Geometry L2 Name:

Geometry L2 Name: _________________________________ Midterm Exam Review The midterm exam will cover the topics listed below from Units 1, 2 and 3. A formula sheet will be provided. Unit 1 – Transformations Pythagorean Theorem Simplifying radicals and solving application problems Distance and Midpoint on the Coordinate Plane

Distance Formula: 𝑑 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2

Midpoint Formula 𝑀 (𝑥1+𝑥2

2,

𝑦1+𝑦2

2)

Partition a Segment Find a partition point given a fraction, ratio, or percentage Isometries Translations Using Mapping Notation and Vector Notation

Naming Vectors, Component Form, Length of a Vector: √𝑎2 + 𝑏2 Reflections Over x-axis, y-axis, y = x, y = -x Over vertical lines (x = ____); Over horizontal lines (y = ____) Rotations 90°, 180°, 270° about origin on coordinate plane Composition of Transformations on the coordinate plane Dilations – Graph a figure and its dilation on the coordinate plane Symmetry - Determine Line Symmetry and Rotational Symmetry of figures Unit 2 – Congruence and Proof Angles Formed by Parallel Lines and Transversals Corresponding, Alternate Interior, Alternate Exterior, Consecutive (Same-side) Interior Vertical Angles, Linear Pairs, Complementary and Supplementary Angles Triangle Sum Theorem Exterior Angle Theorem Congruent Triangles/Congruent Polygons - Corresponding Parts

Proving Triangles Congruent – SSS, SAS, ASA, AAS, HL CPCTC Two-Column Proofs involving congruent triangles and CPCTC Isosceles and Equilateral Triangles – Properties and Theorems

Unit 3 - Polygons Polygon Angle Sum Theorems Interior Angle Sum: (n – 2) 180° Exterior Angle Sum: 360°

Regular Polygons: Each Interior Angle: (𝑛−2)180°

𝑛

Each Exterior Angle: 360°

𝑛

The best way to prepare for your Geometry Midyear Exam is to complete the problems in this review packet and to study your tests and quizzes from units 1, 2, and 3. I. Use the Pythagorean Theorem to find the length of the missing side in each triangle. Write answers in simplest radical form. 1. 2.

x=___________ x=___________ 3. 4.

x=___________ x = ___________

8

5

x

12

√5

√11

x

x 4√3

8

x 12

5. Find the height and the area of a rectangle which has a diagonal of length 26 and a base of length 24. Height = ________ Area = __________ 6. An isosceles triangle has legs of length 34 and a base of length 32. Find its height and its area.

height = __________ Area = ____________ II. Distance and Midpoint Formulas 7. On the coordinate plane, 𝑅𝑆̅̅̅̅ has coordinates R (2, - 7) and S (-4, 1).

a. Find the length of . b. Find the midpoint (M) of .

c. Use the distance formula to prove that M is the midpoint of . 8. M is the midpoint of 𝑋𝑌̅̅ ̅̅ . If X has coordinates (-3, 5) and M has coordinates (1, 0), find the coordinates of point Y.

RS RS

RS

III. Partition A Segment 9. Point A has coordinates (-3, 2). Point B has coordinates (3, -7).

Point C is located 2/3 of the way from A to B. Find the coordinates of point C.

10. Point G has coordinates (4, 3). Point H has coordinates (-1, -7).

Point J divides GH in a 1:4 ratio. Find the coordinates of point J. IV. Transformations 11. Use the translation (x, y) → (x + 3, y – 4):

a. What is the image of D (4, 7)? b. What is the pre-image of M’ (-5, 3)?

12. The vertices of Δ MNO are M (-2, 4), N (-1, 1), and O (3, 3). Graph Δ MNO and its image using prime notation after

the translation (x, y) → (x + 4, y – 2):

M’: ________ N’: ________ O’: ________

13. ΔR’S’T’ is the image of ΔRST after a translation. Write a rule for the translation in mapping notation and in vector

notation. Mapping Notation: Vector Notation:

14. Name the vector, write its component form, and find its length:

a. b.

15. Write the component form of the vector that describes the translation from S (-3, 2) to S’ (6, -4). 16. The vertices of ΔABC are A (0, 4), B (2, 1) and C (4, 3). Graph and label the coordinates of ΔA’B’C’ after each

transformation. a. Translate ΔABC using the vector ⟨−3, 1⟩. b. Reflect ΔABC over the x-axis.

R

S

T S’

T’

R’

c. Reflect ΔABC over the line y = - x. d. Reflect ΔABC over the line y = -1.

17. The vertices of ΔABC are A (-3, 1), B (1, 1) and C (1, -2). Reflect ΔABC over the line x = 2. Then reflect ΔA’B’C’ over

the line y = -3. Graph ΔABC, ΔA’B’C’, and ΔA”B”C”. State the coordinates of ΔA”B”C”.

A”:________ B”:________ C”:________

18. The coordinates of ∆ABC are A (0, 4), B (3, 6), and C (5, 2). Graph ∆ABC. Rotate ∆ABC 90°, 180°, and 270° counterclockwise about the origin. Record the coordinates after each rotation.

After a 90° Rotation: A’ ________ B’ _______ C’ ________

After a 180° Rotation: A’ ________ B’ _______ C’ ________

After a 270° Rotation: A’ ________ B’ _______ C’ ________

19. List the image of each of the following points after the specified composition of transformations:

a. If point A (-2, 5) is reflected in the y-axis,

and then point A’ is reflected in the x-axis, the coordinates of point A’’ are _________.

b. If point B (-4, -2) is reflected over the line y =- x, and then point B’ is rotated 90° counterclockwise about the origin, the coordinates of B’’ are _________.

c. If point C (6, -3) is reflected over the line y = x, and then point C’ is rotated 270° counterclockwise about the origin, the coordinates of C’’ are _________.

d. If point D (-2, 10) is rotated 180° about the origin, and then point D’ is reflected over the line y=-5, then the coordinates of D’’ are ________.

20. Point P (-6, 2) is transformed to point P’ (2, 6). What is the transformation that maps P into P’? Explain. 21. The vertices of ∆ABC are A (2, 4), B (7, 6) and C (5, 2). Graph the image of ∆ABC after a composition of the

transformations in the order they are listed.

Transformation: (𝑥, 𝑦) → (𝑥 + 2, 𝑦 – 4) Rotation: 180° about the origin Record the coordinates of ∆A”B”C”: A” _________ B” __________ C” __________

22. The coordinates of ∆PQR are: P (6, 2), Q (-6, 4), R (0, -4).

a. Draw the dilation of ∆PQR centered at the origin with a scale factor of 2

P’ ________ Q’ _________ R’ __________

b. Draw the dilation of ∆PQR centered at the origin with a scale factor of ½

P’ ________ Q’ _________ R’ __________

V. Symmetry 23. State the number of lines of symmetry and the angle(s) of rotational symmetry of each of the following figures: a. Rectangle b. Isosceles Triangle c. Square # of Lines:_______ # of Lines:_______ # of Lines:_______ Angle(s):____________ Angle(s):_____________ Angle(s):_____________ d. Regular Hexagon e. Equilateral Triangle f. Parallelogram # of Lines:_______ # of Lines:_______ # of Lines:_______ Angle(s):____________ Angle(s):_____________ Angle(s):_____________

4

2

3

5

m

n

6 7

8

125°

x 35° 85°

y

33° x°

y° 25°

23°

VI. Parallel Lines Intersected by a Transversal 24. Name the following pairs of angles in the diagram below:

Corresponding Angles (4 pairs) –

Alternate Interior Angles (2 pairs) –

Alternate Exterior Angles (2 pairs) –

Consecutive (Same – Side) Interior Angles (2 pairs) -

Vertical Angles (4 pairs) –

Linear Pairs (name 4 of the 8 in the diagram) –

25. In the figure above, if line m is parallel to line n, and the measure of angle 1 is 106°, find the measures of the other seven angles:

m < 2 = ______ m < 3 = ________ m < 4 = ________ m < 5 = ________

m < 6 = ______ m < 7 = ________ m < 8 = ________

VII. Angles of Triangles

Find the missing angle(s) in each of the following figures:

26. 27.

28. 29.

1

A

D B C

C

5x°

7x+42° 18x°

y

115° 135°

x

y

30. 31.

32. 33.

34. Angle DBA is an exterior angle of ∆ABC. Find the measure of angle ABC.

m < ABC = ________

35. In ΔDEF, m < D = 7x + 10°, m < E = 9x -1°, and m < F = 3x + 38°. Find the measures of the angles of the triangle. What type of triangle is ∆DEF?

m < D = _______

m < E = _______

m < F = _______

Triangle Type: _________________

A

B C

P

Q R

90°

23°

12

5

VIII. Congruent Triangles

36. ∆𝐴𝐵𝐶 ≅ ∆𝑃𝑄𝑅 m < P = _______ m < Q = _______ PQ = ________ QR = ________

Determine the theorem (if any) which can be used to prove the triangles congruent.

37. ___________ 38. ___________ 39. ____________

40. ___________ 41. ___________ 42. ___________

Determine the third congruence which is needed to prove the triangles congruent by the indicated method.

43. SAS 44. ASA 45. AAS 46. HL

A

C B

E F

D

Statements Reasons

Statements Reasons

Statements Reasons

IX. Two – Column Proofs

47. Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐷̅̅ ̅̅

𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝐷̅̅ ̅̅

Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶

48. Given: 𝑊𝑉̅̅ ̅̅ ̅ 𝑎𝑛𝑑 𝑋𝑇̅̅ ̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟

Prove: ∆𝑋𝑊𝑌 ≅ ∆𝑇𝑉𝑌

49. Given: 𝐷𝐵̅̅ ̅̅ ⊥ 𝐴𝐶̅̅ ̅̅

𝐴𝐵̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅

Prove: < 𝐴 ≅< 𝐶

Statements Reasons

N

T

M

Q

x

50. Given: 𝑁𝑇̅̅ ̅̅ ≅ 𝑀𝑄̅̅ ̅̅ ̅ 𝑁𝑇̅̅ ̅̅ || 𝑀𝑄̅̅ ̅̅ ̅ Prove: < 𝑄 ≅ < 𝑁

X. Isosceles and Equilateral Triangles

51. 52. x = ______

y = ______

𝑚 < 1 = ____ 𝑚 < 2 = ______

𝑚 < 3 = ______ 𝑚 < 4 = ________

53. 54.

a= _______ x = ________ b = ________

c = _______ d = ________

130°

y x

A

B C

S

Q R

R Q

S

55. In ΔABC, m < A = 4x + 22°, m < B = 12x - 2°, and m < C = 8x + 16°. Is ΔABC isosceles? Explain why or why not.

56. ΔQRS is isosceles with base 𝑄𝑅̅̅ ̅̅ . If SQ = 14x – 3 cm, SR = 8x + 9 cm, and QR = 5x + 13 cm, find the perimeter of the triangle.

Perimeter = ____________

57. ΔQRS is isosceles with base 𝑄𝑅̅̅ ̅̅ . If m < Q = (6x + 3)° and m < R = (3x +21)°, find the measures of the angles of the triangle.

m < Q = _______

m < R = _______

m < S = _______

XI. Angles of Polygons

58. Find the interior angle sum and the exterior angle sum of a 15-gon.

Interior Angle Sum = ________

Exterior Angle Sum = _______

59. A polygon has an interior angle sum of 2880°. How many sides does it have?

60. Find the measure of each interior angle and each exterior angle of a regular nonagon.

Each Interior Angle = _________

Each Exterior Angle = _________

61. Each exterior angle of a regular polygon has a measure of 30°. Name the polygon.

62. Each interior angle of a regular polygon has a measure of 135°. Name the polygon.