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Name ______________________________________________ Date _______ End of Module 2 Review Sheet (2.1 – 2.28) CC Geometry 1. In the diagram below, 𝑆𝑄̅̅̅̅ and 𝑃𝑅̅̅ ̅̅ intersect at T, 𝑃𝑄̅̅ ̅̅ is drawn, and 𝑃𝑆̅̅̅̅ || 𝑄𝑅̅̅ ̅̅ . Prove that ∆𝑃𝑆𝑇 ~ ∆𝑅𝑄𝑇.

2. Simplify each of the following:

A) yx275 B) 87285 yx C) 42 452 cddc�

3. Are the triangles shown below similar? Explain. If the triangles are similar, write a similarity statement. A) B) C) D) E)

Statement Reason SQ and PR intersect at T, PQ Given Is drawn and PS ll QR <STP = <QTR Vertical angles are congruent <SPT = <QRT If parallel lines are cut by a transversal then alternate interior angles are congruent PST = RQT AA=AA

Answer key

9. The scale factor between two triangles is 2. If the area of the larger triangle is 36 cm2, find the area of the smaller triangle.

10. Given ∆𝐼𝐽𝐾~∆𝐿𝑀𝑁, find the length of 𝐼�̅� and the length of 𝐼𝐾̅̅ ̅.

11. Given ∆𝐾𝐷𝐶~∆𝐾𝐿𝑀, solve for x.

12. Sam wants to enlarge a triangle with sides 3, 6 and 6 inches. If the shortest side of the new triangle is 13

inches, how long will the other two sides be? 13. Given ∆𝐾𝐷𝐶~∆𝐾𝐿𝑀, solve for x.

14. Sam wants to enlarge a triangle with sides 3, 6 and 6 inches. If the shortest side of the new triangle is 13

inches, how long will the other two sides be?

Repeat question

Repeat question

15. Given ∆𝐽𝐾𝐿~∆𝐺𝐾𝐻, solve for x. 16. Given ∆𝐴𝐵𝐶~∆𝐴𝐷𝐸, find the length of CE and BD. 17. Given ∆𝐴𝐵𝐶~∆𝑋𝑌𝑍, solve for x and y.

18. Given ∆𝐴𝐵𝐶~∆𝑋𝑌𝑍, solve for x and y.

19. Find the center of dilation for the following:

20. In the following diagram, 65

''CBDAreaABDArea . If CD = 12, find: (a) AD (b) AC

21. Triangle ABC has a right angle at A and AD is the altitude from A to hypotenuse BC. Solve for x.

A) B)

22. The diagram below shows ∆𝐴𝐵𝐶, with 𝐴𝐸𝐵̅̅ ̅̅ ̅̅ , 𝐴𝐷𝐶̅̅ ̅̅ ̅̅ , and . Prove that ∆𝐴𝐵𝐶 is similar to ∆𝐴𝐷𝐸.

23. In the diagram below, , , , and . Prove that .

Statement Reason <ACB=<AED Given <A=<A Reflexive Property ABC ADE AA=AA

Statement Reason BFCE, AB l BE, DE l BE. Given <BFD=<ECA <B and <E are right angles Perpendicular lines form right angles <B =<E All right angles are congruent <BFD + <DFE = 180 Linear pairs form supplementary <ECA + <ACB = 180 angles <BFD+<DFE =<ECA+ Substitution <ACB <DFE=<ACB Subtraction Postulate ABC DEF AA=AA

24. Dean constructed the composition of dilations shown below. Triangle A’B’C’ is 1/2 times the size of triangle ABC, and triangle A”B”C” is 5

2 the size of triangle A’B’C’.

a) Determine the scale factor from ∆𝐴𝐵𝐶 to ∆𝐴"𝐵"𝐶".

b) Find the center of dilation mapping ∆𝐴𝐵𝐶 to ∆𝐴′𝐵′𝐶 and label O1.

c) Find the center of dilation mapping ∆𝐴′𝐵′𝐶′ to ∆𝐴"𝐵"𝐶" and label O2.

d) What is the relationship between the centers of dilations found is parts (b) through (c)?

25. A similarity transformation for triangle 𝐴𝐵𝐶 is described by (𝐷𝑂,2 (𝑟𝐷𝐺 ⃡ (∆𝐴𝐵𝐶))) . Locate and label the

image of triangle 𝐴𝐵𝐶 under the similarity.

26. In the diagram below, 𝐴𝐵̅̅ ̅̅ || 𝐷𝐸̅̅ ̅̅ . If AB = 20 inches, DE = 12 inches, and BC = 15 inches, what is the length of 𝐷𝐶̅̅ ̅̅ ?

27. The scale factor between two triangles is 5. If the area of the smaller triangle is 20 in2, find the area of the

larger triangle. 28. Triangle ABC has a right angle at A and AD is the altitude from A to hypotenuse BC. Solve for x.

29. In the diagram to the right, 𝐴𝐶 = 35. If the ratio of the areas of the triangles is Area ∆𝐴𝐵𝑇Area ∆𝐶𝐵𝑇

= 25, find x and y.

30. Find the missing length indicated for the diagrams below.

A) B) C)

31. Triangle ABC has a right angle at A and AD is the altitude from A to hypotenuse BC. Solve for x. A) B)

32. Solve for the missing side lengths and express your answers in simplest radical form.

A) B) C) D) E) F)

33. Find the value of each trig ratio

A) Tan Z B) Cos C C) Sin C

34. Find the measure of the indicated angle to the nearest tenth of a degree. A) B) C)

35. Find the missing side to the nearest hundredth.

A) B) C) D) 36. In the accompanying diagram, x represents the length of a ladder that is leaning against a wall of a

building, and y represents the distance from the foot of the ladder to the base of the wall. The ladder makes a 60° angle with the ground and reaches a point on the wall 17 feet above the ground. Find, to the nearest foot, the measure of x and y.

37. A 10-foot ladder is placed against the side of a building as shown in figure 1 below. The bottom of the ladder is 8 feet from the base of the building. In order to increase the reach of the ladder against the building, it is moved 4 feet closer to the base of the building as shown in figure 2. To the nearest foot, how much further up the building does the ladder now reach?

38. A cable 20 feet long connects the top of a flagpole to a point on the ground that is 16 feet from the base of the pole. How tall is the flagpole?

39. Find the value of x that makes each statement true.

a. Sin (x + 2) = Cos (2x – 11) d. Sin (x) = Cos 54

40. Calculate the missing side length, CB, to the nearest hundredth.

41. Calculate the missing side length, CB, to the nearest hundredth.

42. If cos 𝜃 = 45, find sinT and tanT in simplest radical form if necessary.

43. If sin 𝜃 = 3

4, find cosT and tanT in simplest radical form if necessary.

44. If tan 𝜃 = 10, find sinT and cosT in simplest radical form if necessary.

45. If cos 𝜃 = 22 , find sinT and tanT in simplest radical form if necessary.

46. If sin 𝜃 = 66 , find cosT and tanT in simplest radical form if necessary.

47. a. State whether each triangle should be solved by using the Law of Sines or the Law of Cosines. b. Find the value of x in each triangle. Round your answers to the nearest tenth. A) B) C)

48. Triangle MNO has adjacent sides with length 12 and 17 and an included angle of 41o. Find the area of the

triangle to the nearest hundredth.

49. Find to the nearest integer the area of an isosceles triangle if the measure of the vertex angle is q42 and the measure of each of the congruent sides is 12.

50. In the accompanying diagram, the base of a 15-foot ladder rests on the ground 4 feet from a 6-foot fence.

a.) If the ladder touches the top of the fence, and the side of the building, what angle to the nearest degree does the ladder make with the ground?

b.) Using your answer from part a, find how high to the nearest tenth of a foot the ladder reaches up the side of the building.

51. For parallelogram ABCD below, find acute angle of <DBC to the nearest whole degree.

52. For the figure below, CD = 15, find BC to the nearest whole number.

53. A lighthouse is built on the edge of a cliff near the ocean, as shown in the accompanying diagram. From a boat located 200 feet from the base of the cliff, the angle of elevation to the top of the cliff is 18o and the angle of elevation to the top of the lighthouse is 28o. What is the height of the lighthouse, x, to the nearest tenth of a foot?