UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 2: Constructing Lines, Segments, and Angles Instruction

Transcription

1 Prerequisite Skills This lesson requires the use of the following skills: using a compass understanding the geometry terms line, segment, ray, and angle 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. 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. s allow you to draw accurate segments and angles, segment and angle bisectors, and parallel and perpendicular lines. Key Concepts 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. s are different from drawings or sketches. 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. arkings 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 segment is a part of a line that is noted by two endpoints. U1-58

2 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 below and on the next page to copy a segment and an angle. Copying a Segment Using a Compass 1. To copy AB, first make an endpoint on your paper. Label the endpoint C. 2. Put the sharp point of your compass on endpoint A. Open the compass until the pencil end touches endpoint B. 3. Without changing your compass setting, put the sharp point of your compass on endpoint C. ake a large arc. 4. Use your straightedge to connect endpoint C to any point on your arc. 5. Label the point of intersection of the arc and your segment D. Do not erase any of your markings. AB is congruent to CD. Copying a Segment Using Patty Paper 1. To copy AB, place your sheet of patty paper over the segment. 2. ark the endpoints of the segment on the patty paper. Label the endpoints C and D. 3. Use your straightedge to connect points C and D. AB is congruent to CD. U1-59

3 Copying an Angle Using a Compass 1. To copy A, first make a point to represent the vertex A on your paper. Label the vertex E. 2. From point E, draw a ray of any length. This will be one side of the constructed angle. 3. 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. 4. Draw an arc across both sides of A. Label where the arc intersects the angle as points B and C. 5. Without changing the compass setting, put the sharp point of the compass on point E. Draw a large arc that intersects the ray. Label the point of intersection as F. 6. Put the sharp point of the compass on point B of the original angle and set the width of the compass so it touches point C. 7. Without changing the compass setting, put the sharp point of the compass on point F and make an arc that intersects the arc in step 5. Label the point of intersection as D. 8. Draw a ray from point E to point D. Do not erase any of your markings. A is congruent to E. Copying an Angle Using Patty Paper 1. To copy A, place your sheet of patty paper over the angle. 2. ark the vertex of the angle. Label the vertex E. 3. Use your straightedge to trace each side of A. A is congruent to E. Common Errors/isconceptions inappropriately changing the compass setting moving the patty paper before completing the construction attempting to measure lengths and angles with rulers and protractors U1-60

4 Guided Practice Example 1 Copy the following segment using only a compass and a straightedge. N 1. ake an endpoint on your paper. Label the endpoint P. Original segment N P 2. Put the sharp point of your compass on endpoint. Open the compass until the pencil end touches endpoint N. Original segment N P U1-61

5 3. Without changing your compass setting, put the sharp point of your compass on endpoint P. ake a large arc. Original segment N P 4. Use your straightedge to connect endpoint P to any point on your arc. Original segment N P 5. Label the point of intersection of the arc and your segment Q. Original segment N Q Do not erase any of your markings. N is congruent to PQ. P U1-62

6 Example 2 Copy the following angle using only a compass and a straightedge. 1. ake a point to represent vertex. Label the vertex R. R 2. From point R, draw a ray of any length. This will be one side of the constructed angle. R U1-63

7 3. Put the sharp point of the compass on vertex of the original angle. Set the compass to any width that will cross both sides of the original angle. R 4. Draw an arc across both sides of. Label where the arc intersects the angle as points K and L. K L R 5. Without changing the compass setting, put the sharp point of the compass on point R. Draw a large arc that intersects the ray. Label the point of intersection as S. K L S R U1-64

8 6. Put the sharp point of the compass on point L of the original angle and set the width of the compass so it touches point K. K L S R 7. Without changing the compass setting, put the sharp point of the compass on point S and make an arc that intersects the arc you drew in step 5. Label the point of intersection as T. K T L S R 8. Draw a ray from point R to point T. K T L S Do not erase any of your markings. is congruent to R. R U1-65

9 Example 3 Use the given line segment to construct a new line segment with length 2AB. A B 1. Use your straightedge to draw a long ray. Label the endpoint C. C 2. Put the sharp point of your compass on endpoint A of the original segment. Open the compass until the pencil end touches B. 3. Without changing your compass setting, put the sharp point of your compass on C and make a large arc that intersects your ray. C 4. ark the point of intersection as point D. C D U1-66

10 5. Without changing your compass setting, put the sharp point of your compass on D and make a large arc that intersects your ray. C D 6. ark the point of intersection as point E. C D E Do not erase any of your markings. CE = 2AB U1-67

11 Example 4 Use the given angle to construct a new angle equal to A + A. A 1. Follow the steps from Example 2 to copy A. Label the vertex of the copied angle G. 2. 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. 3. Draw an arc across both sides of A. Label where the arc intersects the angle as points B and C. C A B U1-68

12 4. 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. G H 5. Put the sharp point of the compass on C of the original angle and set the width of the compass so it touches B. 6. 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. H G U1-69

13 7. Draw a ray from point G to point. H G Do not erase any of your markings. G= A+ A Example 5 Use the given segments to construct a new segment equal to AB CD. A C D B 1. Draw a ray longer than that of AB. Label the endpoint. 2. Follow the steps from Example 3 to copy AB onto the ray. Label the second endpoint P. P U1-70

14 3. Put the sharp point of the compass on endpoint of the ray. Copy segment CD onto the same ray. Label the endpoint N. N P Do not erase any of your markings. NP = AB CD Angles can be subtracted in the same way. U1-71