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CCGPS UNIT 1A – Semester 1 ANALYTIC GEOMETRY Page 1 of 35 Similarity Congruence and Proofs Name: _________________ Date: ________________ Understand similarity in terms of similarity transformations MCC9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. 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. MCC9-12.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. MCC9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Prove theorems involving similarity MCC9-12.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. MCC9-12.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 MCC9-12.G.CO.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. MCC9-12.G.CO.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. MCC9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Prove geometric theorems MCC9-12.G.CO.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. MCC9-12.G.CO.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. MCC9-12.G.CO.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 MCC9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying 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. MCC9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Lesson 1.0 Definitions Adjacent Angles: Angles in the same plane that have a common vertex and a common side, but no common interior points. Alternate Exterior Angles: Alternate 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. Alternate Interior Angles: Alternate 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. Angle: Angles are created by two distinct rays that share a common endpoint (also known as a vertex). ABC or B denote angles with vertex B. Bisector: A bisector divides a segment or angle into two equal parts. Centroid: The point of concurrency of the medians of a triangle. Circumcenter: The point of concurrency of the perpendicular bisectors of the sides of a triangle. Coincidental: Two equivalent linear equations overlap when graphed. Complementary Angles: Two angles whose sum is 90 degrees. Congruent: Having the same size, shape and measure. Two figures are congruent if all of their corresponding measures are equal. Congruent Figures: Figures that have the same size and shape. Corresponding Angles: Angles that have the same relative positions in geometric figures. Corresponding Sides: Sides that have the same relative positions in geometric figures Dilation: Transformation that changes the size of a figure, but not the shape.

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  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 1 of 35

    Similarity Congruence and Proofs

    Name: _________________

    Date: ________________

    Understand similarity in terms of similarity transformations MCC9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. 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. MCC9-12.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.

    MCC9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

    Prove theorems involving similarity MCC9-12.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. MCC9-12.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 MCC9-12.G.CO.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. MCC9-12.G.CO.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.

    MCC9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

    Prove geometric theorems MCC9-12.G.CO.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 segments endpoints.

    MCC9-12.G.CO.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. MCC9-12.G.CO.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 MCC9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding,

    dynamic geometric software, etc.). Copying 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. MCC9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

    Lesson 1.0 Definitions

    Adjacent Angles: Angles in the same plane that have a common vertex and a common side, but no common interior points.

    Alternate Exterior Angles: Alternate 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.

    Alternate Interior Angles: Alternate 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. Angle: Angles are created by two distinct rays that share a common endpoint (also known as a vertex). ABC or B denote angles with vertex B.

    Bisector: A bisector divides a segment or angle into two equal parts.

    Centroid: The point of concurrency of the medians of a triangle.

    Circumcenter: The point of concurrency of the perpendicular bisectors of the sides of a triangle.

    Coincidental: Two equivalent linear equations overlap when graphed.

    Complementary Angles: Two angles whose sum is 90 degrees.

    Congruent: Having the same size, shape and measure. Two figures are congruent if all of their corresponding measures are equal.

    Congruent Figures: Figures that have the same size and shape.

    Corresponding Angles: Angles that have the same relative positions in geometric figures.

    Corresponding Sides: Sides that have the same relative positions in geometric figures

    Dilation: Transformation that changes the size of a figure, but not the shape.

  • CCGPS UNIT 1A Semester 1 ANALYTIC 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 Angle 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. AB denotes a line that passes through point A and B.

    Line Segment or Segment: The part of a line between two points on the line. AB denotes a line segment between the points A and B.

    Linear Pair: Adjacent, supplementary angles. Excluding their common side, a linear pair forms a straight line.

    Measure of each Interior Angle 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: An equation which states that two ratios are equal.

    Ratio: Comparison of two quantities by division and may be written as r/s, r:s, or r to s.

    Ray: A ray begins at a point and goes on forever in one direction.

    Reflection: A transformation that "flips" a figure over a line of reflection

    Reflection Line: A 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: A polygon that is both equilateral and equiangular.

    Remote Interior Angles of a Triangle: the two angles non-adjacent to the exterior angle.

    Rotation: A transformation that turns a figure about a fixed point through a given angle and a given direction.

    Same-Side Interior Angles: 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 Angles: 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 Angles of a Convex Polygon: 180(n 2).

    Supplementary Angles: 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: A transformation that "slides" each point of a figure the same distance in the same direction

    Transversal: A line that crosses two or more lines.

    Vertical Angles: Two nonadjacent angles formed by intersecting lines or segments. Also called opposite angles.

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 3 of 35

    Lesson 1.1 Dilations

    A dilation is a transformation that produces an image that is the same shape as the original, but is a different

    size. A 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.

    A 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

    ABC with the center of dilation at the origin ABCDE with the center of dilation at the origin and

    And a scale factor of 2. a scale factor of 1/3.

  • CCGPS UNIT 1A Semester 1 ANALYTIC 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.

    PROBLEMS 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

    Before Dilation: _____ _____ _____ _____ _____ _____ _____

    After Dilation: _____ _____ _____ _____ _____ _____ _____

    3. Dilate by a factor of 2 4. Dilate by a factor of 3

    Before Dilation: _____ _____ _____ _____ _____ _____ _____

    After Dilation: _____ _____ _____ _____ _____ _____ _____

    5. Dilate by a factor of 6. Dilate by a factor of

    Before Dilation: _____ _____ _____ _____ _____ _____ _____

    After Dilation: _____ _____ _____ _____ _____ _____ _____

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 5 of 35

    7. Dilate by a factor of 8. Dilate by a factor of 4

    Before Dilation: _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____

    After Dilation: _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____

    9. Dilate by a factor of 2 10. Dilate by a factor of

    Before Dilation: _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____

    After Dilation: _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____

    11. Dilate by a factor of 12. Dilate by a factor of 2

    Before Dilation: _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____

    After Dilation: _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 6 of 35

    When the center of dilation is not the origin you have to proceed as outlines below.

    PROBLEM: Draw the dilation image of rectangle EFGH with the center of dilation at point E and a scale factor of 1/2. OBSERVE: 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: Be sure to measure distances for this problem.

    PROBLEMS:

    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 , dilation center (2,4).

    11. Dilate by a factor of , dilation center (10,0) 12. Dilate by a factor of , dilation center (2,6)

  • CCGPS UNIT 1A Semester 1 ANALYTIC 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. Connect the points with line segments as follows:

    For Set 1, connect the points in order. Connect the last point in the set to the first point in

    the set.

    For Set 2, connect the points in order. Connect 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 2

    10

    10

    10

    10

    20

    20

  • CCGPS UNIT 1A Semester 1 ANALYTIC 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 4 20

    20

    10

    10

    10

    10

  • CCGPS UNIT 1A Semester 1 ANALYTIC 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)

    After 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.

    Figure 5 Figure 6

    10 10

    10 10

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 10 of 35

    4. 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 ?

    6. The sketch below shows two triangles, ABC and EFG. ABC has an area of 12 square units, and its base

    (AB) is equal to 8 units. The base of EFG is equal to 24 units. G

    C

    30 70 30 70

    A 8 B 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 EFG? Explain your reasoning.

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 11 of 35

    d. What is the relationship between the area of ABC and the area of 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 Jeannies measurements:

    Length of shadow cast by goal post and backboard: 5 ft. 9 in.

    Length of yardsticks shadow: 1 ft. 6 in.

    a. Draw and label a picture to illustrate Jeannies 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 Jeannies house compare to the one in the gym? Justify your answer.

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 12 of 35

    How to tell if triangles are similar

    Any triangle is defined by six measures (three sides, three angles). But 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. AAA (angle angle angle)

    All three pairs of corresponding angles are the same.

    2. SSS in same proportion (side side side)

    All three pairs of corresponding sides are in the same proportion.

    Since corresponding sides have the same ratios, 15 12 6

    10 8 4 , both triangles are similar.

    3. SAS (side angle side) Two pairs of sides in the same proportion and the included angle equal.

    The ratios of 16 20

    1.312 15

    are the same. Since the angles between intersecting lines are congruent (vertical angles

    theorem), we have side-angle-side similarity.

    70

    80

    40

    40 80

    70

    = 16 = 20

    = 12 = 15

    A 4

    8 10

    8

    12 15

    D F

    E

    B

    C

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 13 of 35

    As 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 SAS similarity (angle congruent and sides proportional).

    You can also show similarity by using the Angle-Angle Similarity Postulate ( AA~ ). If two angles of one

    triangle are congruent to two angles of another triangle then the triangles are similar.

    Examples of AA~

    PROBLEMS

    For problems 8 13. Are the triangles similar? If so state why and which method you used.

    8.

    The GHJ ~ GMK are similar by AA~ because

    1) H and M are congruent by Corresponding Angles Postulate. 2) G and G are congruent since they are the same angle.

    The ABC ~ XZY are similar by AA~ because

    1) They are both right triangles; therefore they both have a 90

    degree angle.

    2) All triangles add up to 180 degrees, since angle C is 40 degrees

    in ABC angle A will be 50 degrees. Therefore, A and X are congruent.

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 14 of 35

    9.

    10.

    11.

    12.

    E

    D

    10

    12

    18

    5

    6

    9

    A

    B C

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 15 of 35

    13.

    14.

    15.

    16.

    6

    3 3.5

    7

    A B

    C D

    E

    50 130

    A

    E

    D

    C

    B

    B

    A

    12

    9

    8 B

    B

    B

    D

    A

    C

    20

    16 24

    36 B

    30

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 16 of 35

    17. Calculate the distance across the lake by using the similar triangle theorem.

    18. Both triangles are similar. What value is x?

    19.

    44

    31 50

    DISTANCE

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 17 of 35

    Lesson 1.2 Pythagorean Theorem using Triangle Similarity

    The figure below shows the right triangle ABC. Its right angle is angle C.

    1. Draw a perpendicular line from C to AB. Label that point D. Use a straight edge.

    2. How many triangles do you see in the figure above?

    3. Why are the triangles ABC and BCD similar to each other? (Hint: Measure the angles with a protractor)

    4. Why are the triangles ABC and ACD similar to each other? (Hint: Measure the angles with a protractor)

    5. Why are the triangles BCD and ACD similar to each other? (Hint: Measure the angles with a protractor)

    6. What is the ratio of triangle BCD to ACD? (Measure and compare corresponding sides)

    7. What is the ratio of triangle ACD to ABC? (Measure and compare corresponding sides)

    8. What is the ratio of triangle BCD to ABC? (Measure and compare corresponding sides)

    C

    B A

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 18 of 35

    Example Find the missing length indicated

    Solution: First solve for AB = 100 36 = 64.

    Knowing that triangle ADB is similar to triangle BCD, we can validate the following

    ratios of corresponding sides:

    AD x

    x DB Ratios of corresponding sides of similar triangles are the same

    64

    36

    x

    x Plug in the values

    2 64 36x Cross-multiply

    48x Solve for x

    PROBLEMS

    Find the missing length indicated

    1.

    2.

    X

    36 100

    A B

    C

    D

    X

    9

    25

    C

    D

    A B

    X

    9

    25

    C

    D

    A B

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 19 of 35

    3.

    4.

    5.

    6.

    X

    45

    81

    C

    D

    A B

    48 X

    64

    C

    D

    A B

    X 9

    7

    C

    D

    A B

    12 16

    X

    C

    D

    A B

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 20 of 35

    Lesson 1.3 Angle Geometry

    Two angles are vertical angles if their sides form two pairs of opposite rays.

    How do you know that vertical angles are congruent?

    m1 + m3 = 180 because the Linear Pair postulate m2 + m3 = 180 because the Linear Pair postulate Set the two equations equal to each other since they both equal 180 degrees. m2 + m3 = m1 + m3

    - m3 = - m3 Subtract m3from both sides m2 = m1

    Therefore:2 1

    PROBLEMS

    1. Prove that 3 4 using a similar method.

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 21 of 35

    Transversal Lines

    When a transversal crosses parallel lines, there are several pairs of special angles.

    Corresponding Angle 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?

    By using the Alternate Interior Angle 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?

    SameSide Interior Angle Theorem: If two parallel lines are cut by a transversal, then sameside interior angles are supplementary.

    Example 1 Complete the statement given that 90m AGF .

    a. ?CGD A B C

    b. 113 , ?if m BGF then m DGE G

    F E D

    Solution:

    a. Because CGD and AGF are vertical angles, CGD = AGF . By definition of

    congruent angles, .m CGD m AGF So, 90 .CGD

    b. By the angle addition postulate .if m BGF m AGF m AGB Substitute to get

    113 90 .m AGB Therefore 23 .m AGB Because DGE and AGB are

    vertical angles, .DGE AGB By definition of congruent angles,

    .m DGE m AGB Therefore 23 .m DGE

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 22 of 35

    PROBLEMS

    Copy and complete the statement given that 90 .m BHD m CHE

    1. _____m AHG B C D

    2. _____m CHA

    3. 31 , _____If m CHD thenm DHE A H E

    4. 48 , _____If m BHA thenm EHF G F

    5. 38 , _____If m GHF thenm AHB

    If 90 26m BGD and m CGD , find

    6. 1m B C

    7. 2m

    8. 3m A G D

    9. m FGE F E

    10. m DGE

    Lines L and M are parallel, find

    11. 1m

    12. 2m

    13. 3m

    14. 4m

    15. 5m

    16. 6m

    17. 7m

    18. 8m

    65 1m

    2m

    8m 7m

    5m

    3m

    4m

    1m

    3m

    2m

    16

    L

    M

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 23 of 35

    PROBLEMS:

    11. Solve for x in the diagram

    B

    A H E

    F

    12. Solve for x in the diagram

    B

    A H E

    F

    Use the diagram to decide whether the statement is true or false.

    13. 1 47 , 2 43 .If m thenm

    14. 1 47 , 3 47 .If m thenm 1 1 2 15. 1 3 2 4.m m m m 3 4

    16. 1 4 2 3.m m m m

    Find the value of the variable.

    1 (13 9)x 2(3 25)y (4 2)y (15 1)x

    128

    (3 6)x

    (X+5)

    84

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 24 of 35

    17. Find the value of x, y, and z (the horizontal lines are parallel to each other)

    x=_______

    y= ______

    z= ______

    18. Find the value of x, y, and z (the horizontal lines are parallel to each other)

    x=_______

    y= ______

    z= ______

    19. Find the value of x, y, and z (the horizontal lines are parallel to each other)

    x=_______

    y= ______

    z= ______

    20. Find the value of x, y, and z (the horizontal lines are parallel to each other)

    x=_______

    y= ______

    z= ______

    Y

    116

    Z X

    Y 70

    X Z

    135

    Y 66

    X

    Z

    Y

    75

    X Z

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 25 of 35

    Lesson 1.4 Triangle Proportionality

    Look at the shape (BD is parallel to AE).

    1. Identify the triangles CAE and CBD.

    2. Measure the segments and perimeters (use millimeters) and record in the table below.

    Side 1 Side 2 Side 3 Perimeter

    CBD CB CD BD CBDPerim

    CAE CA AE AE CAEPerim

    Ratio CB

    CA

    BD

    AE

    BD

    AE CBD

    CAE

    Perim

    Perim

    3. Explain why AB (illustrated in figure 2) is a transversal (look at the beginning of packet for definition).

    4. Explain why segments CB and CA 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?

    A

    B

    E

    D

    C

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 26 of 35

    Based on the information you gathered on the previous page compute the correct length of the variables for the shapes below. Assume that lines which look parallel are indeed parallel.

    6.

    7.

    8.

    B= _________

    A=__________

    Y= _______

    X= _______

    L=_______

    M= ______

    N= ______

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 27 of 35

    Based 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.

    PROBLEMS

    9. What is the length of NR?

    10. What is the length of DF?

    11. Are MN and PQ parallel to each other?

    Assume that EF and BC are parallel. Then the following

    holds true:

    1 2 3 4

    AE AF

    EB FC

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 28 of 35

    Lesson 1.5 Geometric Constructions

    Common Core State Standards

    MCC9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and

    straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying 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 Ancient 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:

    A straight edge without any markings

    A compass

    1. Your First Challenge: Can you copy a line segment? Step 1 Construct a circle with a compass on the sheet below.

    Step 2 Mark the center of the circle and label it point A.

    Step 3 Mark a point on the circle and label it point B.

    Step 4 Draw .AB

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 29 of 35

    2. Your Second Challenge: Can you copy any line segment?

    Below is a line segment AB . Using only an unmarked straight edge and compass, can you construct another line

    segment the same length beginning at point C? Write instructions that explain the steps you used to complete

    the construction. (Hint: An ancient geometer would require you to cut off from the greater of two lines a line

    segment equal to a given segment.)

    A B

    C

    3. Your Third Challenge: Can 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. Be prepared to share your ideas with the class.

    A

    D

    4. Your Fourth Challenge: Can you bisect a segment? 1. Draw a segment AB 2. Place the compass at point A. Adjust the compass radius so that it is more than ()AB. Draw an arc

    as shown. 3. Without changing the compass radius, place the compass on point B. Draw an arc intersecting the

    previously drawn arc. Label the midpoint C.

    A B

    C

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 30 of 35

    5. Construct the perpendicular bisector of the segments. Mark congruent segments and right angles. Check your work with a protractor.

    a. A B

    b. A B

    c. A B

    d. A

    B

    e. B

    A

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 31 of 35

    6. Bisecting 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 .

    7. a.

    b.

    c.

    P R

    Q W

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 32 of 35

    Constructing Parallel and Perpendicular Lines Example:

    Construct 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 CD to draw arcs which intersect above and below T. 3. Connect the intersections of the arcs to draw a line which goes through T and is perpendicular to line CD. PROBLEMS:

    Construct a line that is perpendicular to line m and passes through point .X 1. m X 2. m X 3. m X

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 33 of 35

    Example: Construct a line parallel to the given line m and through the given point C. C q m 1. Draw an arc from C 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 C. 3. Draw a circle around C. 4. Use the intersection points of the circle and the line, m, to draw arcs. 5. Connect the intersection points of the arcs. This line (q) should be parallel to line m and go through point C. PROBLEMS: Construct a line parallel to the given line m and through the given point C. 4. C m 5. C m

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 34 of 35

    Constructing 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. Call 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. Call

    this point R.

    5. Continue 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.

  • CCGPS UNIT 1A Semester 1 ANALYTIC GEOMETRY Page 35 of 35

    6. Construct 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?