Geometry 2012 – 2013Name ______

1st Semester Exam Review Answers

I. Definitions

  1. conjecture – an educated guess based on known information
  2. segment - a measurable part of a line that consists of 2 endpoints and all of the points between them
  3. ray –a segment that continues on in one direction
  4. postulate –a statement that is accepted as true without proof
  5. acute angle –an angle whose measure is less than 90
  6. right angle – an angle whose measure is exactly 90
  7. obtuse angle – an angle whose measure is more than 90
  8. straight angle – an angle whose measure is exactly 180
  9. segment bisector – a segment, line, or plane that intersects a segment at its midpoint
  10. angle bisector – a ray that divides an angle into two congruent angles
  11. vertical angles – two nonadjacent angles formed by two intersecting lines
  12. complementary angles – two angles whose sum is 90
  13. supplementary angles – two angles whose sum is 180
  14. counterexample – an example used to disprove a statement
  15. linear pair – a pair of adjacent angles that form a line
  16. converse – the statement formed by switching the hypothesis and conclusion of a conditional statement
  1. inverse – the statement formed negating the hypothesis and conclusion of a conditional statement
  2. contrapositive – the statement formed by switching and negating the hypothesis and conclusion of a conditional statement
  1. perpendicular lines – lines that intersect to form right angles
  2. parallel lines – lines that do not intersect
  3. skew lines – lines that do not intersect and are not the same plane
  4. transversal – a line that intersects two or more lines at different points
  5. alternate interior angles – a pair of interior angles that are on opposite sides of the transversal
  6. alternate exterior angles – a pair of exterior angles that are on opposite sides of the transversal
  7. consecutive interior angles – a pair of interior angles that are on the same side of the transversal
  8. midpoint – the point on a segment exactly halfway between the endpoints of a segment
  9. isosceles triangle – a triangle with at least two sides congruent
  10. right triangle – a triangle with a right angle and two acute angles
  11. obtuse triangle – a triangle with an obtuse angle and two acute angles
  12. acute triangle – a triangle with all 3 angles acute
  13. scalene triangle – a triangle with no sides congruent
  14. equilateral triangle – a triangle with all 3 sides congruent
  15. polygon – a closed figure
  16. perpendicular bisector – a segment that is perpendicular to a side of a triangle at the midpoint
  17. concurrent lines – three or more lines that intersect in the same point
  18. circumcenter – the point of concurrency of the perpendicular bisector of a triangle
  19. incenter – the point of concurrency of the angle bisectors of a triangle
  20. median – segment whose endpoints are a vertex and the midpoint of the opposite side
  21. centroid – the point of concurrency of the medians of a triangle
  22. altitude – a perpendicular segment drawn from a vertex to the opposite side
  23. orthocenter – the point of concurrency of the altitudes of a triangle
  24. convex – a polygon whose sides do not cave in
  25. concave – a polygon whose sides cave in
  26. regular – a polygon that is equiangular and equilateral
  27. parallelogram – a quadrilateral with both pairs of opposite sides parallel
  28. rhombus – a parallelogram with 4 congruent sides
  29. square – a parallelogram with 4 congruent sides and 4 right angles
  30. rectangle – a parallelogram with 4 right angles
  31. trapezoid – a quadrilateral with exactly one pair of parallel sides
  32. isosceles trapezoid – a trapezoid whose legs are congruent

II. Quadrilateral Properties

Parallelogram / Rectangle / Rhombus / Square
50. Opposite sides are parallel. / X / X / X / X
51. Opposite sides are . / X / X / X / X
52. Opposite angles are . / X / X / X / X
53. Consecutive interior angles supplementary. / X / X / X / X
54. Diagonals bisect each other. / X / X / X / X
55. All 4 angles are right angles. / X / X
56. Diagonals are . / X / X
57. All 4 sides are . / X / X
58. Diagonals bisect opposite angles. / X / X
59. Diagonals are . / X / X

III. Proofs

Given: AB = BC

Prove: ½ AC = BC

Statements / Reasons
1. AB = BC / 60. Given
2. AC = AB + BC / 61. Segment Addition
3. AC = BC + BC / 62. Substitution
4. AC = 2 BC / 63. Substitution
5. ½ AC = BC / 64. Multiplication/Division

Given: 1 and 3 are a linear pair 1 3

2 and 3 are a linear pair 4 2

Prove: m1 = m2

Statements / Reasons
1. 1 and 3 are a linear pair
2 and 3 are a linear pair / 65. Given
2. 1 and 3 are supplementary
2 and 3 are supplementary / 66. Supplement Thm
3. m 1 + m 3 = 180
m 2 + m 3 = 180 / 67. Def of Supplementary Angles
4. m 1 = m 2 / 68. Congruent SuppThm

Given: AB  BC

ABC is bisected by BD

Prove: ∆ABD  ∆CBD

Statements / Reasons
1. AB  BC / 1. Given
69. ABD = CBD / 2. Definition of  Bisector
3. BD  BD / 70. Reflexive
4. ∆ABD  ∆CBD / 71. SAS

Given: ∆DGC  ∆DGE, ∆GCF  ∆GEF

Prove: ∆DFC  ∆DFE

Statements / Reasons
1. ∆DGC  ∆DGE, ∆GCF  ∆GEF / 1. Given
2. CDG EDG; CD  ED; CFD EFD / 72. CPCTC
3. ∆DFC  ∆DFE / 73. ASA

IV. Problems

Find the measure of each variable.

74. x = 6875. y = 12076. x = 24

Find the measure of each angle.

77. 178. 279. 380. 481. 5

59 78 102 22 68

82. 6

34

Determine whether the following triangles are congruent. (SSS, SAS, ASA, AAS, cannot be determined)

83. ASA84. CBD85. CBD

Use the conditional statement to identify the following.

If an angle measures less than 90, then it is an acute angle.

86. Hypothesis: an angle measures less than 90

87. Conclusion: it is an acute angle

88. Converse: If an angle is an acute angle, then it measures less than 90.

89. Inverse: If an angle does not measure less than 90, then it is not an acute angle.

90. Contrapositive: If an angle is not an acute angle, then it does not measure less than 90.

G is the centroid of ABC, AD = 15, CG = 13 and AD  CB.

91. Find the length of AG. 10

92. Find the length of GD.5

93. Find the length of GE.6.5

94. Find the length of GB.13

List the angles of the triangle in order from least to greatest.

95. I, G, H96. L, K, J

Find the possible measures for the third side of XYZ.

97. XZ = 6, YZ = 898. XZ = 9, YZ = 5

2 < x < 14 4 < x < 14

Use the figure below to determine if the segments are parallel, skew, or perpendicular.

99. AB and AHperpendicular

100. EF and ACskew

G

101. DF and BGparallel

F

CD

Use the figure to identify the special angle pair. (alt. int., alt. ext., cons. int., corr., linear pair)

102. 1 & 8alternate exterior

103. 5 & 6linear pair

104. 2 & 6corresponding

105. 4 & 5alternate interior

106. 4 & 6consecutive interior

Find the value of the variables.

107. x = 20108. x = 26.4

Use the diagram to answer the following questions.

109. Name a point collinear to K. M, L, or R

110. Name a point coplanar to P.O, M, N, Q, R, or L

111. x = 35, y = 50112. x = 31, y = 11113. x = 16, y = 10

Find the missing measure(s) for the given trapezoid.

114. For trapezoid ADFC, B and E are 115. For trapezoid WXYZ, P and Q are

midpoints of the legs. Find AD. midpoints of the legs. Find WX.

AD =58WX = 5

116. For trapezoid DEFG, T and U are 117. For isosceles trapezoid QRST, find

midpoints of the legs. Find TU, mE. AB, mQ, and mS.

mG.

TU = 28, mE = 95, mG = 145 AB = 42.5, mQ = 125, mS = 55

PRST is a rectangle. Find each measure if m1 = 50.

118. m2 = 40119. m3 = 40120. m4 = 50

121. m5 = 100122. m6 = 40123. m7 = 80

124. ABCD is a rectangle. If AD = x2 – 7 and BC = 4x + 5, find AD.

x = -2 or x = 6

AD = 29