describes a ray whose endpoint is point A. TRUE g. A plane has no thickness. TRUE h. Symbols XY and YX describe the same line. TRUE i.

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1 Geometry Ms. H. Ray, 010 NSWRS TO TH RVIW FOR TH GOMTRY MITRM XM. 1. True or False? e prepared to explain your answer. a. efinitions and theorems are very important in mathematics but every mathematical system must contain some undefined terms and some unproved statements called postulates. TRU b. The terms that we have left undefined in developing our geometrical system are: point, line, and angle. FLS c. Through any two points you can draw just one line. TRU d. Through any three points in a plane you can draw a triangle. FLS (Need non-collinear) e. If two planes intersect, their intersection is a line. TRU f. Symbol describes a ray whose endpoint is point. TRU g. plane has no thickness. TRU h. Symbols XY and YX describe the same line. TRU i. Symbols and describe the same ray. FLS j. If two lines in the same plane are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel. TRU k. If angles form a linear pair then they are supplementary. TRU l. If two angles are vertical then they are supplementary. FLS m. The sum of the angles in a triangle is 10 o. TRU n. If two lines are cut by a transversal then the alternate interior angles are congruent. FLS (True only if the lines are parallel). o. If two parallel lines are cut by a transversal, any two of the angles at the intersections are either supplementary or congruent. TRU p. The sum of the angles in a quadrilateral is 360. TRU q. If two angles are supplementary then they are congruent. FLS r = 3 3. TRU s. If two lines do not intersect then they are parallel. FLS (an be in different planes and skew) t. triangle can have at most one obtuse angle. TRU u. In a right triangle, the right angle is the largest of the three angles. TRU v. If two angles are complementary, then their sum is 90 o. TRU w. Vertical angles are congruent. TRU x. Horizontal lines have no slope. FLS (slope is 0) y. If two angles are supplementary and congruent, then each is a right angle. TRU z. If two lines in a plane are parallel to a third line in the plane then they are parallel to one another. TRU aa. lines alt. int. s are TRU bb. If two lines in a plane are cut by a transversal and the same side interior angles are congruent, then the lines are parallel. FLS cc. n isosceles triangle has at least two congruent sides. TRU dd. The following are ways to prove that two triangles are congruent:, SSS, SS, S, SS, S. FLS ee. ( 5 +1) = 6 FLS ( FOIL ) ff. The reflection of point (3, -1) in the x axis is point (3, 1). TRU gg. Point (7, 3) lies on the circle (x 3) + y = 5 TRU hh. triangle whose sides are, 3, is a right triangle. TRU ii. Parallel lines have the same slope (or no slope if they are both vertical). TRU

2 Geometry Ms. H. Ray, 010 jj. In a triangle, if two angles are congruent then the sides opposite those angles are also congruent. TRU. In each diagram find the values of the variables (x, y, z etc.) and state the theorem(s) or postulate(s) or definition(s) that you used to get the equations involving the variables a. b. H j k F I M J x quation: + 10 = 0 Theorem(s) or postulate(s) or definition(s): If two parallel lines are cut by a transversal, the alternate exterior angles are congruent. Solution: x = 10 c. quation: y + 10 = 1, x = 1 Theorem(s) or postulate(s) or definition(s): The Pythagorean Theorem: If a and b are the legs, and c is the hypotenuse of a right triangle, then a + b = c VHMI VHMJ by SS, x = 1 by PT Solution: y =, y = = 11 x = 1 d. T quation: x = y = 0 Theorem(s) or postulate(s) or definition(s): Vertical angles are congruent; angles in a linear pair are supplementary Solution: x = y = 0 quation: x = y; x+ y + = 10 Theorem(s) or postulate(s) or definition(s): The sum of the angles in a triangle is 10. In an isosceles triangle, the angles opposite congruent sides are congruent. Solution: x = y = 6

3 Geometry Ms. H. Ray, Given that F, find the measure of each marked angle: x = 35 F y = 110 z = 70 w = 35 v = 0 s = 70. M is the midpoint of, M = x and M = 3 x x + 3 = x, x = Simplify: a. 50 = 5 b. = 3 c. 1 = 3 6. Perform each operation and simplify the answers a. = 3 +1 b. 1 7 = 3 c. ( 3) = 3 d. ( 5) = 0 e. 5( ) = f. ( +1)( 3 + 6) = What are the coordinates of the midpoint of the line segment with endpoints (-6, ) and (, 6)? 6 +, + 6 = ( 1,5 ). What is the length of the line 6 segment joining the points (-6, ) and (, 6)? (alculator OK). 1 + = 00 = What is the slope of the line segment joining the points (-6, ) and (, 6)? rise run = 6 ( 6) = 1 =

4 Geometry Ms. H. Ray, For each diagram state the relationship between 1 and (vertical, adjacent, linear pair, no relationship) a. b. c adjacent no relationship linear pair 11. Find an approximate location of 15 on the number line below. 9 < 15 < Given that OG TIP, complete each statement: a. P G b. G TP c. PTI GO 13. Given the diagram and the information below, find the measure of each numbered angle. No justification is necessary; it is OK to write just numbers for answers. j k 1 = 36 o 7 = 65 o = 1 o 3 = 65 o j = 36 o 5 = 65 o k 6 = 115 o =115 o 9 = 79 o = 79 o 11= 115 o

5 Geometry Ms. H. Ray, For what values of x are lines in the diagram parallel? Justify your answer. a. b. (x - 60) (0 + 10x) G quation: 1 x + 57 = 3x + 7 Theorem/postulate that makes the equation true: If corresponding angles are congruent, then lines are parallel. Solution: x = 0 quation: x x = 10 Theorem/postulate that makes the equation true: If same side interior angles are supplementary, then lines are parallel. Solution: x = 10, reject x = Find the equation of each circle in the diagram: : x + (y ) = 9 : (x ) + (y + ) = 16 : (x ) + (y ) = 1 : (x + 5) + (y +1) = 1

6 Geometry Ms. H. Ray, Find the equation in any form that you like of each line in the diagram: K 6 M 1 : y = x + 3 : y = x GH : y = 6 M : y 3 = (x 7) KL : x = 11 L G H onsider the line segment that joins points (-6, -) and (, -) a. What is the slope of? m = b. What are the coordinates of the midpoint of? ( ), = ( 1, 3) suu r c. Write an equation of in point-slope form. Use point. y ( ) = 1 (x ( 6)) 5 y + = 1 (x + 6) d. Transform your equation from above into a slope-intercept form. y = 1 5 x Solve the following system of equations by substitution. y = 3x 7 y = 3 x +1 a. y = 1 x b. y = 6x 7 x =, y = 1 x = 3, y =

7 Geometry Ms. H. Ray, Solve the following systems of equations by linear combination (addition). 3x + y = 5x + 3y = a. x + 6y = 3 x = 3, y = 1 b. x +1y = 0 x =, y = 3 0. o segments whose lengths are 3, 5, make a right triangle? Prove your answer. Yes: ( 3) + ( 5 ) = = = ( ) 1. o segments whose lengths are,, 3 make a right triangle? Prove your answer. No: ( ) + ( ) = + = 10 ( 3 ). Point M is the midpoint of JF. JF =, JM = 3x+. arefully make a diagram and find x. ( 3x + ) =, 6x + =, 6x = 1, x = 3 3. In a right triangle, the hypotenuse is 1 and one leg is 6. How long is the other leg?. 6 + x = 1, x = 10 = P H R T R S ongruent? Yes M Z ongruent? Yes N RST HP by SSS RNZ RMZ by SSS F.5 M F ongruent? Yes U ongruent? Not congruent ( U = 0 o ) F by SS

8 Geometry Ms. H. Ray, H G K J ongruent? I No conclusion ongruent? Yes If congruent, complete the congruence statement and justify it: by S 30. Is? Justify your answer. 31. Is HG FGH? Justify your answer. 1 H F 1 1 V VF by SS by PT 3. Is? Justify your answer. G F VHG VGFH by SS HG FGH by PT 33. Is?Justify your answer V V by S by PT V V by S by PT

9 Geometry Ms. H. Ray, Prove that with vertices (, ) ) (-3, 0) and (1, -3) is scalene. (Hint: what is the definition of a scalene triangle? alculate the length of each side and compare them.) = + 5 = 9 = 3 + = 5 = 5 = = 6 no two sides are congruent Prove that with vertices (, ) (3, -3) and (-, 1) is not a right triangle. (Hint: alculate the length of each side and see if the Pythagorean relationship holds.) = = 6 = 7 + = 65 = = 37 ( 6 ) + 37 ( ) = 63 ( 65 ) n angle is 35 degrees smaller than its complement. What is its measure? x + (x + 35) = 90, x = 55, x = Graph and label the following circles (all three in the same grid) 37. Graph and label the following lines:

10 Geometry Ms. H. Ray, Find the equation of the image of the circle (x ) + (y + 5) = under the transformations: a. Translation to the left 3 units. b. Reflection in the line y = c. d. Reflection in the x axis "mirror" (x +1) + (y + 5) = (x ) + (y 9) = - - (x ) + (y 5) = 0. What is the measure of each exterior angle of a regular decagon (10-gon)? What is the sum of the angles of a convex 11-gon? 9 10 = 160. If the sum of the angles of a polygon is 30, how many sides does the polygon have? If one of the complementary angles measures 6x and the other 1x, what is x and what is the measure of each angle? 6x + 1x = 90, x = 5, smaller angle measures 30, larger 60.. If two complementary angles have measure (x + 1) and (x 30), what is x and what is the measure of each angle? x x 30 = 90, 3x = 10, x = 3. ngles are 5 and 3. uuur 5. bisects. measures. If measures (x + 3x) o, what is x, and what is the measure of? x + 3x =, x + 3x = 0, (x + 7)(x ) = 0, x = or x = 7 = = What is the image of point (3, 7) under the following transformation: a. Reflection in the x axis. b. Reflection in the y axis. b (-3, 7) 6 c. Rotation around the origin 10 o counterclockwise? 7. True or False? a. right triangle cannot be acute. TRU b. n acute triangle cannot be scalene. FLS c. n isosceles triangle cannot be scalene. TRU -6 c (-3, -7) a (3. -7) d. If angles form a linear pair then they are adjacent -

11 Geometry Ms. H. Ray, 010 angles. TRU e. If angles are adjacent than they form a linear pair. FLS f. If a product of two numbers is zero, one or both of these numbers must be zero. TRU g. The Pythagorean Theorem applies only to right triangles. TRU F F F = 66. What is the measure of H? 1. H F 9. M is the midpoint of, M = x and M = 1 x + 3. raw a diagram, and set up and solve an equation to find the value of x and the length of. x = 1 x + 3, x = 5, = Place the number in an appropriate place on a numberline. ecide to which of the number sets (natural numbers, whole numbers, integers, rational real, irrational real) each number belongs natural number, -7 integer, 7 rational, 0 whole number, 7 rational, 3 irrational, 3 irrational, 36 = 6 natural 51. Give an example of a system of linear equations that is inconsistent. y = 3x + and y = 3x 1 Find x: 5. 1 x + = x x = x 3 5. x 15x = x =, x = -1 x = 1, x = ± 1 = ±3 x = 11, x =

12 Geometry Ms. H. Ray, re the two triangles congruent? YS If the triangles are congruent, state the reason why you can conclude that they are. SSS omplete the congruence statement VPIG VNOT State the reason why the equation is true. State an equation to find x. x = x + 5 Solve the equation for x. x = 30 orresponding parts of congruent triangles are congruent omplete: O = 60 o I = 60 o T = 50 o G = 50 o 56. Find x, y and z. (Note: there are no parallel lines in the diagram.) x = = 15, z = 55, y + 0 = 55, y = State the definition of an altitude of a triangle n altitude is a line segment from a vertex in a triangle to the opposite side that is perpendicular to the opposite side (extended if necessary). 10 z x ( y+0) 5. Is it possible for an altitude not to be contained within the triangle? xplain. Yes, all obtuse triangles have two altitudes that are outside of the triangle. 59. (corrected) The three angles in a triangle measure 70, 60 and (x 5x). What is the value of x? x = 10 or x = n exterior angle of regular hexagon (6-gon) measures (x x). What is the value of x? x = 10 or x = If a polygon has 11 sides, what is the sum of its interior angles? What is the sum of its exterior angles? Interior angles sum: 160 xterior angles sum: 360 w

13 Geometry Ms. H. Ray, 010

14 Geometry Ms. H. Ray, 010