MFAS Geometry CPALMS Review Packet. Congruency Similarity and Right Triangles

Transcription

1 MFAS Geometry CPALMS Review Packet Congruency Similarity and Right Triangles

2 Table of Contents MAFS.912.G-CO Definition of an Angle... 6 Definition of Perpendicular Lines... 6 Definition of Parallel Lines... 6 Definition of Line Segment... 7 Definition of a Circle... 7 MAFS.912.G-CO Demonstrating Rotations... 8 Demonstrating Reflections... 9 Transformations And Functions... 9 Comparing Transformations Demonstrating Translations MAFS.912.G-CO Define a Rotation Define a Reflection Define a Translation MAFS.912.G-CO Two Triangles Reflect a Semicircle Indicate the Transformations Rotation of a Quadrilateral MAFS.912.G-CO Transformations of Trapezoids Transformations of Regular Polygons Transformations of Rectangles and Squares Transformations of Parallelograms and Rhombi MAFS.912.G-CO Repeated Reflections and Rotations Transform This Congruent Trapezoids MAFS.912.G-CO Congruence Implies Congruent Corresponding Parts Showing Congruence Using Corresponding Parts Showing Congruence Using Corresponding Parts Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 2

3 Proving Congruence Using Corresponding Parts Showing Triangles Congruent Using Rigid Motion MAFS.912.G-CO Justifying SSS Congruence Justifying SAS Congruence Justifying ASA Congruence MAFS.912.G-CO Proving the Vertical Angles Theorem Proving Alternate Interior Angles Congruent Equidistant Points MAFS.912.G-CO Isosceles Triangle Proof Triangle Midsegment Proof Triangle Sum Proof Median Concurrence Proof MAFS.912.G-CO Proving Parallelogram Side Congruence Proving Parallelogram Diagonals Bisect Proving a Rectangle Is a Parallelogram Proving Congruent Diagonals MAFS.912.G-CO Constructing a Congruent Segment Constructions for Parallel Lines Constructions for Perpendicular Lines Constructing a Congruent Angle MAFS.912.G-CO Construct the Center of a Circle Regular Hexagon in a Circle Equilateral Triangle in a Circle Square in a Circle MAFS.912.G-SRT Dilation of a Line: Center on the Line Dilation of a Line: Factor of Two Dilation of a Line: Factor of One Half Dilation of a Line Segment Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 3

4 MAFS.912.G-SRT To Be or Not To Be Similar Showing Similarity The Consequences of Similarity MAFS.912.G-SRT Describe the AA Similarity Theorem Justifying a Proof of the AA Similarity Theorem Prove the AA Similarity Theorem MAFS.912.G-SRT Triangle Proportionality Theorem Converse of the Triangle Proportionality Theorem Pythagorean Theorem Proof Geometric Mean Proof MAFS.912.G-SRT Basketball Goal County Fair Similar Triangles Prove Rhombus Diagonals Bisect Angles Similar Triangles MAFS.912.G-SRT Will It Fit? TV Size River Width Washington Monument Holiday Lights Step Up Perilous Plunge Lighthouse Keeper MAFS.912.G-SRT The Sine of The Cosine Ratio MAFS.912.G-SRT Patterns in the Table Finding Sine Right Triangle Relationships Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 4

5 Sine and Cosine Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 5

6 MAFS.912.G-CO.1.1 Level 2 Level 3 Level 4 Level 5 uses definitions to choose examples and non-examples uses precise definitions that are based on the undefined notions of point, line, distance along a line, and distance around a circular arc analyzes possible definitions to determine mathematical accuracy explains whether a possible definition is valid by using precise definitions Definition of an Angle 1. Draw and label ABC. 2. Define the term angle as clearly and precisely as you can. Definition of Perpendicular Lines 1. Draw and label a pair of perpendicular lines. 2. Define perpendicular lines as clearly and precisely as you can. Definition of Parallel Lines 1. Draw a pair of parallel lines. 2. Define parallel lines as clearly and precisely as you can. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 6

7 Definition of Line Segment MFAS Geometry EOC Review 1. Draw and label AAAA. Clearly indicate what part of your drawing is the line segment. 2. Define the term line segment as clearly and precisely as you can. Definition of a Circle 1. Draw and label a circle. 2. Define the term circle as clearly and precisely as you can. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 7

8 MAFS.912.G-CO.1.2 Level 2 Level 3 Level 4 Level 5 uses definitions to choose examples and non-examples uses precise definitions that are based on the undefined notions of point, line, distance along a line, and distance around a circular arc analyzes possible definitions to determine mathematical accuracy explains whether a possible definition is valid by using precise definitions Demonstrating Rotations Trace the figure onto a transparency or tracing paper. 1. Use the original and the traced version to demonstrate how to rotate quadrilateral EEEEEEEE about point A 90 clockwise. Explain how you rotated the figure. 2. Draw and label the rotated image as EE FF GG HH on the grid below. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 8

9 Demonstrating Reflections 1. Trace the figure onto a transparency or tracing paper. 2. Use the original and the traced version to demonstrate how to reflect quadrilateral EEEEEEEE across line m. 3. Draw and label the reflected image as EE FF GG HH on the grid below. Transformations And Functions Three transformations of points in the plane are described below. Consider each point in the plane as an input and its image under a transformation as its output. Determine whether or not each transformation is a function. Explain. 1. Transformation T translates each point in the plane three units to the left and four units up. 2. Transformation R reflects each point in the plane across the y-axis. 3. Transformation O rotates each point in the plane about the origin 90 clockwise. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 9

10 Comparing Transformations Determine whether or not each transformation, in general, preserves distance and angle measure. Explain. 1. Dilations 2. Reflections Demonstrating Translations 1. Trace the figure onto a transparency or tracing paper. 2. Use the original and the traced version to demonstrate how to translate quadrilateral EEEEEEEE according to vector v shown below. 3. Draw and label the translated image as EE FF GG HH on the grid below. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 10

11 MAFS.912.G-CO.1.4 Level 2 Level 3 Level 4 Level 5 represents transformations in the plane; determines transformations that preserve distance and angle to those that do not uses transformations to develop definitions of angles, perpendicular lines, parallel lines; describes translations as functions uses transformations to develop definitions of circles and line segments; describes rotations and reflections as functions [intentionally left blank] Define a Rotation A. C. 1. Rotate point A 90 clockwise around point C. Then describe the sequence of steps you used to rotate this point. 2. Develop a definition of rotation in terms of any of the following: angles, circles, perpendicular lines, parallel lines, and line segments. Write your definition so that it is general enough to use for a rotation of any degree measure, but make it detailed enough that it can be used to perform rotations. Define a Reflection 1. Reflect point C across. AAAA 2. Develop a definition of reflection in terms of any of the following: angles, circles, perpendicular lines, parallel lines, and line segments. Write your definition so that it is general enough to use for any reflection, but make it detailed enough that it can be used to perform reflections. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 11

12 Define a Translation 3. Translate point A according to. CCCC Then, describe the sequence of steps you used to translate this point. 4. Develop a definition of translation in terms of any of the following: angles, circles, perpendicular lines, parallel lines, and line segments. Write your definition so that it is general enough to use for any translation but make it detailed enough that it can be used to perform translations. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 12

13 MAFS.912.G-CO.1.5 Level 2 Level 3 Level 4 Level 5 chooses a sequence of two transformations that will carry a given figure onto itself or onto another figure uses transformations that will carry a given figure onto itself or onto another figure uses algebraic descriptions to describe rotations and/or reflections that will carry a figure onto itself or onto another figure applies transformations that will carry a figure onto another figure or onto itself, given coordinates of the geometric figure in the stem Two Triangles 1. Clearly describe a sequence of transformations that will map ABC to DFE. You may assume that all vertices are located at the intersections of grid lines. Reflect a Semicircle 1. Draw the image of the semicircle after a reflection across line l. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 13

14 Indicate the Transformations MFAS Geometry EOC Review 1. Clearly describe a sequence of transformations that will map ABC to DEF. You may assume that all vertices are located at the intersections of grid lines. Rotation of a Quadrilateral 1. Draw the image of quadrilateral BCDE after a 90 clockwise rotation about point A. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 14

15 MAFS.912.G-CO.1.3 Level 2 Level 3 Level 4 Level 5 chooses a sequence of two transformations that will carry a given figure onto itself or onto another figure uses transformations that will carry a given figure onto itself or onto another figure uses algebraic descriptions to describe rotations and/or reflections that will carry a figure onto itself or onto another figure applies transformations that will carry a figure onto another figure or onto itself, given coordinates of the geometric figure in the stem Transformations of Trapezoids Use the trapezoid at the right to answer the following questions. 1. Describe the rotation(s) that carry the trapezoid onto itself. 2. Describe the reflection(s) that carry the trapezoid onto itself. Draw any line(s) of reflection on the trapezoid. Use the isosceles trapezoid at the right to answer the following questions. 3. Describe the rotation(s) that carry the isosceles trapezoid onto itself. 4. Describe the reflection(s) that carry the isosceles trapezoid onto itself. Draw any line(s) of reflection on the trapezoid. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 15

16 Transformations of Regular Polygons Use the regular hexagon at the right to answer the following questions. 5. Describe the rotation(s) that carry the regular hexagon onto itself. 6. Describe the reflection(s) that carry the regular hexagon onto itself. Use the regular pentagon at the right to answer the following questions. 7. Describe the rotation(s) that carry the regular pentagon onto itself. 8. Describe the reflection(s) that carry the regular pentagon onto itself. 9. Based on your responses to Questions 1-4, how would you describe the rotations and reflections that carry a regular n-gon onto itself? Transformations of Rectangles and Squares Use the rectangle to answer the following questions. 1. Describe the rotation(s) that carry the rectangle onto itself. 2. Describe the reflection(s) that carry the rectangle onto itself. Draw the line(s) of reflection on the rectangle. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 16

17 Use the square at the right to answer the following questions. 3. Describe the rotation(s) that carry the square onto itself. 4. Describe the reflection(s) that carry the square onto itself. Draw the line(s) of reflection on the square. Transformations of Parallelograms and Rhombi Use the parallelogram to answer the following questions. 1. Describe the rotation(s) that carry the parallelogram onto itself. 2. Describe the reflection(s) that carry the parallelogram onto itself. Draw the line(s) of reflection on the parallelogram. Use the rhombus at the right to answer the following questions. 3. Describe the rotation(s) that carry the rhombus onto itself. 4. Describe the reflection(s) that carry the rhombus onto itself. Draw the line(s) of reflection on the rhombus. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 17

18 MAFS.912.G-CO.2.6 Level 2 Level 3 Level 4 Level 5 determines if a sequence of transformations will result in congruent figures [intentionally left blank] Repeated Reflections and Rotations uses the definition of congruence in terms of rigid motions to determine if two figures are congruent; uses rigid motions to transform figures explains that two figures are congruent using the definition of congruence based on rigid motions 1. Describe what happens to DEF after it is reflected across line m two times in succession, then rotated 90 around point C four times in succession. Explain. Transform This 1. Sketch the triangle formed when AAAAAA is translated using the rule, (x, y) (x 6, y + 2). Name the image DDDDDD. Is AAAAAA DDDDDD? Explain. 2. Sketch the image of AAAAAA after a 90 clockwise rotation around the origin. Name the image GHI. Is AAAAAA GGGGGG? Explain. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 18

19 Congruent Trapezoids MFAS Geometry EOC Review Use the definition of congruence in terms of rigid motion to determine whether or not the two trapezoids are congruent. Clearly justify your decision. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 19

20 MAFS.912.G-CO.2.7 Level 2 Level 3 Level 4 Level 5 identifies corresponding parts of two congruent triangles shows that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent using the definition of congruence in terms of rigid motions; applies congruence to solve problems; uses rigid motions to show ASA, SAS, SSS, or HL is true for two triangles shows and explains, using the definition of congruence in terms of rigid motions, the congruence of two triangles; uses algebraic descriptions to describe rigid motion that will show ASA, SAS, SSS, or HL is true for two triangles justifies steps of a proof given algebraic descriptions of triangles, using the definition of congruence in terms of rigid motions that the triangles are congruent using ASA, SAS, SSS, or HL Congruence Implies Congruent Corresponding Parts AAAAAA DDDDDD. The lengths of the sides and the measures of the angles of AAAAAA are shown in the diagram. 1. Determine the lengths of the sides and the measures of the angles of DDDDDD. 2. Use the definition of congruence in terms of rigid motion to justify your reasoning. 3. Explain clearly how this reasoning can be applied to any two congruent triangles. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 20

21 Showing Congruence Using Corresponding Parts 1 MFAS Geometry EOC Review Given: AAAA, DDDD AAAA, DDDD BBBB, EEEE AA DD, BB EE, and AAAAAA DDDDDD. 1. Use the definition of congruence in terms of rigid motion to show that AAAAAA DDDDDD. Showing Congruence Using Corresponding Parts 2 The lengths of the sides and the measures of the angles of AAAAAA and DDDDDD are indicated below. 1. Use the definition of congruence in terms of rigid motion to show that AAAAAA DDDDDD. Proving Congruence Using Corresponding Parts Given: AAAAAA and DDDDDD in which AAAA, DDDD AAAA, DDDD BBBB, EEEE AA DD, BB EE, and CC FF. 1. Use the definition of congruence in terms of rigid motion to prove AAAAAA DDDDDD. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 21

22 Showing Triangles Congruent Using Rigid Motion MFAS Geometry EOC Review 1. Given AAAAAA with vertices AA ( 4, 3), BB (0, 0), CC (2, 3) and DDDDDD with vertices DD (3, 1), EE (6, 3), FF (3, 5), use the definition of congruence in terms of rigid motion to show that AAAAAA DDDDDD. Describe each rigid motion in terms of coordinates (x, y). Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 22

23 MAFS.912.G-CO.2.8 Level 2 Level 3 Level 4 Level 5 identifies corresponding parts of two congruent triangles shows that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent using the definition of congruence in terms of rigid motions; applies congruence to solve problems; uses rigid motions to show ASA, SAS, SSS, or HL is true for two triangles shows and explains, using the definition of congruence in terms of rigid motions, the congruence of two triangles; uses algebraic descriptions to describe rigid motion that will show ASA, SAS, SSS, or HL is true for two triangles justifies steps of a proof given algebraic descriptions of triangles, using the definition of congruence in terms of rigid motions that the triangles are congruent using ASA, SAS, SSS, or HL Justifying SSS Congruence In the diagram, AAAA, DDDD AAAA DDDD and BBBB. EEEE 1. Using rigid motion, explain in detail, why triangle ABC must be congruent to triangle DEF. Justifying SAS Congruence In the diagram, AAAA, DDDD BB EE and BBBB. EEEE 1. Using rigid motion, explain in detail, why triangle ABC must be congruent to triangle DEF. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 23

24 Justifying ASA Congruence In the diagram, AA DD, AAAA DDDD and BB EE. 1. Using rigid motion, explain in detail, why triangle ABC must be congruent to triangle DEF. MFAS Geometry EOC Review Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 24

25 MAFS.912.G-CO.3.9 Level 2 Level 3 Level 4 Level 5 uses theorems about parallel lines with one transversal to solve problems; uses the vertical angles theorem to solve problems completes no more than two steps of a proof using theorems about lines and angles; solves problems using parallel lines with two to three transversals; solves problems about angles using algebra completes a proof for vertical angles are congruent, alternate interior angles are congruent, and corresponding angles are congruent creates a proof, given statements and reasons, for points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints Proving the Vertical Angles Theorem 1. Identify a pair of vertical angles. 2. Prove the vertical angles you identified are congruent. Proving Alternate Interior Angles Congruent Transversal t intersects parallel lines a and b. 1. Identify a pair of alternate interior angles. 2. Prove that these alternate interior angles are congruent. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 25

26 Equidistant Points PPPP is the perpendicular bisector of AAAA. Prove that point P is equidistant from the endpoints of AAAA. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 26

27 MAFS.912.G-CO.3.10 Level 2 Level 3 Level 4 Level 5 uses theorems about interior angles of a triangle, exterior angle of a triangle completes no more than two steps in a proof using theorems (measures of interior angles of a triangle sum to 180,; 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) about triangles; solves problems about triangles using algebra; solves problems using the triangle inequality and the Hinge theorem completes a proof for theorems about triangles; solves problems by applying algebra using the triangle inequality and the Hinge theorem; solves problems for the midsegment of a triangle, concurrency of angle bisectors, and concurrency of perpendicular bisectors completes proofs using the medians of a triangle meet at a point; solves problems by applying algebra for the midsegment of a triangle, concurrency of angle bisectors, and concurrency of perpendicular bisectors Isosceles Triangle Proof The diagram below shows isosceles AAAAAA with AAAA. BBBB 1. In the space below prove that AA CC. Triangle Midsegment Proof The diagram below shows AAAAAA. Point D is the midpoint of AAAA and point E is the midpoint of. BBBB 1. In the space below prove that DDDD is parallel to AAAA and DDDD = 1 AAAA. 2 Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 27

28 Triangle Sum Proof The diagram below shows AAAAAA in which AAAA is parallel to line BBBB. 1. In the space below prove that the sum of the interior angles of AAAAAA is 180, that is, prove that mm 1 + mm 2 + mm 3 = 180. Median Concurrence Proof Given: AAAAAA, in which points D, E and F are the midpoints of sides AAAA, CCCC, and, AAAA respectively. 1. Draw the three medians of AAAAAA and prove that they intersect in a single point. Label this point of intersection, point G. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 28

29 MAFS.912.G-CO.3.11 Level 2 Level 3 Level 4 Level 5 uses properties of parallelograms to find numerical values of a missing side or angle or select a true statement about a parallelogram completes no more than two steps in a proof for opposite sides of a parallelogram are congruent and opposite angles of a parallelogram are congruent; uses theorems about parallelograms to solve problems using algebra creates proofs to show the diagonals of a parallelogram bisect each other, given statements and reasons proves that rectangles and rhombuses are parallelograms, given statements and reasons Proving Parallelogram Side Congruence 1. Prove that the opposite sides of parallelogram WXYZ are congruent. Proving Parallelogram Diagonals Bisect 1. Prove that the diagonals of parallelogram WXYZ bisect each other. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 29

30 Proving a Rectangle Is a Parallelogram 1. Prove that rectangle WXYZ is a parallelogram. MFAS Geometry EOC Review Proving Congruent Diagonals 1. Draw the diagonals of rectangle WXYZ and prove that they are congruent. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 30

31 MAFS.912.G-CO.4.12 Level 2 Level 3 Level 4 Level 5 chooses a visual or written step in a construction Constructing a Congruent Segment identifies, sequences, or reorders steps in a construction: 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 identifies sequences or reorders steps in a construction of an equilateral triangle, a square, and a regular hexagon inscribed in a circle explains steps in a construction 1. Using a compass and straight edge, construct (PQ) so that (PQ) (AB). Explain the steps of your construction. Constructions for Parallel Lines 1. Use a compass and a straightedge to construct line p so that line p contains point M and is parallel to line n. M n 2. Which definition, postulate, or theorem justifies your construction method and ensures that the line you constructed is parallel to line n? Explain. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 31

32 Constructions for Perpendicular Lines MFAS Geometry EOC Review 1. Use a compass and a straightedge to construct line p so that line p contains point M and is perpendicular to line n. M n 2. Use a compass and a straightedge to construct line q so that line q is perpendicular to line r at point S. r S Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 32

33 Constructing a Congruent Angle MFAS Geometry EOC Review Using a compass and straight edge, construct DDDDDD so that DDDDDD AAAAAA. Explain the steps of your construction. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 33

34 MAFS.912.G-CO.4.13 Level 2 Level 3 Level 4 Level 5 chooses a visual or written step in a construction identifies, sequences, or reorders steps in a construction: 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 identifies sequences or reorders steps in a construction of an equilateral triangle, a square, and a regular hexagon inscribed in a circle explains steps in a construction Construct the Center of a Circle 1. Using a compass and straightedge, construct the center of the circle. Leave all necessary construction marks as justification of your process. Regular Hexagon in a Circle 1. Using a compass and straightedge, construct a regular hexagon inscribed in the circle. Leave all necessary construction marks as justification of your process. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 34

35 Equilateral Triangle in a Circle MFAS Geometry EOC Review Using a compass and straightedge, construct an equilateral triangle inscribed in the circle. Leave all necessary construction marks as justification of your process. Square in a Circle Using a compass and straightedge, construct a square inscribed in the circle. Leave all necessary construction marks as justification of your process. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 35

36 MAFS.912.G-SRT.1.1 Level 2 Level 3 Level 4 Level 5 identifies the scale factors of dilations chooses the properties of dilations when a dilation is presented on a coordinate plane, as a set of ordered pairs, as a diagram, or as a narrative; properties are: 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; the dilation of a line segment is longer or shorter in the ratio given by the scale factor explains why a dilation takes a line not passing through the center of dilation to a parallel line and leaves a line passing through the center unchanged or that the dilation of a line segment is longer or shorter in ratio given by the scale factor explains whether a dilation presented on a coordinate plane, as a set of ordered pairs, as a diagram, or as a narrative correctly verifies the properties of dilations Dilation of a Line: Center on the Line In the figure, points A, B, and C are collinear. 1. Graph the images of points A, B, and C as a result of dilation with center at point C and scale factor of 1.5. Label the images of A, B, and C as AA, B, and C, respectively. 2. Describe the image of AAAA as a result of this dilation. In general, what is the relationship between a line and its image after dilating about a center on the line? Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 36

37 Dilation of a Line: Factor of Two In the figure, the points A, B, and C are collinear. MFAS Geometry EOC Review 1. Graph the images of points A, B, and C as a result of dilation with center at point D and scale factor equal to 2. Label the images of A, B, and C as AA, B, and C, respectively. 2. Describe the image of AAAA as a result of the same dilation. In general, what is the relationship between a line and its image after dilating about a center not on the line? Dilation of a Line: Factor of One Half In the figure, the points A, B, C are collinear. 1. Graph the images of points A, B, C as a result of a dilation with center at point D and scale factor equal to 0.5. Label the images of A, B, and C as AA, B, and C, respectively. 2. Describe the image of AAAA as a result of the same dilation. In general, what is the relationship between a line and its image after dilating about a center not on the line? Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 37

38 Dilation of a Line Segment MFAS Geometry EOC Review 1. Given, AAAA draw the image of AAAA as a result of the dilation with center at point C and scale factor equal to Describe the relationship between AAAA and its image. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 38

39 MAFS.912.G-SRT.1.2 Level 2 Level 3 Level 4 Level 5 determines if two given figures are similar uses the definition of similarity in terms of similarity transformations to decide if two figures are similar; determines if given information is sufficient to determine similarity shows that corresponding angles of two similar figures are congruent and that their corresponding sides are proportional explains using the definition of similarity in terms of similarity transformations that corresponding angles of two figures are congruent and that corresponding sides of two figures are proportional To Be or Not To Be Similar Use the definition of similarity in terms of similiarity transformations to determine whether or not AAAAAA~ DDDDDD. Justify your answer by describing the sequence of similiarity transformations you used. Showing Similarity Use the definition of similarity in terms of transformations to show that quadrilateral ABCD is similar to quadrilateral EFGH. Justify your answer by describing the sequence of similiarity transformations you used. Be sure to indicate the coordinates of the images of the vertices after each step of your transformation. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 39

40 The Consequences of Similarity MFAS Geometry EOC Review The definition of similarity in terms of similarity transformations states that two figures are similar if and only if there is a compositon of rigid motion and dilation that maps one figure to the other. Suppose AAAAAA~ DDDDDD. Explain how this definiton ensures: 1. The equality of all corresponding pairs of angles. 2. The proportionality of all corresponding pairs of sides. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 40

41 MAFS.912.G-SRT.1.3 Level 2 Level 3 Level 4 Level 5 identifies that two triangles are similar using the AA criterion establishes the AA criterion for two triangles to be similar by using the properties of similarity transformations proves that two triangles are similar if two angles of one triangle are congruent to two angles of the other triangle, using the properties of similarity transformations; uses triangle similarity to prove theorems about triangles proves the Pythagorean theorem using similarity Describe the AA Similarity Theorem Describe the AA Similarity Theorem. Include: 1. A statement of the theorem 2. The assumptions along with a diagram that illustrates the assumptions. 3. The conclusion. 4. The definition of similarity in terms of similarity transformations. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 41

42 Justifying a Proof of the AA Similarity Theorem Assume that AA AA and CCCCCC CC BB AA. MFAS Geometry EOC Review The following illustrates the statements of a proof of the AA Similarity Theorem (i.e., a proof of the statement that AAAAAA is similar to AA BB CC ). Explain and justify each numbered statement. Let BB be the point on AAAA so that AABB = AA BB. Denote the dilation with center A and scale factor r = AA BB (which is also AAAA equal to AABB ) by D, and let CC be the point on AAAA such that D(C) = CC. Explain why: AAAA 1. BB"CC" is parallel to. BBBB 2. AABB CC AAAAAA. 3. ΔAA BB CC AABB CC by congruence G. 4. AAAAAA~ AABB CC. 5. AAAAAA ~ AA BB CC. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 42

43 Prove the AA Similarity Theorem The lengths of the sides of AAAAAA and AA BB CC are given in the figure. 1. Describe the relationship between the lengths of the sides of the two triangles. 2. Prove that this relationship guarantees that the triangles are similar. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 43

44 MAFS.912.G-SRT.2.4 Level 2 Level 3 Level 4 Level 5 identifies that two triangles are similar using the AA criterion establishes the AA criterion for two triangles to be similar by using the properties of similarity transformations proves that two triangles are similar if two angles of one triangle are congruent to two angles of the other triangle, using the properties of similarity transformations; uses triangle similarity to prove theorems about triangles proves the Pythagorean theorem using similarity Triangle Proportionality Theorem Prove the Triangle Proportionality Theorem, that is, given AAAAAA and FFFF (as shown) such that FFFF BBBB, prove that AAAA AAAA GGGG. FFFF = Converse of the Triangle Proportionality Theorem In AAAAAA, suppose AAAA = AAAA FFFF GGGG. Prove thatffff BBBB. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 44

45 Pythagorean Theorem Proof MFAS Geometry EOC Review 1. Show that AAAAAA~ CCCCCC and AAAAAA~ AAAAAA. Then use these similarities to prove the Pythagorean Theorem (a 2 + b 2 = c 2 ). Geometric Mean Proof 1. Explain why AAAAAA~ CCCCCC. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 45

46 MAFS.912.G-SRT.2.5 Level 2 Level 3 Level 4 Level 5 finds measures of sides and angles of congruent and similar triangles when given a diagram solves problems involving triangles, using congruence and similarity criteria; provides justifications about relationships using congruence and similarity criteria completes proofs about relationships in geometric figures by using congruence and similarity criteria for triangles proves conjectures about congruence or similarity in geometric figures, using congruence and similarity criteria Basketball Goal The basketball coach is refurbishing the outdoor courts at his school and is wondering if the goals are at the regulation height. The regulation height is 10 feet, measured from the ground to the rim. One afternoon the gym teacher, who is 6 feet tall, measured his own shadow at 5 feet long. He measured the shadow of the basketball goal (to the rim) as 8 feet long. Use this information to determine if the basketball goal is at the regulation height. Show all of your work and explain your answer. County Fair The diagram below models the layout at the county fair. Suppose the two triangles in the diagram are similar. Refreshmen t 100 Petting Zoo Restrooms Park Entrance Ride 1. How far is the park entrance from the rides? Show/explain your work to justify your solution process. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 46

47 Similar Triangles 1 In the figure below, ABCD is a parallelogram. MFAS Geometry EOC Review 8 units 3 units 7 units 13 units 1. Identify a pair of similar triangles in the diagram. 2. Explain why the triangles you named are similar. 3. Find the values of x and y. Show all of your work and leave your answers exact. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 47

48 Prove Rhombus Diagonals Bisect Angles MFAS Geometry EOC Review Quadrilateral ABCD is a rhombus. Prove that both AA and CC are bisected by diagonal AAAA. Similar Triangles 2 Quadrilateral ABCD is a parallelogram. The length of EEEE is 5 units Identify a pair of similar triangles in the diagram. 2. Explain why the triangles you named are similar. 3. Find the length of. FFFF Show all of your work. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 48

49 MAFS.912.G-SRT.3.8 Level 2 Level 3 Level 4 Level 5 calculates unknown side lengths using the Pythagorean theorem given a picture of a right triangle; recognizes the sine, cosine, or tangent ratio when given a picture of a right triangle with two sides and an angle labeled solves for sides of right triangles using trigonometric ratios and the Pythagorean theorem in applied problems; uses the relationship between sine and cosine of complementary angles assimilates that the ratio of two sides in one triangle is equal to the ratio of the corresponding two sides of all other similar triangles leading to definitions of trigonometric ratios for acute angles; explains the relationship between the sine and cosine of complementary angles; solves for missing angles of right triangles using sine, cosine, and tangent uses the modeling context to solve problems that require more than one trigonometric ratio and/or the Pythagorean theorem; solves for sides of right triangles using trigonometric ratios and the Pythagorean theorem when side lengths and/or angles are given using variables Will It Fit? 1. Jan is moving into her new house. She has a circular tabletop that is 7.5 feet in diameter. The door to her house is 7 feet high by 3 feet wide. If she angles the tabletop diagonally, will it fit through the doorway? Why or why not? Show all of your work. TV Size 1. Joey won a new flat screen TV with integrated speakers in a school raffle. The outside dimensions are 33.5 inches high and 59 inches wide. Each speaker, located on the sides of the screen, measures 4.5 inches in width. TV sizes are determined by the length of the diagonal of the screen. Find the size of the TV showing all supporting work. Round your answer to the nearest inch. 59 inches 33.5 inches 4.5 i h Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 49

50 River Width MFAS Geometry EOC Review 1. A farmer needs to find the width of a river that flows through his pasture. He places a stake (Stake 1) on one side of the river across from a tree stump. He then places a second stake 50 yards to the right of the first (Stake 2). The angle formed by the line from Stake 1 to Stake 2 and the line from Stake 2 to the tree stump is 72º. Find the width of the river to the nearest yard. Show your work and/or explain how you got your answer. Tree Stump 72º Stake 1 Stake 2 Washington Monument 1. The Washington Monument in Washington, D.C. is surrounded by a circle of 50 American flags that are each 100 feet from the base of the monument. The distance from the base of a flag pole to the top of the monument is 564 feet. What is the angle of elevation from the base of a flag pole to the top of the monument? Label the diagram with the lengths given in the problem, showing all of your work and calculations, and round your answer to the nearest degree. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 50

51 Holiday Lights MFAS Geometry EOC Review 1. Mr. Peabody wants to hang holiday lights from the roof on the front of his house. His house is 24 feet wide and 35 feet tall at the highest point. The lowest point of his roof is 24 feet off the ground. What is the total length of lights he will need to purchase? Show all of your calculations and round your answer to the nearest foot. lights lights 35 feet 24 feet 24 feet Step Up The diagram below shows stairs leading up to a building. The stringer is the board upon which the stairs are built and is represented by segment ED in the diagram. In the diagram, each riser is 7½ inches and each tread is 9 inches. 1. Assume that the tread, EEEE, meets the wall at a right angle. Explain how the angle at which the stringer meets the wall (see shaded angle) relates to the acute angles of BCE. Wall Stringer Tread 2. Find the angle at which the stringer meets the wall (the shaded angle), to the nearest degree. Show all of your calculations. Ground Riser Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 51

52 Perilous Plunge MFAS Geometry EOC Review 1. Perilous Plunge water ride in California ranks as one of the highest and steepest water rides in the country! The vertical height of the ride is 115 feet. The angle of elevation from the bottom of the drop to the top is 75. What is the distance a rider would travel on the major drop of the flume ride? 2. Label the diagram, show all of your work and calculations, and round your answer to the nearest tenth of a foot. Top Vertical Height Major Drop Lighthouse Keeper From the top of a 210-foot tall lighthouse, a keeper sights two boats coming into the harbor, one behind the other. The angle of depression to the more distant boat is 25 and the angle of depression to the closer boat is 36. Draw and label a diagram that models this situation. Then determine the distance between the two boats showing all of your work and calculations. Round your answer to the nearest foot. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 52

53 MAFS.912.G-SRT.3.6 Level 2 Level 3 Level 4 Level 5 calculates unknown side lengths using the Pythagorean theorem given a picture of a right triangle; recognizes the sine, cosine, or tangent ratio when given a picture of a right triangle with two sides and an angle labeled solves for sides of right triangles using trigonometric ratios and the Pythagorean theorem in applied problems; uses the relationship between sine and cosine of complementary angles assimilates that the ratio of two sides in one triangle is equal to the ratio of the corresponding two sides of all other similar triangles leading to definitions of trigonometric ratios for acute angles; explains the relationship between the sine and cosine of complementary angles; solves for missing angles of right triangles using sine, cosine, and tangent uses the modeling context to solve problems that require more than one trigonometric ratio and/or the Pythagorean theorem; solves for sides of right triangles using trigonometric ratios and the Pythagorean theorem when side lengths and/or angles are given using variables The Sine of 57 In right triangle AAAAAA, m A = 57. Sophia finds the sin 57 using her calculator and determines it to be approximately Explain what the sin 57 = indicates about AAAAAA 2. Does the sine of every 57 angle have the same value in every right triangle that contains an acute angle of 57? Why or why not? Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 53

54 The Cosine Ratio In the right triangles shown below, αα = ββ. b 2 b 3 a 2 a 3 a 1 α ββ 1. Use >, <, or = to compare the ratios aa 1 aa 3 and bb 1 bb 3. Explain and justify your answer. b 1 2. How is the relationship between these ratios related to the cosine of αα and the cosine of ββ? Explain. 3. Suppose DDDDDD is a right triangle and E is one of its acute angles. Also, PPPPPP is a right triangle and Q is one of its acute angles. If cos (E) = cos (Q), what must be true of DDDDDD and PPPPPP? Explain. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 54

55 MAFS.912.G-SRT.3.7 Level 2 Level 3 Level 4 Level 5 calculates unknown side lengths using the Pythagorean theorem given a picture of a right triangle; recognizes the sine, cosine, or tangent ratio when given a picture of a right triangle with two sides and an angle labeled solves for sides of right triangles using trigonometric ratios and the Pythagorean theorem in applied problems; uses the relationship between sine and cosine of complementary angles assimilates that the ratio of two sides in one triangle is equal to the ratio of the corresponding two sides of all other similar triangles leading to definitions of trigonometric ratios for acute angles; explains the relationship between the sine and cosine of complementary angles; solves for missing angles of right triangles using sine, cosine, and tangent uses the modeling context to solve problems that require more than one trigonometric ratio and/or the Pythagorean theorem; solves for sides of right triangles using trigonometric ratios and the Pythagorean theorem when side lengths and/or angles are given using variables Patterns in the Table Use the given triangle to complete the table below. Do not use a calculator. Leave your answers in simplest radical form. A C 1 60 B 1. Describe the relationship between sin 30 and cos Describe the relationship between sin 60 and cos Why do you think this relationship occurs? Explain clearly and concisely. sin 30 sin 60 cos 30 cos 60 Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 55

56 Finding Sine αα β 1. If cos β = 3, what is sin α? Explain your reasoning If cos β = sin α, what must be true about α and β? Explain your reasoning. Right Triangle Relationships 1. Suppose that sin αα = What is the value of cos (90 - αα)? Explain. 2. Suppose that cos ββ = What is the value of sin (90 - ββ)? Explain. 3. Suppose that sin A = 0.41 and cos B = What is the relationship between A and B? Explain. Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 56

57 Sine and Cosine 1. Use the triangle to explain why sin α = cos β. MFAS Geometry EOC Review αα b c a β Congruency, Similarity, Right Triangles, and Trigonometry Student Packet 57