CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 1 of 35. Similarity Congruence and Proofs Name:

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1 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 1 of 35 Similarity ongruence and Proofs Name: Date: Understand similarity in terms of similarity transformations M9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. M9-12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. M9-12.G.SRT.3 Use the properties of similarity transformations to establish the criterion for two triangles to be similar. Prove theorems involving similarity M9-12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. M9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Understand congruence in terms of rigid motions M9-12.G.O.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. M9-12.G.O.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. M9-12.G.O.8 Explain how the criteria for triangle congruence (S, SS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems M9-12.G.O.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. M9-12.G.O.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. M9-12.G.O.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Make geometric constructions M9-12.G.O.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). opying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. M9-12.G.O.13 onstruct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Lesson 1.0 Definitions djacent ngles: ngles in the same plane that have a common vertex and a common side, but no common interior points. lternate Exterior ngles: lternate exterior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are outside the other two lines. When the two other lines are parallel, the alternate exterior angles are equal. lternate Interior ngles: lternate interior angles are pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on opposite sides of the transversal and are in between the other two lines. When the two other lines are parallel, the alternate interior angles are equal. ngle: ngles are created by two distinct rays that share a common endpoint (also known as a vertex). isector: bisector divides a segment or angle into two equal parts. entroid: The point of concurrency of the medians of a triangle. ircumcenter: The point of concurrency of the perpendicular bisectors of the sides of a triangle. oincidental: Two equivalent linear equations overlap when graphed. omplementary ngles: Two angles whose sum is 90 degrees. or denote angles with vertex. ongruent: Having the same size, shape and measure. Two figures are congruent if all of their corresponding measures are equal. ongruent Figures: Figures that have the same size and shape. orresponding ngles: ngles that have the same relative positions in geometric figures. orresponding Sides: Sides that have the same relative positions in geometric figures Dilation: Transformation that changes the size of a figure, but not the shape.

2 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 2 of 35 Endpoints: The points at an end of a line segment Equiangular: The property of a polygon whose angles are all congruent. Equilateral: The property of a polygon whose sides are all congruent. Exterior ngle of a Polygon: an angle that forms a linear pair with one of the angles of the polygon. Incenter: The point of concurrency of the bisectors of the angles of a triangle. Intersecting Lines: Two lines in a plane that cross each other. Unless two lines are coincidental, parallel, or skew, they will intersect at one point. Intersection: The point at which two or more lines intersect or cross. Line: One of the basic undefined terms of geometry. Traditionally thought of as a set of points that has no thickness but its length goes on forever in two opposite directions. denotes a line that passes through point and. Line Segment or Segment: The part of a line between two points on the line. denotes a line segment between the points and. Linear Pair: djacent, supplementary angles. Excluding their common side, a linear pair forms a straight line. Measure of each Interior ngle of a Regular n-gon: n180 (n 2) Orthocenter: The point of concurrency of the altitudes of a triangle. Parallel Lines: Two lines are parallel if they lie in the same plane and they do not intersect. Perpendicular Lines: Two lines are perpendicular if they intersect at a right angle. Plane: One of the basic undefined terms of geometry. Traditionally thought of as going on forever in all directions (in two-dimensions) and is flat (i.e., it has no thickness). Point: One of the basic undefined terms of geometry. Traditionally thought of as having no length, width, or thickness, and often a dot is used to represent it. Proportion: n equation which states that two ratios are equal. Ratio: omparison of two quantities by division and may be written as r/s, r:s, or r to s. Ray: ray begins at a point and goes on forever in one direction. Reflection: transformation that "flips" a figure over a line of reflection Reflection Line: line that is the perpendicular bisector of the segment with endpoints at a pre-image point and the image of that point after a reflection. Regular Polygon: polygon that is both equilateral and equiangular. Remote Interior ngles of a Triangle: the two angles non-adjacent to the exterior angle. Rotation: transformation that turns a figure about a fixed point through a given angle and a given direction. Same-Side Interior ngles: Pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on the same side of the transversal and are between the other two lines. When the two other lines are parallel, same-side interior angles are supplementary. Same-Side Exterior ngles: Pairs of angles formed when a third line (a transversal) crosses two other lines. These angles are on the same side of the transversal and are outside the other two lines. When the two other lines are parallel, same-side exterior angles are supplementary. Scale Factor: The ratio of any two corresponding lengths of the sides of two similar figures. Similar Figures: Figures that have the same shape but not necessarily the same size. Skew Lines: Two lines that do not lie in the same plane (therefore, they cannot be parallel or intersect). Sum of the Measures of the Interior ngles of a onvex Polygon: 180º(n 2). Supplementary ngles: Two angles whose sum is 180 degrees. Transformation: The mapping, or movement, of all the points of a figure in a plane according to a common operation. Translation: transformation that "slides" each point of a figure the same distance in the same direction Transversal: line that crosses two or more lines. Vertical ngles: Two nonadjacent angles formed by intersecting lines or segments. lso called opposite angles.

3 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 3 of 35 Lesson 1.1 Dilations dilation is a transformation that produces an image that is the same shape as the original, but is a different size. dilation stretches or shrinks the original figure. The center of dilation is a fixed point in the plane about which all points are expanded or contracted. It is the only invariant point under a dilation. dilation of scalar factor k whose center of dilation is the origin may be written: Dk (x, y) = (kx, ky). If the scale factor, k, is greater than 1, the image is an enlargement (a stretch). If the scale factor is between 0 and 1, the image is a reduction (a shrink). Properties preserved (invariant) under a dilation: 1. angle measures (remain the same) 2. parallelism (parallel lines remain parallel) 3. colinearity (points stay on the same lines) 4. midpoint (midpoints remain the same in each figure) 5. orientation (lettering order remains the same) Dilations create similar figures. Example 1: Draw the dilation image of triangle Example 2: Draw the dilation image of pentagon with the center of dilation at the origin DE with the center of dilation at the origin and nd a scale factor of 2. a scale factor of 1/3.

4 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 4 of 35 Given a center and a scale factor, verify experimentally, that when dilating a figure in a coordinate plane, a segment of the pre-image that does not pass through the center of the dilation, is parallel to its image when the dilation is performed. However, a segment that passes through the center remains unchanged. PROLEMS Given the figure dilate the figure by the factor shown. The center of dilation is the origin. Label all points before and after the dilation. 1. Dilate by a factor 2 2. Dilate by a factor ½ efore Dilation: fter Dilation: 3. Dilate by a factor of 2 4. Dilate by a factor of 3 efore Dilation: fter Dilation: 5. Dilate by a factor of ½ 6. Dilate by a factor of ½ efore Dilation: fter Dilation:

5 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 5 of Dilate by a factor of ⅓ 8. Dilate by a factor of 4 efore Dilation: fter Dilation: 9. Dilate by a factor of Dilate by a factor of ½ efore Dilation: fter Dilation: 11. Dilate by a factor of ½ 12. Dilate by a factor of 2 efore Dilation: fter Dilation:

6 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 6 of 35 When the center of dilation is not the origin you have to proceed as outlines below. PROLEM: Draw the dilation image of rectangle EFGH with the center of dilation at point E and a scale factor of 1/2. OSERVE: Point E and its image are the same. It is important to observe the distance from the center of the dilation, E, to the other points of the figure. Notice EF = 6 and E'F' = 3. HINT: e sure to measure distances for this problem. PROLEMS: Given the figure and the center of dilation dilate the figure by the factor shown. 9. Dilation factor 2, dilation center (4,0). 10. Dilation factor ½, dilation center (2,4). 11. Dilate by a factor of ½, dilation center (10,0) 12. Dilate by a factor of ½, dilation center (2,6)

7 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 7 of 35 Lesson 1.1 Introduction to Similarity Plot the ordered pairs given in the table to make six different figures. Draw each figure on a separate sheet of graph paper. onnect the points with line segments as follows: For Set 1, connect the points in order. onnect the last point in the set to the first point in the set. For Set 2, connect the points in order. onnect the last point in the set to the first point in the set. For Set 3, connect the points in order. Do not connect the last point in the set to the first point in the set. For Set 4, make a dot at each point (do not connect the dots). Figure 1 Figure 2 Set 1 Set 1 (6,4) (12, 8) (6,-4) (12, -8) (-6,-4) (-12, -8) (-6,4) (-12, 8) Set 2 Set 2 (1, 1) (2, 2) (1, -1) (2, -2) (-1, -1) (-2, -2) (-1, 1) (-2, 2) Set 3 Set 3 (4, -2) (8, -4) (3, -3) (6, -6) (-3, -3) (-6, -6) (-4, -2) (-8, -4) Set 4 Set 4 (4, 2) (8, 4) (-4, 2) (-8, 4) Figure 1 Figure

8 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 8 of 35 Figure 3 Figure 4 Set 1 Set 1 (18, 4) (18, 12) (18, -4) (18, -12) (-18, -4) (-18, -12) (-18, 4) (-18, 12) Set 2 Set 2 (3, 1) (3, 3) (3, -1) (3, -3) (-3, -1) (-3, -3) (-3, 1) (-3, 3) Set 3 Set 3 (12, -2) (12, -6) (9, -3) (9, -9) (-9, -3) (-9, -9) (-12, -2) (-12, -6) Set 4 Set 4 (12, 2) (12, 6) (-12, 2) (-12, 6) Figure 3 Figure

9 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 9 of 35 Figure 5 Figure 6 Set 1 Set 1 (6, 12) (8, 6) (6, -12) (8, -2) (-6, -12) (-4, -2) (-6, 12) (-4, 6) Set 2 Set 2 (1, 3) (3, 3) (1, -3) (3, 1) (-1, -3) (1, 1) (-1, 3) (1, 3) Set 3 Set 3 (4, -6) (6, 0) (3, -9) (5, -1) (-3, -9) (-1, -1) (-4, -6) (-2, 0) Set 4 Set 4 (4, 6) (6, 4) (-4, 6) (-2, 4) Figure 5 Figure fter drawing the six figures, compare Figure 1 to each of the other figures and answer the following questions. 1. Which figures are similar? Use the definition of similar figures to justify your response. 2. Describe any similarities and/or differences between Figure 1 and each of the similar figures. Describe how corresponding sides compare. Describe how corresponding angles compare. 3. How do the coordinates of each similar figure compare to the coordinates of Figure 1? Write general rules for making the similar figures.

10 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 10 of Is having the same angle measurement enough to make two figures similar? Why or why not? 5. What would be the effect of multiplying each of the coordinates in Figure 1 by ½? 6. The sketch below shows two triangles, Δ and ΔEFG. Δ has an area of 12 square units, and its base () is equal to 8 units. The base of ΔEFG is equal to 24 units. G E 24 F a. How do you know that the triangles are similar? b. Name the pairs of corresponding sides and the pairs of corresponding angles. How are the corresponding sides related and how are the corresponding angles related? Why is this true? c. What is the area of Δ EFG? Explain your reasoning.

11 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 11 of 35 d. What is the relationship between the area of Δ and the area of Δ EFG? What is the relationship between the scale factor and the ratio of the areas of the two triangles? 7. Jeannie is practicing on the basketball goal outside her house. She thinks that the goal seems lower than the 10 ft. goal she plays on in the gym. She wonders how far the goal is from the ground. Jeannie can not reach the goal to measure the distance to the ground, but she remembers something from math class that may help. First, she needs to estimate the distance from the bottom of the goal post to the top of the backboard. To do this, Jeannie measures the length of the shadow cast by the goal post and backboard. She then stands a yardstick on the ground so that it is perpendicular to the ground, and measures the length of the shadow cast by the yardstick. Here are Jeannie s measurements: Length of shadow cast by goal post and backboard: 5 ft. 9 in. Length of yardstick s shadow: 1 ft. 6 in. a. Draw and label a picture to illustrate Jeannie s experiment. b. Using her measurements, determine the height from the bottom of the goal post to the top of the backboard. c. If the goal is approximately 24 inches from the top of the backboard, how does the height of the basketball goal outside Jeannie s house compare to the one in the gym? Justify your answer.

12 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 12 of 35 How to tell if triangles are similar ny triangle is defined by six measures (three sides, three angles). ut you don't need to know all of them to show that two triangles are similar. Various groups of three will do. Triangles are similar if: 1. (angle angle angle) ll three pairs of corresponding angles are the same SSS in same proportion (side side side) ll three pairs of corresponding sides are in the same proportion. E D 4 Since corresponding sides have the same ratios, F , both triangles are similar SS (side angle side) Two pairs of sides in the same proportion and the included angle equal. = 16 = 20 = 15 = 12 The ratios of are the same. Since the angles between intersecting lines are congruent (vertical angles theorem), we have side-angle-side similarity.

13 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 13 of 35 s seen on the previous page we can show similarity by: 1.) Showing that corresponding angles are congruent. 2.) Showing that corresponding sides are proportional. 3.) Showing SS similarity (angle congruent and sides proportional). You can also show similarity by using the ngle-ngle Similarity Postulate ( ~ ). If two angles of one triangle are congruent to two angles of another triangle then the triangles are similar. Examples of ~ The ΔGHJ ~ ΔGMK are similar by ~ because 1) H and M are congruent by orresponding ngles Postulate. 2) G and G are congruent since they are the same angle. The Δ ~ ΔXZY are similar by ~ because 1) They are both right triangles; therefore they both have a 90 degree angle. 2) ll triangles add up to 180 degrees, since angle is 40 degrees PROLEMS For problems re the triangles similar? If so state why and which method you used. 8.

14 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 14 of D E

15 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 15 of E D E D D

16 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 16 of alculate the distance across the lake by using the similar triangle theorem. DISTNE oth triangles are similar. What value is x? 19.

17 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 17 of 35 Lesson 1.2 Pythagorean Theorem using Triangle Similarity The figure below shows the right triangle. Its right angle is angle. 1. Draw a perpendicular line from to. Label that point D. Use a straight edge. 2. How many triangles do you see in the figure above? 3. Why are the triangles and D similar to each other? (Hint: Measure the angles with a protractor) 4. Why are the triangles and D similar to each other? (Hint: Measure the angles with a protractor) 5. Why are the triangles D and D similar to each other? (Hint: Measure the angles with a protractor) 6. What is the ratio of triangle D to D? (Measure and compare corresponding sides) 7. What is the ratio of triangle D to? (Measure and compare corresponding sides) 8. What is the ratio of triangle D to? (Measure and compare corresponding sides)

18 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 18 of 35 Example Find the missing length indicated 100 D 36 X Solution: First solve for = = 64. Knowing that triangle D is similar to triangle D, we can validate the following ratios of corresponding sides: D x Ratios of corresponding sides of similar triangles are the same x D 64 x Plug in the values x 36 2 x ross-multiply x 48 Solve for x PROLEMS Find the missing length indicated D 9 X D 9 X

19 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 19 of D 45 X D 48 X 5. 7 D X 9 6. X D 12 16

20 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 20 of 35 Lesson 1.3 ngle Geometry Two angles are vertical angles if their sides form two pairs of opposite rays. How do you know that vertical angles are congruent? m 1 + m 3 = 180 because the Linear Pair postulate m 2 + m 3 = 180 because the Linear Pair postulate Set the two equations equal to each other since they both equal 180 degrees. m 2 + m 3 = m 1 + m 3 - m 3 = - m 3 Subtract m 3from both sides m 2 = m 1 Therefore: 2 1 PROLEMS 1. Prove that 3 4 using a similar method.

21 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 21 of 35 Transversal Lines When a transversal crosses parallel lines, there are several pairs of special angles. orresponding ngle Postulate: If two parallel lines are cut by a transversal, then corresponding angles are congruent. Using this postulate, name a pair of congruent angles. How do we know that 3 6? y using the lternate Interior ngle Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent. How do we know that 3 5 are supplementary? Same Side Interior ngle Theorem: If two parallel lines are cut by a transversal, then same side interior angles are supplementary. Example 1 omplete the statement given that m GF 90. a. GD? b. if m GF 113, then m DGE? G Solution: F E D a. ecause GD and GF are vertical angles, GD = GF. y definition of congruent angles, m GD m GF. So, GD 90. b. y the angle addition postulate if m GF m GF m G. Substitute to get m G. Therefore m G 23. ecause DGE and G are vertical angles, DGE G. y definition of congruent angles, m DGE m G. Therefore m DGE 23.

22 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 22 of 35 PROLEMS opy and complete the statement given that m HD m HE m HG D 2. m H 3. If m HD 31, then m DHE H E 4. If m H 48, then m EHF G F 5. If m GHF 38, then m H 16 If m GD 90 and m GD 26, find 6. m 1 7. m 2 8. m 3 m 3 G D 9. m FGE F E 10. m DGE m 2 m 1 Lines L and M are parallel, find 11. m m m m m m m m 8 M L m 8 m 7 m 4 m 5 m 1 m 2 65 m 3

23 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 23 of 35 PROLEMS: 11. Solve for x in the diagram (X+5) 128 H E F 12. Solve for x in the diagram 84 (3x 6) H E F Use the diagram to decide whether the statement is true or false. 13. If m 1 47, thenm If m 1 47, thenm m 1 m 3 m 2 m m 1 m 4 m 2 m 3. Find the value of the variable. 1 (13x 9) 2(3y 25) (4y 2) (15 x 1)

24 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 24 of Find the value of x, y, and z (the horizontal lines are parallel to each other) Z x= X Y y= z= Find the value of x, y, and z (the horizontal lines are parallel to each other) x= y= X Z 135 z= 70 Y 19. Find the value of x, y, and z (the horizontal lines are parallel to each other) x= y= X Z z= 66 Y 20. Find the value of x, y, and z (the horizontal lines are parallel to each other) x= y= z= Z Y X 75

25 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 25 of 35 Lesson 1.4 Triangle Proportionality Look at the shape (D is parallel to E). D E 1. Identify the triangles E and D. 2. Measure the segments and perimeters (use millimeters) and record in the table below. Side 1 Side 2 Side 3 Perimeter D D D Perim D E E E Perim E Ratio D E D E Perim Perim 3. Explain why (illustrated in figure 2) is a transversal (look at the beginning of packet for definition). D E 4. Explain why segments and are called corresponding segments. 5. In view of the last row of results in the table, what appears to be true about the ratio of lengths defined by two transversals intersecting parallel lines?

26 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 26 of 35 ased on the information you gathered on the previous page compute the correct length of the variables for the shapes below. ssume that lines which look parallel are indeed parallel. 6. = = 7. X= Y= 8. L= M= N=

27 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 27 of 35 ased on the results of the previous pages we can deduce and formulate the following theorem: Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. ssume that EF and are parallel. Then the following holds true: E F E F PROLEMS 9. What is the length of NR? 10. What is the length of DF? 11. re MN and PQ parallel to each other?

28 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 28 of 35 Lesson 1.5 Geometric onstructions ommon ore State Standards M9-12.G.O.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). opying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Materials compass and straightedge Mira or reflective mirror graph paper patty paper or tracing paper (optional) The study of Geometry was born in ncient Greece, where mathematics was thought to be embedded in everything from music to art to the governing of the universe. Plato, an ancient philosopher and teacher, had the statement, Let no man ignorant of geometry enter here, placed at the entrance of his school. This illustrates the importance of the study of shapes and logic during that era. Everyone who learned geometry was challenged to construct geometric objects using two simple tools, known as Euclidean tools: straight edge without any markings compass 1. Your First hallenge: an you copy a line segment? Step 1 onstruct a circle with a compass on the sheet below. Step 2 Mark the center of the circle and label it point. Step 3 Mark a point on the circle and label it point. Step 4 Draw.

29 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 29 of Your Second hallenge: an you copy any line segment? elow is a line segment. Using only an unmarked straight edge and compass, can you construct another line segment the same length beginning at point? Write instructions that explain the steps you used to complete the construction. (Hint: n ancient geometer would require you to cut off from the greater of two lines a line segment equal to a given segment.) 3. Your Third hallenge: an you copy an angle? Now that you know how to copy a segment, copying an angle is easy. How would you construct a copy of an angle at a new point? Think about what congruent triangles are imbedded in your construction and use them to justify why your construction works. e prepared to share your ideas with the class. D 4. Your Fourth hallenge: an you bisect a segment? 1. Draw a segment 2. Place the compass at point. djust the compass radius so that it is more than (½). Draw an arc as shown. 3. Without changing the compass radius, place the compass on point. Draw an arc intersecting the previously drawn arc. Label the midpoint.

30 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 30 of onstruct the perpendicular bisector of the segments. Mark congruent segments and right angles. heck your work with a protractor. a. b. c. d. e.

31 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 31 of isecting an angle 1. Let point P be the vertex of the angle. Place the compass on point P and draw an arc across both sides of the angle. Label the intersection points Q and R. 2. Place the compass on point Q and draw an arc across the interior of the angle. 3. Without changing the radius of the compass, place it on point R and draw an arc intersecting the one drawn in the previous step. Label the intersection point W. 4. Using the straightedge, draw ray PW. This is the bisector of QPR. Q W P R 7. a. b. c.

32 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 32 of 35 onstructing Parallel and Perpendicular Lines Example: onstruct a line that is perpendicular to line m and passes through point T. m T 1. Draw an arc around T 2. Use the intersections of the arc around T and the line D to draw arcs which intersect above and below T. 3. onnect the intersections of the arcs to draw a line which goes through T and is perpendicular to line D. PROLEMS: onstruct a line that is perpendicular to line m and passes through point X. 1. m X 2. m X 3. m X

33 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 33 of 35 Example: onstruct a line parallel to the given line m and through the given point. q m 1. Draw an arc from so that it passes through line m. 2. Use the intersection points and draw a line, m, between the points. The line should also hit point. 3. Draw a circle around. 4. Use the intersection points of the circle and the line, m, to draw arcs. 5. onnect the intersection points of the arcs. This line (q) should be parallel to line m and go through point. PROLEMS: onstruct a line parallel to the given line m and through the given point. 4. m 5. m

34 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 34 of 35 onstructing a Hexagon 1. Mark a point anywhere on the circle. Label this point P. This will be the first vertex of the hexagon. 2. Set the compass on point P and set the width of the compass to the center of the circle O. The compass is now set to the radius of the circle OP. 3. Make an arc across the circle. This will be the next vertex of the hexagon. all this point Q. (It turns out that the side length of a hexagon is equal to its circumradius - the distance from the center to a vertex). 4. Move the compass on to the next vertex Q and draw another arc. This is the third vertex of the hexagon. all this point R. 5. ontinue in this way until you have all six vertices. PQRSTU 6. Draw a line between each successive pairs of vertices, for a total of six lines.

35 GPS UNIT 1 Semester 1 NLYTI GEOMETRY Page 35 of onstruct the largest regular hexagon that will fit in the circle below based on the steps of the previous page. 7. How would you construct an equilateral triangle (all sides are congruent) inscribed in the given circle below?

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