2. The pentagon shown is regular. Name Geometry Semester 1 Review Guide Hints: (transformation unit)

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1 Name Geometry Semester 1 Review Guide Jen and Beth are graphing triangles on this coordinate grid. Beth graphed her triangle as shown. Jen must now graph the reflection of Beth s triangle over the x-axis. Hints: (transformation unit) A. B. C. D. 2. The pentagon shown is regular and has rotational symmetry. What is the angle of rotation? A. 45 B. 72 C. 90 D. 60 How many lines of reflection are there? (transformation unit) Angle of rotation: where n is number of sides Number of lines of reflection = n

2 3. Trapezoid is drawn on the coordinate grid. If you reflect the trapezoid over the dashed line, what would be the new coordinates of trapezoid? (transformation unit)if the reflection line is diagonal, count on the diagonal! A. S (-1, -6) W (5, 0) I (1, 1) M (-2, -2) B. S (-4, -3) W (2, 3) I (3, -2) M (0, -5) C. S (1, -6) W (-5, 0) I (5, -3) M (2, -6) D. S (3, 5) W (-4, 9) I (-3, 5) M (-6, 2) 4. Find the new coordinates of when it is rotated 90 clockwise about the origin. (transformation unit) Re-plot on a bigger graph! A. M (-6, 1) Q (-2, 1) H (-4, 5) B. M (6, -1) Q (2, -1) H (4, -5) C. M (5, 4) Q (-1, 6) H (-2, 1) D. M (8, 7) Q (8, 3) H (5, 4)

3 5. (transformation unit) *Assume the rotation is counterclockwise unless otherwise stated. A. B. C. D. Angle of rotation = where n is number of sides. How many turns is in a 270 rotation? Where will EI end up if you turn the figure that many counterclockwise? 6. (transformation unit)angle of rotation = where n is number of sides. How many turns is in a 240 rotation? Where will X end up if you turn the figure that many counterclockwise? 7. Which of the following is not a rigid transformation? A. Reflection C. Translation (transformation unit) A rigid transformation occurs when the preimage and the image are congruent. A rigid transformation preserves: B. Rotation D. Dilation Distance (lengths of sides are the same Angle measure (angles are congruent) Shape (parallel sides remain parallel, shape does not change) 8. Will the rotation of a pair of parallel lines always result in another pair of parallel lines? (transformation unit) A rotation is an example of a rigid transformation! A. yes, they will remain parallel to each other through any rotation B. Only if the lines are rotated 180 or 360 C. No, they will always result in intersecting lines D. Bo, they will always result in perpendicular lines

4 9. Square ABCD, shown at the right, is translated up 3 units and right 2 units to produce rectangle A B C D. (transformation unit) A translation is an example of a rigid transformation!! Which statement is true? A B C D and and 2 and 3 2 and What would be the image point B after a reflection over the line y = 2 and a translation 4 units right and 2 units down? y A. (1, 4) B. (11, -4) C. (1, -4) D. (1, 0) B x (transformation unit) The line y = 2 is a horizontal line through 2 on the y-axis.

5 12. Steve created this design for a wall mural. (transformation unit) Choose one of the three vertices in figure 1, and count to find its corresponding vertex in image 2. Which describes the translation from figure 1 to figure 2? A. up 3 units and right 1 unit C. up 4 units and right 4 units B. down 1 unit and left 3 units D. down 4 units and left 4 units 13. Given: Quadrilateral ABCD on this graph. (transformation unit) Use patty paper Which graph shows the reflection of ABCD over line l? A. C. B. D.

6 (transformation unit) Example of dilation: 14. Which description MOST accurately describes a dilation? A. When a shape is dilated, the parallel lines remain parallel B. When a shape is dilated, the angles within the shape change C. When a shape is dilated, the orientation of the shape changes D. When a shape is dilated, the distance between the lines stays the same. 15. Mary Ann drew two lines on the chalkboard. Her two lines lie in the same plane but share no common points. Which could be a description of Mary Ann s drawing? (Language of Geometry) A. A set of skew lines C. A set of adjacent rays B. A set of parallel lines D. A set of collinear points 16. Lines and are parallel. The, and the. (Language of Geometry) Which statement explains why you can use the equation to solve for? A. Alternate Exterior Angles are congruent B. Alternate Interior Angles are congruent Corresponding example: 1 & 5 Alternate Interior example: 3 & 6 Alternate Exterior example: 1 & 8 C. Complementary Angles are congruent D. Corresponding Angles are congruent

7 17. Greg used a compass and straight edge to draw the construction below. Which of these is shown by this construction? A. The medians of a triangle are concurrent B. The altitudes of a triangle are concurrent C. The angle bisectors of a triangle are concurrent D. The perpendicular bisectors of a triangle are concurrent 18. A segment has endpoints (-4, 8) and (3, -2). What are the coordinates of the midpoint? A. (-3.5, 6) C. (-0.5, 3) (Triangles) Vocab: Concurrent: meet at a point Median: line from the vertex of a triangle to midpoint of opposite side Altitide: line starting from the vertex of a triangle and creating a right angle with the opposite side (height) Angle bisector: line that divides an angle into two congruent parts Perpendicular bisector: line that may or may not start at the vertex, but goes to the midpoint of the opposite side at a right angle. (Language of Geometry) (, ) B. (-1, 6) D. (-1.5, 3) 19. line Q is perpendicular to the line for the equation: y = - x 4 What is the slope of line Q? (Language of Geometry) Parallel lines have the same slope. Perpendicular lines have opposite reciprocal slopes. A. C. - B. - D.

8 20. Points P, Q, and R are shown below. If these points are all vertices of a parallelogram then which point would represent the coordinates of the fourth vertex of parallelogram PRQS. (Quadrilaterals) In a parallelogram, opposide sides are parallel. Use slope to help find the last point. A (4,6) P Q R B (8,-1) C (5,2) D (9,1) 21. What is the coordinate of the midpoint of? (Language of Geometry) Where is the middle of -4 and 2? A. -2 B. -1 C. 3 D Which of the following is unneccessary to prove that the 2 base angles of an isosceles triangle are congruent? (Triangles) Isosceles triangles: A. The angle sum theorem for triangles B. SAS postulate C. The definition of angle bisector D. The definition of congruent triangles

9 23. What is the perimeter of the triangle? A. 18 B. 20 C. 52 (Language of Geometry) Perimeter = add up all sides Use pythagorean theorem to find the side lengths. D Square EFGH is shown below. A dilation of 2 centered at (2,2) is performed. The resulting square is labeled E F G H. What is the length of? (Transformatons) The center is the fixed point. The other vertices will be twice as far from (2, 2) A. 2 units B. 3 units C. 5 units D. 6 units 25. Which of the following is necessary to prove that a triangle s exterior angle equals the sum of the two remote interior angles? A. the definition of complementary angles B. The definition of angle bisector 3 is exterior angle. 1 and 2 are remote interior angles C. The definition of supplementary angles D. The definition of congruent triangles

10 26. A segment connects the midpoints on two sides of a triangle. What is true about this segment (the midsegment)? (Triangles ) is a midsegment = A. it is always horizontal and half the length of each side. B. It is always perpendicular to the two sides it joins and forms an isosceles triangle with the portions of the sides above it. C. It is always parallel to the third side and half the as long as the third side. D. It is always half the length of those two sides and parallel to the third. 27. When proving the exterior angle sum theorem, the first step is to show that an exterior angle of a polygon and an interior angle of a polyhon add together to equal 180. (Triangles) The polygon exterior angle theorem: the sum of the exterior angles of any polygon = 360 Which angle classification justifies this step? A. Vertical angles B. Corresponding angles C. Complementary angles D. Linear Pairs of angles 28. and are the midpoints of and, respectively. (Triangles) is the midsegment! See #26 What does the midpoint theorem tell us about the relationship between and? A. is ½ the length of B. is twice the length of C. and are not related by that theorem D. and are equal in length

11 29. Which diagram shows a triangle drawn so that it is congruent to? (Triangles) Congruent means equal in size/measure. Use patty paper if necessary. A. C. B. D. 30. In the diagram below, what is the measure of if the triangles and are similar? A. 30 (Triangles) **In similar triangles, corresponding angles are congruent! B. 60 C. 90 D. 70

12 31. Triangle is rotated to become triangle. Without the use of any measurement devices, which could be used to prove that triangle is congruent to triangle? A. SAS beause both are right triangles, JK is congruent to J K and KL is congruent to K L B. ASA because both are right triangles and JL is congruent to J L (Triangles) You must have 3 pairs of congruent angles and sides to use ASA or SAS. The sides and angles must be corresponding!! Try highlighting the information in the multiple choices to see if the sides and angles correspond. C. SAS because both are right triangles, JL is congruent to J L and JK is congruent to K L D. ASA because both are right triangles and angle J is congruent to angle L and JK is conguent to K L 32. Triangle is rotated, reflected, and translated to yield triangle. (Triangles) Be careful on the order of the sides and angles! Which statement proves that the two triangles are congruent? A. is taken to, and is taken to. B. is taken to, and is taken to. C. is taken to, is taken to, and is taken to. D. is taken to, is taken to, and is taken to.

13 33. Which of the triangles below is similar to ΔXYZ? A. C. (Triangles) In similar figures, sides are proportional. Write pairs of corresponding sides as fractions and simplify. If they all reduce to the same number, this is the similarity ratio. If they do nt all reduce to the same number, the triangles are not similar. Be careful when you write the corresponding sides as fractions be consistant on which triangle is the numerator and which is the denominator! B. D. all of the above

14 34. Which statement correctly completes the sentence below? If two distinct pairs of angles in two triangles are congruent, then (Triangles) There is no AAA congruence theorem! A. The pair of included sides must also be congruent and the triangles must be congruent B. The pair of included sides must also be congruent and the triangles must be similar C. The third pair of angles must also be congruent and the triangles must be congruent D. The third pair of angles must also be congruent and the triangles must be similar. 35. The dotted lines in the figure below show how Jenny inscribed circle in right triangle on a practice test. What should Jenny have done differently to answer the question correctly? A. Jenny could have used the altitudes of the triangle to find the incenter. B. Jenny should have constructed an inscribed circle. Circle O is circumscribed. C. Jenny should have used the bisectors of angles X, Y, and Z to find the incenter. (Triangles) Vocab: Incircle/inscribed circle: inside the polygon Circumcircle/circumscribed circle: around the polygon Medians are concurrent at centroid Angle bisectors are concurrent at incenter Perpendicular bisectors are concurrent at circumcenter Altitudes are concurrent at orthocenter D. Jenny should have put points A, B, and C at the midpoints of the triangle.

15 36. Solve for x and y: A. x = 99 y = 74 B. x = 74 y = 99 (Quadrilaterals) If a quadrilateral is inscribed in a circle, then opposite angles are supplementary: C. x = 81 y = 106 D. x = 106 y = True or false: If a figure is a parallelogram then it is a rectnagle. 38. Which of the following quadrilaterals DOES NOT have perpendicular diagonals? (Quadrilaterals) A. rhombus B. square C. kite D. parallelogram 39. Which of the following have congruent diagonals? (Quadrilaterals) Draw the figures A. rectangle, rhombus, square B. rectangle, square, isosceles trapezoid C. kite, parallelogram, trapezoid D. parallelogram, rectangle, rhombus, square

16 40. What are the coordinates of G? (Quadrilaterals) A. (2a, b) B. (-2a, -b) C. (2a, -b) D. (-2a, b) 41. The translation vector 8, 12 means to go: A. left 8 down 12 B. right 8 down 12 (Transformations), X is the horizontal slide Y is the vertical slide C. left 8 up 12 D. right 8 up What is the line of reflection? A. x = -2 B. x axis C. y = -2 (Transformations) Hoy Vux Horizontal lines are of the form y = k Vertical lines are of the form x = h D. y- axis

17 43. Which is the correct list of undefined terms in geometry? (Language of Geometry) A. point, line, ray B. point, line, plane C. segment, angle, point D. plane, helicopter, jet 44. Line A passes through (-3, 5) and (1, -3). What is the slope of the line that is parallel to line A? A. 2 C. -2 (Language of Geometry) Parallel lines have the same slope. Perpendicular lines have opposite reciprocal slopes. B. D. - Graph the points and count or use the formula 45. Line A passes through (2, 5) and (-1, 1). What is the slope of the line that is perpendicular to line A? (Language of Geometry) Parallel lines have the same slope. A. C. - Perpendicular lines have opposite reciprocal slopes. B. D. - Graph the points and count or use the formula

18 UCS Geometry SEMESTER 1 REVIEW GUIDE #2 1. The following picture is a reflection of the image and preimage over what line? 2. Rotate the figure How many reflection lines exist to take the octagon onto itself?

19 4. For a circus act, a trapeze artist swings along the path shown in the coordinate grid. Next, the trapeze artist swings along the same path, but 12 feet higher from the original starting point. A C B D 5. Terry made this quilt with the pattern shown. Which transformations best describe the relationship between block A and block B? A Two flips C A flip and a slide B Two slides D None of the above

20 6. Which transformation, when performed individually, would transform the rectangle below onto itself? A A clockwise rotation 180 about the origin C A reflection across the x-axis, followed by a translation up 5 units B A clockwise rotation 180 about the point 7, 3 D A reflection across the y-axis, followed by a translation left 12 units 7.. Look at figure and its image,. Which best describes the series of transformations that were performed on figure? A a rotation of and a translation of units down C a reflection across the y-axis and a reflection across the x-axis B a reflection across the y-axis and a translation of units down D a rotation of counterclockwise and a translation of units to the right

21 8. The F below is going to be translated 4 units down and 3 units to the right and then translated 2 units up and 1 unit to the right. y A Which of the following single transformations would have the same effect on the F as performing both of the transformations listed above? Translating the F 2 units down and 4 units to the right x B C D Translating the F 6 units down and 2 units to the right Rotating the F 180 counterclockwise about the origin Reflecting the F over the x-axis 9. The point (2, 3) is reflected over the x-axis and then translated 4 units to the left and 2 units up. What are the new coordinates of the point?

22 10. Paul and Janelle are getting ready to play a board game. Paul sets up his pieces as shown in this diagram. Janelle s pieces will be set up the same way but reflected across the line 5. Which of these coordinates represents one of Janelle s pieces? A (4,7) C (9,10) B (3,8) D (7,6) 11. Rectangle KLMN is shown below. If KLMN undergoes a dilation of 2 centered on the origin to produce K L M N, which statement is correct? A B C D The equation of the line passing through points N and K is the same as the equation of the line passing through N and K. The equation of the line passing through points M and N is the same as the equation of the line passing through M and N. The equation of the line passing through points L and M is the same as the equation of the line passing through L and M. The equation of the line passing through points K and L is the same as the equation of the line passing through K and L.

23 12. Pentagon PENTA undergoes a dilation of 2.5 to produce pentagon P E N T A. Which ratio is equivalent to the ratio of the length of to the length of? 13. Which terms does NOT have a formal geometric definition? A Angle C Plane B Segment D Triangle 14. Lines q and r are parallel. The measure of 8 67 and the measure of Why can you use the equation to solve for? 15. What is the first step in constructing a line segment perpendicular to line segment AB that passes through the point P as shown below? A P B

24 16. Construct an equilateral triangle inscribed in a circle. 17. Find the equation of the line passing through the point ( 3, 4) and is parallel to the line having the equation: y = Write an equation for line t can be written as 8. Perpendicular to line t is line u, which passes through the point (-8,1). What is the equation of line u?

25 19. What is the midpoint of if the coordinate of A is (3, -4) and the coordinate of B is (-7, 10)? 20. In a triangle with coordinates (1, 4), (2, 8), and (5, 4), what would be the perimeter rounded to the nearest hundredth? 21. How could a student determine that a triangle and its transformed image are congruent? A They are congruent if and only if the triangles are right triangles. B They are congruent if and only if the transformed figure was not rotated. C They are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. D They are congruent if and only if corresponding pairs of sides are similar and corresponding pairs of angles are similar.

26 22. Which translation from solid-lined figure to dashed-lined figure is 3 units left and 3 units up? Careful, the units are by 2 s. A C B D 23.

27 24. is reflected across the y-axis. Which set of congruence statements explains why the triangles are congruent? A and ; : SAS B and ; : ASA C ; ; : SSS D ; ; : AAA 25. What is necessary to prove that a triangle's exterior angle equals the sum of the two remote interior angles?

28 26. Two frame houses are built, a taller one next to a shorter one. If the frame houses are to be similar in their construction, what should the dimensions of the bigger house be? 27. Which steps correctly state how to construct the circumscribed circle of a triangle? A B C D To find the center of the circumscribed circle, find the point of concurrence of the three internal angle bisectors. The radius of the circle is the segment that joins the center and one side of the triangle and is perpendicular to the side. To find the center of the circumscribed circle, find the point of concurrence of the three medians of the triangle. The radius of the circle is the segment that joins the center and one of the vertices of the triangle. To find the center of the circumscribed circle, find the point of concurrence of the three altitudes. The radius of the circle is the segment that joins the center and one of the vertices of the triangle. To find the center of the circumscribed circle, find the point of concurrence of the three perpendicular bisectors. The radius of the circle is the segment that joins the center and one of the vertices of the triangle.

29 28. Which of the other triangles is similar to ΔABC and why? A C B D None of the above 29. A dilation with a scale factor of 2 is applied to to produce. If measure of 35 and the measure of 102 what are the measures of,,? 30. is a right triangle. CH is a perpendicular bisector that goes from vertex C to the hypotenuse AB of the triangle. How many similar triangles are there?

30 31. Find the missing side lengths of BC and DC if ~ B A C 26 D E 42.5 F 32. Quadrilateral ABCD is inscribed circle O What is true about and? What is the measure of in the diagram above?

31 34. Mrs. Douglas drew a quadrilateral on the chalkboard. She wanted her class to prove it was a rectangle. What conditions must be met for the quadrilateral to be a rectangle? 35. What is the perimeter of the triangle shown below? 36. Jeremy was given, shown below. Show how Jeremy could construct the perpendicular bisector of. Explain each step. Describe how Jeremy could draw the perpendicular bisector of. Describe how Jeremy could sketch the perpendicular bisector of. Use words, numbers, and/or pictures to show your work. Write your answer(s) on the paper provided.

32 37. Solve for a, x, and b:

33 Name Geometry Semester 1 Review Guide Jen and Beth are graphing triangles on this coordinate grid. Beth graphed her triangle as shown. Jen must now graph the reflection of Beth s triangle over the x-axis. Hints: (transformation unit) A. B. C. D. 2. The pentagon shown is regular and has rotational symmetry. What is the angle of rotation? A. 45 B. 72 C. 90 D. 60 How many lines of reflection are there? (transformation unit) Angle of rotation: where n is number of ways it could look the same

34 3. Trapezoid is drawn on the coordinate grid. If you reflect the trapezoid over the dashed line, what would be the new coordinates of trapezoid? (transformation unit) If the reflection line is diagonal, count on the diagonal! A. S (-1, -6) W (5, 0) I (1, 1) M (-2, -2) B. S (-4, -3) W (2, 3) I (3, -2) M (0, -5) C. S (1, -6) W (-5, 0) I (5, -3) M (2, -6) D. S (3, 5) W (-4, 9) I (-3, 5) M (-6, 2) 4. Find the new coordinates of when it is rotated 90 clockwise about the origin. A. M (-6, 1) Q (-2, 1) H (-4, 5) (transformation unit) Re-plot on a bigger graph! B. M (6, -1) Q (2, -1) H (4, -5) C. M (5, 4) Q (-1, 6) H (-2, 1) D. M (8, 7) Q (8, 3) H (5, 4) 5. Regular octagon EIGHTSUP is divided into eight congruent triangles. Find the image of each point or segment for the given rotation. A. (transformation unit) *Assume the rotation is counterclockwise unless otherwise stated. Angle of rotation = B. C. D.

35 6. Name the image of X for a 240 ⁰ counterclockwise rotation about the center of the regular hexagon. A. A B. G C. O D. H (transformation unit) Angle of rotation = 7. Which of the following is not a rigid transformation? A. Reflection C. Translation B. Rotation D. Dilation 8. Will the rotation of a pair of parallel lines always result in another pair of parallel lines? A. yes, they will remain parallel to each other through any rotation (transformation unit) A rigid transformation occurs when the preimage and the image are congruent. A rigid transformation preserves: Distance (lengths of sides are the same Angle measure (angles are congruent) Shape (parallel sides remain parallel, shape does not change) (transformation unit) B. Only if the lines are rotated 180 or 360 C. No, they will always result in intersecting lines D. No, they will always result in perpendicular lines 9. Square ABCD, shown at the right, is translated up 3 units and right 2 units to produce rectangle A B C D. (transformation unit) Which statement is true? A B C D and and 2 and 3 2 and 3

36 10. What would be the image point B after a reflection over the line y = 2 and a translation 4 units right and 2 units down? y A. (1, 4) B. (11, -4) C. (1, -4) D. (1, 0) B x (transformation unit) The line y = 2 is a horizontal line through 2 on the y-axis. 11. Steve created this design for a wall mural. Which describes the translation from figure 1 to figure 2? A. up 3 units and right 1 unit B. down 1 unit and left 3 units C. up 4 units and right 4 units D. down 4 units and left 4 units (transformation unit) Choose one of the three vertices in figure 1, and count to find its corresponding vertex in image Which description MOST accurately describes a dilation? (transformation unit) Example of dilation: A. When a shape is dilated, the parallel lines remain parallel B. When a shape is dilated, the angles within the shape change C. When a shape is dilated, the orientation of the shape changes D. When a shape is dilated, the distance between the lines stays the same.

37 13. Given: Quadrilateral ABCD on this graph. (transformation unit) Use patty paper Which graph shows the reflection of ABCD over line l? A. C. B. D. 14. Lines and are parallel. The, and the. (Language of Geometry) Corresponding: Same place Alternate Interior: Z shaped, interior angles Which statement explains why you can use the equation to solve for x? Alternate Exterior: Z shaped, exterior angles Complementary: Add up to 90 A. Alternate Exterior Angles are congruent B. Alternate Interior Angles are congruent C. Complementary Angles are congruent D. Corresponding Angles are congruent

38 15. Mary Ann drew two lines on the chalkboard. Her two lines lie in the same plane but share no common points. Which could be a description of Mary Ann s drawing? (Language of Geometry) A. A set of skew lines C. A set of adjacent rays B. A set of parallel lines D. A set of collinear points 16. Greg used a compass and straight edge to draw the construction below. Which of these is shown by this construction? A. The medians of a triangle are concurrent B. The altitudes of a triangle are concurrent C. The angle bisectors of a triangle are concurrent D. The perpendicular bisectors of a triangle are concurrent (Triangles) Vocab: Concurrent: meet at a point Median: line from the vertex of a triangle to midpoint of opposite side Altitide: line starting from the vertex of a triangle and creating a right angle with the opposite side (height) Angle bisector: line that divides an angle into two congruent parts Perpendicular bisector: line that may or may not start at the vertex, but goes to the midpoint of the opposite side at a right angle. 17. A segment has endpoints (-4, 8) and (3, -2). What are the coordinates of the midpoint? A. (-3.5, 6) C. (-0.5, 3) (Language of Geometry) (, ) B. (-1, 6) D. (-1.5, 3) 18. line Q is perpendicular to the line for the equation: y = - x 4 What is the slope of line Q? (Language of Geometry) Parallel lines have the same slope. Perpendicular lines have opposite reciprocal slopes. A. C. - B. - D.

39 19. Points P, Q, and R are shown below. If these points are all vertices of a parallelogram then which point would represent the coordinates of the fourth vertex of parallelogram PRQS. (Quadrilaterals) In a parallelogram, opposide sides are parallel. Use slope to help find the last point. A (4,6) B (8,-1) C (5,2) D (9,1) 20. What is the coordinate of the midpoint of? (Language of Geometry) Where is the middle of -4 and 2? A. -2 B. -1 C. 3 D Which of the following is unneccessary to prove that the 2 base angles of an isosceles triangle are congruent? (Triangles) Isosceles triangles: A. The angle sum theorem for triangles B. SAS postulate C. The definition of angle bisector D. The definition of congruent triangles

40 22. What is the perimeter of the triangle? A. 18 (Language of Geometry) Perimeter = add up all sides B. 20 C. 52 D Square EFGH is shown below. A dilation of 2 centered at (2,2) is performed. The resulting square is labeled E F G H. What is the length of? (Transformatons) The center is the fixed point. The other vertices will be twice as far from (2, 2) A. 2 units B. 3 units C. 5 units D. 6 units 24. Which of the following is necessary to prove that a triangle s exterior angle equals the sum of the two remote interior angles? 3 is exterior angle. 1 and 2 are remote interior angles A. the definition of complementary angles B. The definition of angle bisector C. The definition of supplementary angles D. The definition of congruent triangles

41 25. A segment connects the midpoints on two sides of a triangle. What is true about this segment (the midsegment)? (Triangles ) A. it is always horizontal and half the length of each side. B. It is always perpendicular to the two sides it joins and forms an isosceles triangle with the portions of the sides above it. C. It is always parallel to the third side and half the as long as the third side. D. It is always half the length of those two sides and parallel to the third. 26. When proving the exterior angle sum theorem, the first step is to show that an exterior angle of a polygon and an interior angle of a polyhon add together to equal 180. Which angle classification justifies this step? (Triangles) The polygon exterior angle theorem: the sum of the exterior angles of any polygon = 360 A. Vertical angles B. Corresponding angles C. Complementary angles D. Linear Pairs of angles 27. R and are the midpoints of and, respectively. What does the midpoint theorem tell us about the relationship between and? (Triangles) is the midsegment A. is ½ the length of B. is twice the length of C. and are not related by that theorem D. and are equal in length

42 28. Which diagram shows a triangle drawn so that it is congruent to? (Triangles) Congruent means equal in size/measure. Use patty paper if necessary. A. C. B. D. 29. In the diagram below, what is the measure of if the triangles and are similar? A. 30 B. 60 (Triangles) **In similar triangles, corresponding angles are congruent! C. 90 D. 70

43 30. Triangle is rotated to become triangle. Without the use of any measurement devices, which could be used to prove that triangle is congruent to triangle? A. SAS beause both are right triangles, JK is congruent to J K and KL is congruent to K L B. ASA because both are right triangles and JL is congruent to J L C. SAS because both are right triangles, JL is congruent to J L and JK is congruent to K L (Triangles) You must have 3 pairs of congruent angles and sides to use ASA or SAS. The sides and angles must be corresponding!! Try highlighting the information in the multiple choices to see if the sides and angles correspond. D. ASA because both are right triangles and angle J is congruent to angle L and JK is conguent to K L 31. Triangle is rotated, reflected, and translated to yield triangle. Which statement proves that the two triangles are (Triangles) Be careful on the order of the sides and angles! congruent? A. is taken to, and is taken to. B. is taken to, and is taken to. C. is taken to, is taken to, and is taken to. D. is taken to, is taken to, and is taken to.

44 32. Which of the triangles below is similar to ΔXYZ? A. C. (Triangles) In similar figures, sides are proportional. Write pairs of corresponding sides as fractions and simplify. If they all reduce to the same number, this is the similarity ratio. If they do nt all reduce to the same number, the triangles are not similar. Be careful when you write the corresponding sides as fractions be consistant on which triangle is the numerator and which is the denominator! B. D. all of the above

45 33. Which statement correctly completes the sentence below? If two distinct pairs of angles in two triangles are congruent, then (Triangles) There is no AAA congruence theorem! A. The pair of included sides must also be congruent and the triangles must be congruent B. The pair of included sides must also be congruent and the triangles must be similar C. The third pair of angles must also be congruent and the triangles must be congruent D. The third pair of angles must also be congruent and the triangles must be similar. 34. The dotted lines in the figure below show how Jenny inscribed circle in right triangle on a practice test. What should Jenny have done differently to answer the question correctly? A. Jenny could have used the altitudes of the triangle to find the incenter. B. Jenny should have constructed an inscribed circle. Circle O is circumscribed. C. Jenny should have used the bisectors of angles X, Y, and Z to find the incenter. D. Jenny should have put points A, B, and C at the midpoints of the triangle. (Triangles) Vocab: Incircle/inscribed circle: inside the polygon Circumcircle/circumscribe d circle: around the polygon Medians are concurrent at centroid Angle bisectors are concurrent at incenter Perpendicular bisectors are concurrent at circumcenter Altitudes are concurrent at orthocenter

46 35. Solve for x and y: A. x = 99 y = 74 B. x = 74 y = 99 (Quadrilaterals) If a quadrilateral is inscribed in a circle, then opposite angles are supplementary: C. x = 81 y = 106 D. x = 106 y = True or false: If a figure is a parallelogram then it is a rectangle. 37. Which of the following quadrilaterals DOES NOT have perpendicular diagonals? A. rhombus (Quadrilaterals) Draw the figures and their diagonals B. square C. kite D. parallelogram 38. Which of the following have congruent diagonals? A. rectangle, rhombus, square (Quadrilaterals) Draw the figures B. rectangle, square, isosceles trapezoid C. kite, parallelogram, trapezoid D. parallelogram, rectangle, rhombus, square

47 39. The translation vector 8, 12 means to go: A. left 8 down 12 B. right 8 down 12 (Transformations), X is the horizontal slide Y is the vertical slide C. left 8 up 12 D. right 8 up What is the line of reflection? A. x = -2 B. x axis (Transformations) Horizontal lines: y = # Vertical lines: x = # C. y = -2 D. y- axis 41. Which is the correct list of undefined terms in geometry? (Language of Geometry) A. point, line, ray B. point, line, plane C. segment, angle, point D. plane, helicopter, jet 42. Line A passes through (-3, 5) and (1, -3). What is the slope of the line that is parallel to line A? A. 2 C. -2 (Language of Geometry) Parallel lines have the same slope. Perpendicular lines have opposite reciprocal slopes. B. D. - Graph the points and count or use the formula

48 43. Line A passes through (2, 5) and (-1, 1). What is the slope of the line that is perpendicular to line A? (Language of Geometry) Parallel lines have the same slope. A. B. C. - D. - Perpendicular lines have opposite reciprocal slopes. Graph the points and count or use the formula 44. A line is perpendicular to the line for the equation: 4. What is the slope of the perpendicular line? (Language of Geometry) Parallel lines have the same slope. A. B. C. D. Perpendicular lines have opposite reciprocal slopes. Graph the points and count or use the formula 45. Which could be the coordinates of the fourth vertex of rectangle if the other coordinates are X (-1,5), Y (6,-2), and Z (3, -5)? (Quadrilateral Unit) Graph A. (-5, 3) B. (-4, 2) C. (0, -2) D. (2, 8)

49 46. Below are three statements about the figure. (Quadrilateral Unit) I. II. III. Which statement can be proven? A. I only B. II only C. I and II D. II and III

50 47. What is the area of in square units? (Triangles Unit) Step #1: Draw a rectangle around the triangle so each vertex of the triangle is on the rectangle. Step #2: Find the area of the rectangle. Step #3: Find the area of all 3 right triangles. Step #4: Subtract the area of the 3 rights triangles from the total area of the rectangle. A B. 72 C D. 144

51 Alternate Interior Angles Angle Bisector 1 2 Centroid Circumcenter 3 4 Complementary Angles Corresponding Angles 5 6 Dilation Incenter 7 8

52 Linear Pair of Angles Median 9 10 Parallel Lines Parallelogram Perimeter Perpendicular Bisector Perpendicular Lines Rectangle 15 16

53 Rhombus Rigid Transformation Similar Figures Slope Square Supplementary Angles Undefined Terms Vertical Angles 23 24

54 scale factor = ½ center E APD CPB

55 Construction: in a triangle: 33, m CMA = 90, M is midpoint of = x 2 52 = x = x P = = MON and PON form a 38 Construction: in a triangle: 39 40

56 41 42 ABC ~ DEF (6, 6)

57 A parallelogram with four right angles. A point is a location. It has no size. 49 Diagonals are congruent Diagonals are bisected Both pairs opposite sides parallel Both pairs opposite sides congruent A line is a series of points that extend in both directions without end. A plane is a flat surface with no thickness. It extends without end. 50 These lines intersect to form right angles. Their slopes are opposite reciprocals, for example 2 and - or and - This is a similarity transformation. It results in an enlargement or reduction of an image. Preserved: Angle measures (A is congruent to A, B is congruent to B, etc) Shape (parallel lines remain parallel, shape remains the same, just different size) orientation One figure is a dilation of the other. Corresponding angles are congruent Sides are proportional 53 Segment connecting the vertex of a triangle to the midpoint of the opposite side. 54

58 A parallelogram with four congruent sides and four right angles. Diagonals congruent Diagonals perpendicular Diagonals bisect opposite angles Diagonals bisect each other Opposite sides parallel 55 A parallelogram with four congruent sides Diagonals bisect each other Diagonals perpendicular Diagonals bisect opposite angles Opposite angles congruent Opposite sides parallel 56 The distance around a figure. Add up the lengths of the sides. You will likely need Pythagorean Theorem or the Distance formula to calculate the side lengths. Tow angles whose sides form two pairs of opposite rays (formed by the intersection of 2 lines). These angles are congruent These transformations are also called isometries. The original figure and its image are congruent. They preserve the following: Two angles whose sum is 90. Angles can be adjacent or nonadjacent. Angle measure (angles are congruent) Distance (side lengths are congruent) Shape (parallel sides remain parallel, shape does not change) 59 60

59 The center of the circle that is circumscribed about a triangle. Point of concurrency of the perpendicular bisectors. This point is equidistant to the vertices of the triangle. 61 Non adjacent interior angles who lie on opposite sides of the transversal. These angles are congruent. 62 The point of concurrency of the medians of a triangle. This is the center of gravity. These angles lie on the same side of the transversal in corresponding positions relative to the parallel lines. They are congruent. 63 Two adjacent, supplementary (sum to 180 ) angles. They form a line. 65 Two angles whose sum is 180. The angles can be adjacent (linear pair) or nonadjacent These are coplanar lines that do not intersect. They have no point in common. These lines have the same slope and different y-intercepts. For example: Y = 3x 5 and y = 3x 2 66 A line that bisects another line or segment into two congruent pieces and forms a right angle. In a triangle, this line may or may not begin at the vertex, but it crosses the opposite side at the midpoint and forms a right angle. All three of these lines in a triangle are concurrent at the circumcenter. 68

60 The center of the circle that is inscribed in a triangle. The point of concurrency of the angle bisectors of a triangle. This point is equidistant to the sides of the triangle. 69 A line that divides an angle into two congruent parts. In a triangle all three of these lines are concurrent at the incenter. 70 A quadrilateral with both pairs of opposite sides parallel. or 71 Diagonals are bisected Opposite angles congruent Opposite sides congruent Y = mx + b The steepness of a line 72 Term Figure Definition Term Figure definition

61 Term Figure Definition Term Figure definition

62 Your exam has 40 questions. If you google the standard, look for the Shmoop.com entry. It will have additional practice problems (sample assignments) G.CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc G.CO.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

63

64 G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself

65 G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 14.

66 G.CO.5:Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

67 17. 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

68 A triangle was rotated 90 degrees counterclockwise and then translated 2 down and 4 left. The final coordinates of the triangle are: (1, -3), (-2, 0), and (3, 2). What were the original coordinates? A (1, 1) (4, 4) (6, -1) B (0, 2) (2, -3) (-3, -1) C (-1, 3) (2, 0) (-3, -2) D (3, 7) (8, -2) (6, 4) G.CO.7 Use the definitions 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. 23.

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

70 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.

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72 33. G.CO.10: Prove theorems about triangles. Theorems include: 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; the medians of a triangle meet at a point. 34.

73 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

74 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

75 41. G.CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.SRT.1: Verify experimentally the properties of dilations given by a center and a scale factor:

76 44. 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. G.SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar

77 47. 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. 48.

78 G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.C.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle A (4, 6) B (6, 4) C (3, 3) D (1, 2)

79 G.GPE.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G.GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio Where should you plot point X so that PX is of the length of PQ? A. (4, 0) B (6, 0) C (7, 0) D (8, 0) Where should you plot X so it divides PQ in a ratio of 3:4? A. (5, 0) B (4, 0) C 4, 0) D (3, 0)

80 G.GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula