You try: What is the definition of an angle bisector? You try: You try: is the bisector of ABC. BD is the bisector of ABC. = /4.MD.

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1 US Geometry 1 What is the definition of a midpoint? midpoint of a line segment is the point that bisects the line segment. That is, M is the midpoint of if M M. 1 What is the definition of an angle bisector? 2 m O = m O =. O m O= = 90. O 1.0/4.M.6 3 is the bisector of. If m = 52, then m + m = 3 is the bisector of. If m = 46, then m + m = is the bisector of m = m = 52. m + m = m = m 104 = m m + m = = /7.G.5,G.O.1 Page 1 of 13 M@WUS (US) 10/31/12

2 US Geometry 4 Points,, and are collinear, but they do not necessarily lie on the same line in the order named. If = 3and = 8, what is? There are two possibilities: 4 Points,, and are collinear, but they do not necessarily lie on the same line in the order named. 3 = 3+ 8 = 11 8 If = 9and = 5, what is? 3 8 = 8 3 = 5 1.0/G.O.1 5 Give an example of the Symmetric Property of ongruence. If, then. 5 Give an example of each of the following properties of angle congruence. a) Reflexive b) Transitive 1.0 c) Symmetric 6 What kind of angles are O & O? 6 Ifm O= 2x 5andm O= 3x+ 10, what is the value of x? O O and O are adjacent angles. They also form a linear pair and are, therefore, supplementary. O m O + m O = /7.G.5 Page 2 of 13 M@WUS (US) 10/31/12

3 US Geometry 7 Given: m = 43, is a complement of and is a supplement of, which of the following is a true statement? ) m = 137 ) m = 43 ) m = 137 ) m = 43 is a complement of, so m = 90 43, = 47 which is not one of the choices. is a supplement of, so m = = 137. ) is the correct choice., 7 Given: m = 27, is a complement of and is a supplement of,, which of the following is a true statement? ) m = 153 ) m = 27 ) m = 153 ) m = 27 ) m = /7.G.5, G.O.1 8 p q. Solve for x. 8 p q. Solve for x. 5x + 18 p p 10x + 12 The angles whose measurements are given are same-side, or consecutive, interior angles. ecause the lines are parallel, these angles are supplementary. Therefore, q 2x + 5 5x 4 p q p (5x+ 18) + (10 x+ 12) = x + 30 = x = 150 x = /7.G.5, G.O.1 Page 3 of 13 M@WUS (US) 10/31/12

4 US Geometry 9 If and are complementary, and m = x, what is m? m + m = 90 x + m = 90 m = 90 x 9 a) What does the Triangle Sum Theorem say about the following triangle? /7.G.5, G.O.1 onstruct a line through P parallel to line l. P l We will construct PW parallel to l using corresponding angles: b) If m = N, m =? raw any line through P and l. We will copy P using only a straightedge and compass. Make an arc from (using as center). Make the same arc from P: Y P 10 o each of the following constructions: a) opy a line segment. b) isect a line segment. c) opy an angle. X l d) isect an angle. P Measure how much opens up by putting the compass point and pencil point at X and Y. Make a mark to show that you measured correctly: Z Y P X Make that same mark from Z, crossing the arc you made from P: Z Y P W X raw PW. PW will be parallel to line l because ZPW Y and are corresponding angles. 16.0/G.O.12 Page 4 of 13 M@WUS (US) 10/31/12

5 US Geometry 11 Given: 11 Given: Prove: Statements Reasons Given as true Reflexive Prop. of. Prove: SSS Post /G.SRT.5 What does the Isosceles Triangle, or ase ngles, Theorem say about? If, then. 12 In the triangle below, what is the value of x? If, then. If, then 1.0/4.G.2 (3x + 1) In, if, then what do you know about,, and? 13 They are all congruent. If below is equiangular, what is the value of x? 5x 3x ,12.0/7.G.5 Page 5 of 13 M@WUS (US) 10/31/12

6 US Geometry 14 F by which postulate or theorem? 14 Name the postulate or theorem by which the congruence statement is true. a) F F The S ongruence Postulate. F b) 5.0/G.SRT.5 c) 15 What is the definition of a median of a triangle? median of a triangle connects a vertex to the midpoint of the opposite side raw an example of each of the following in a triangle. a) altitude b) angle bisector c) median d) perpendicular bisector Page 6 of 13 M@WUS (US) 10/31/12

7 US Geometry Which statement is correct for the diagram above? ) ) ) ) ) both ) and ) y the lternate Interior ngle Theorem, and. Therefore, must correspond to and to. That happens in both ) and ). ) is the correct answer. Which statement is correct for the diagram above? ) ) ) ) ) None of the above. 5.0,7.0/G.SRT.5 17 In the diagram, p q, = 7, and = 3x + 4. What is the value of x? p 17 In the diagram, p q, = 13, and = 2x + 3. What is the value of x? q p = 7= 3x + 4 3= 3x 1 = x q 7.0/8.G.5 Page 7 of 13 M@WUS (US) 10/31/12

8 US Geometry 18 In, which of the following is NOT a possible value for the length of? ) 21 ) 19 ) 11 ) ) oth ) and ) m > m (110º > 53º), so the side opposite is greater than the side opposite. That is, >. = 20, so cannot equal 21. the answer is ). 18 In F, which of the following is NOT a possible value for the length of F? F 12 ) 14 ) 11 ) 9 ) 5 ) 3 6.0/G.O Find the midpoint of the segment whose 2, 7 3, 1. endpoints are ( ) and ( ) The coordinates of the midpoint are the averages of the coordinates of the endpoints: 19 Find the midpoint of the segment whose 3, 9 5, 2. endpoints are ( ) and ( ) x1+ x2 y1+ y2, , 1 8 =, 1 =, /N.N.6 Page 8 of 13 M@WUS (US) 10/31/12

9 US Geometry 20 Find the distance from ( 5, 2) to ( 1, 4). 20 Find the distance from ( 6, 1) to( 4, 6). a ( 1, 4) b c ( 5, 2) a + b = c = c = c 2 40 = c 40 = c 2 10 = c 2 ( ) ( ) d = x x + y y d = ( 1 5) + ( 4 2) = ( 6) + ( 4 + 2) = ( 2) = = 40 = /N.N.6 2 nd of Study Guide Page 9 of 13 M@WUS (US) 10/31/12

10 US Geometry 1 You Try Solutions: n angle bisector is a ray that is equidistant from the sides of an angle, n angle bisector is a ray that bisects an angle. n angle bisector is a ray that cuts an angle into two congruent angles. 5 (nswers may vary.) a. b. If and or, then. c. If, then. 2 3 m O= = 120 is the bisector of. m = 46. y the ngle ddition Postulate, m + m = m = m 92 = m m + m = = y the ngle ddition Postulate, m O+ m O= m O Therefore, 2x 5+ 3x+ 10= 180 5x + 5= 180 5x + 5= 180 5x = 175 x = 35 is a complement of, so m = 90 27, which is not one of = 63 the choices. is a supplement of so m = = 153. ) is the correct choice., 4 There are two possibilities: = 9+ 5= 14 or 8 The angles whose measurements are given are alternate interior angles. ecause the lines are parallel, these angles are congruent. Therefore, 2x+ 5= 5x 4 9= 3x 3 = x = 9 5= 4 Page 10 of 13 M@WUS (US) 10/31/12

11 US Geometry 9 a. m + m + m = 180 b. N m = 180 N + m = 90 m = 90 N c. Given First draw a ray or line and pick a point to call : Then draw an arc using as the center and draw that same arc from : 10 a. Given To copy, first draw a ray or line and pick a point to call : Measure with a compass by putting the point on one end and drawing an arc on the other end. Make that same arc from : will ' ' be congruent to b. Given To bisect, make an arc above and below with the compass at (using as center) and opened 1 more than halfway (with radius > 2 m): d. Given Measure with your compass and use that same setting to make a mark from : raw a ray from through the intersection of the two arcs, and will be congruent to : Make an arc from : o the same thing using as center, using the same radius as the previous step. The intersections of the arcs will be on the perpendicular bisector. raw a line segment, ray, or line on the line that connects the two points of intersection, and you have bisected : bisects. Using the same radius (compass opening), make arcs from the points of intersection, and : F will bisect F : Page 11 of 13 M@WUS (US) 10/31/12

12 US Geometry 11 Statements Reasons 1. Given as true. 2. Vertical angles are congruent. 3. S ongruence Postulate. 15 a. is an altitude because it is drawn from a vertex perpendicular to the opposite side. 12 y the ase ngles, or Isosceles Triangle, Theorem,. 3x + 1 = 61 3x = 60 x = 20 b. is an angle bisector because it cuts into two congruent angles. 13 is equiangular 5x= 3x+ 6 2x = 6 x = 3 c. is a median (it connects a vertex with the midpoint of the opposite side). 14 a. HL ongruence Theorem b. SSS ongruence Postulate c. S ongruence Postulate d. is a perpendicular bisector because it is perpendicular to and it bisects. Page 12 of 13 M@WUS (US) 10/31/12

13 US Geometry 16, so by the HL right Theorem. is the same as because corresponding angles are written in corresponding places. ) is the correct answer. 20 ( 6, 1) = 13 = 2x = 2x 5 = x F must be smaller than F because F is opposite the smaller angle. ll of the numbers are less than 12 except for 14. ) is the correct answer. ( 4, 6) a + b = c = c = c = c 125 = c = c 5 5 = c ( ) ( ) d = x x + y y d = (6 ( 4)) + ( 1 ( 6)) = (6 + 4) + ( 1+ 6) = = = 125 = 25 5 = The coordinates of the midpoint are the averages of the coordinates of the endpoints: x1+ x2 y1+ y2, , 2 11 =, 11 = 1, 2 Page 13 of 13 M@WUS (US) 10/31/12