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.

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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.

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

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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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