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ABC and DCB

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601 Review Part II – Geometry

1. If the diagram were drawn to scale, which side would be the shortest?

2. In each pair of triangles below, like markings indicate congruent parts. Name the congruence postulate or theorem that proves the triangles congruent. If none, write none. a) b) c) d) e) f)

3. Use the information given in the figure to find the value of each of the following: a) x = ________ b) y = ________ c) z = ________ d) w = ________ e) t = ________ f) s = ________ g) v = ________ h) r = _________

4. Explain why the sum of the lengths of altitudes of any triangle is less than the perimeter. State the theorem(s) used to justify your reasoning.

5. In the figure, ∠1 ≅ ∠2. a) If BC = 8x – 7, CD = 6x, and BD = 21, find x. b) If AC = 4y + 2, AD = 5y – 1, and CD = y + 14, find y. c) If

AC bisects

BD , BC = 2z – 15, and BD = z + 9, find BD. d)b If AB = 14 and BD = 21, what can you conclude about AD? e) State the postulate or theorem that justifies the statement. i) m∠ACB > m∠1 ii) ∠ACB and ∠ACD are supplementary iii) Point C is the only point common to

AC and

BD . iv) m∠BAD > m∠1

v) If

AC bisects ∠BAD, then

m!1=12

m!BAD .

6. In ΔABC, D is the midpoint of

AB . A line through D, parallel to

BC , intersects

AC at E. If AE = 4x + 2 and EC = 8x – 10, find x, AE, EC, and AC

7. In right ΔABC,

BD is the median to hypotenuse

AC . If BD = x+3, AD = 6y+1, and DC = x+y+1, find x, y, BD, and AC.

8. If the lengths of the sides of ΔABC are represented by a, b, and c, represent the perimeter of the triangle whose vertices are the midpoints of ΔABC.

9. Find the value of x.

10. JKMP is a rectangle. PK = 0.2x, JM = x – 12. Find PK.

Sometimes/Always/Never

11. Two Δs are ≅ if they have a side and any two angles of one ≅ to the corresponding parts of the other.

12. The point of intersection of the bisectors of the base ∠s of an isosceles Δ and the ends of the base of the isosceles Δ are the vertices of another isosceles Δ.

13. Two right Δs are ≅ if any side and an acute ∠ of one are ≅ to the corresponding parts of the other.

14. If 2 coplanar isosceles Δs have the same base, the line joining the vertices is ⊥ to the base.

15. If 2 ∠s of a polygon are equal, the sides opposite the ∠s are equal.

16. If the altitude drawn to the base of a Δ bisects the base, the Δ is an isosceles Δ.

17. Two Δs are ≅ if 3 ∠s of one are ≅ to 3 ∠s of the other.

18. Two Δs are ≅ if 2 sides and an ∠ of one are ≅ to two sides and the corresponding ∠ of the other.

19. If 2 Δs have equal bases and the altitudes drawn to these bases are also equal, the Δs are ≅.

E

A

C

DB

20. The bisectors of 2 coplanar adjacent supplementary ∠s are perpendicular to each other.

21. If a plane contains one of 2 skew lines, it contains the other.

22. If a line and a plane never meet, they are parallel.

23. If 2 parallel lines lie in different planes, the planes are parallel.

24. If a line is perpendicular to 2 planes, the planes are parallel.

25. If a plane and a line not in the plane are each perpendicular to the same line, then they are parallel to each other.

26. Two lines must either intersect or be parallel.

27. In a plane, 2 lines perpendicular to the same line are parallel.

28. In space, 2 lines perpendicular to the same line are parallel.

29. If a line is perpendicular to a plane, it is perpendicular to every line in the plane.

30. It is possible for two planes to intersect in a point.

31. If a line is perpendicular to a line in a plane, it is perpendicular to the plane.

32. Two lines perpendicular to the same line are parallel.

33. A triangle is a plane figure.

34. A line that is perpendicular to a horizontal line is vertical.

35. Every 4-sided figure is a plane figure.

36. If a quadrilateral is equilateral, then the quadrilateral is a square.

37. The diagonals of a parallelogram divide it into 4 ≅ Δs.

38. In parallelogram ABCD, if ∠A is a right ∠, then diagonal

AC is equal to diagonal

BD .

39. If the diagonals of a quadrilateral are equal, the quadrilateral is a rectangle.

40. One of the exterior ∠s of a right Δ may be an acute ∠.

41. If Δ ABC is divided into 2 ≅ Δs by the median drawn from vertex C, then ΔABC must be: a) isosceles b) equilateral c) right

42. The bisectors of ∠A and ∠B of ΔABC intersect at point P. The bisector of ∠C a) always passes through P b) sometimes passes through P c) never passes through P

43. If two altitudes of a given Δ fall outside then the Δ is a) right b) acute c) obtuse

44. If the point of intersection of the perpendicular bisectors of the sides of a Δ is outside the Δ, the Δ is a) acute b) right c) obtuse

45. The altitudes of a right Δ intersect a) outside the Δ b) inside the Δ c) on a vertex

46. An exterior ∠ at the base of an isosceles Δ is always a) an obtuse ∠ b) an acute ∠ c) a right ∠

47. By definition, a parallelogram is a quadrilateral a) whose diagonals intersect b) with both pairs of opposite sides are = c) with both pairs of opposite sides are parallel

48. Two consecutive ∠s of a parallelogram are always a) equal b) complementary c) supplementary

49. All quadrilaterals whose diagonals bisect each other are a) rectangles b) squares c) rhombuses

50. Which of the following statements is false? a) A parallelogram is a quadrilateral b) A rectangle is a square c) A rectangle is a parallelogram

51. Given:

CA ! CB 52. Given:

AB ! BC

CB / ! AB

AE ! EC

AD bisects !CAB Prove:

AD ! DC (do not use ! "s) Prove:

AD / ! BC 2

43

1

C

B A

D

B

CA

D

P D E

A B

C

P

Z Y

X

W

D E

F

A

CB

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BD E

C

A

F

C

D A

B

53. Given: AB = CD 54. In ΔABC,

BD!AC, AE!BC, AD = CB and

BD ! AE. Prove that Prove: AP = CP ΔABC is isosceles. 55. Given:

AB and CD lie in plane S 56. Given:

EF!CF ,

PT!S, PC " PD

CE ! DE

PA ! PB ∠FCD≅∠FDC Prove: T is the midpoint of Prove:

EF!M

AB and CD 57. Given:

AD and BC intersect at 58. Given:

WX ! WZ E,

AC ⊥ M,

AC ⊥ N,

XY! YZ

BD ⊥ M,

BD ⊥ N Prove: ΔWPZ is a right Δ Prove:

AD! BC 59. Given: F is the midpoint of

BC 60. Given:

CA ! CB

DB ! EC, DB"DF

EG!CA, DF!CB

EC!EF

DF ! EG Prove:

AF!BC Prove: ∠FDA ≅ ∠GEB 61. Given: Quadrilateral ABCD

Prove: DA + AB + BC + CD > DB + CA

T

S

P

D B C

A

M F

E

D C

M

N E

D C

B A