Download Add`l Geometry Review Guides

Name ________________ Geometry Exam Review #1: Constructions and Vocab
Copy an angle:
1. Place your compass on A, make
any arc. Label the intersections
of the arc and the sides of the
angle B and C.
A
2. Compass on A’, make the same
arc from #1. Label the
intersection B’.
3. MEASURE from B to C. you will
have to adjust your compass.
4. with your compass open from
#3, compass on B’, make an arc.
Where this arc crosses the arc
from step 2 label C’. Connect A’
and C’.
A’
Bisect an angle:
1. Compass on A, make any arc.
Label where the arc crosses the
sides of the angle as B and C.
2. compass on B, make an arc
3. do not change compass:
compass on C make an arc
4. label where arcs intersect D
A
Parallel Lines:
1. Place point P above a line. Connect P
to the line on some kind of angle/slant.
Label the intersection of the lines Q.
2. compass on Q. Draw any arc. Label
the intersection points A and B.
3. compass on P. Draw the same arc
from #2. Label the intersection point C.
4. MEASURE from A to B – you will have
to adjust your compass.
5. with your compass open the distance
from A to B, place your compass on C
and make an arc, label the intersection
D.
6. connect P and D.
⃡
⃡
* Q and P are congruent corresponding
angles
Perpendicular Bisector:
1. open compass to a little more than half
the segment. Compass on A, make an arc.
2. do not change compass!! Compass on B,
make an arc.
3. arcs must cross each other!! Label these
points C and D. Connect C and D.
4. label the intersection of ̅̅̅̅ and ̅̅̅̅ E.
̅̅̅̅
̅̅̅̅, ̅̅̅̅
̅̅̅̅
*also, if you place any point along the
perpendicular bisector (̅̅̅̅) it is equidistant
to the endpoints
Perpendicular from a point to a line
1. place point P above the line
2. compass on P, make an arc so it crosses
the line twice. Label these A and B.
3. widen your compass a bit. Compass on
A, make an arc. Without changing
compass, compass on B, make an arc.
4. label where the arcs intersect as C.
Connect to P.
̅̅̅̅
̅̅̅̅
A
B
Incenter:
These all mean the same thing:
Find the incenter
Find the center of the circle that is inscribed
in a triangle
Construct the incircle, inscribed circle
Steps:
Construct each angle bisector of a triangle.
The bisectors are concurrent at the incenter.
Call the incenter C.
Construct a line perpendicular from C to one
of the sides of the triangle. This will be the
radius of the circle.
Compass on C, open to the length of the
radius, draw a circle. It should just graze the
sides of the triangle.
*the incenter is equidistant to the sides of
the triangle because it is the radius of the
circle.
Circumcenter:
These all mean the same thing:
Find the circumcenter
Find the center of the circle that you can
circumscribe about a triangle
Construct the circumcircle, circumscribed circle
Steps:
1. construct the perpendicular bisectors of
each side of the triangle. These lines are
concurrent at the circumcenter. Label the
circumcenter C.
2. Compass on C, open it to one of the vertices
(corners) of the triangle. This is the radius of
your circle. Draw a circle.
*The circumcenter is equidistant to the
vertices of the triangle because it is the
radius of the circle.
Square:
1. Draw the diameter of the circle. Label
the endpoints A and B.
2. Construct the perpendicular bisector
of ̅̅̅̅. Make sure this line goes all the
way through the circle. Label where it
crosses the circle C and D.
3. Connect A, B, C, D
Hexagon:
1. Label the center of the circle O.
Place any point A on the circle.
2. open your compass the length of ̅̅̅̅.
This is the radius of your circle.
3. Compass on A, make an arc crossing
the circle. Label it B.
4. compass on B, make an arc. Label it C.
Continue around the circle.
5. Connect A & B, B & C, and so on.
Equilateral triangle:
Label the center of your circle O. Place any
point A on the circle.
2. open your compass the length of ̅̅̅̅. This
is the radius of your circle.
3. Compass on A, make an arc crossing the
circle. Label it B.
4. compass on B, make an arc. Label it C.
Continue around the circle.
5. Connect EVERY OTHER letter: A and C, C
and E, E and A.
For additional help on constructions go to mathopenref.com
Three undefined terms in geometry:
____________________________
____________________________
____________________________
Three transformations that use rigid motions to produce congruent figures:
___________________________
___________________________
___________________________
Congruent means: _____________________________________________
When figures are congruent, their sides are __________________________
and their angles are ______________________________.
The transformation that produces similar figures: _______________________
When figures are similar, their sides are ______________________________
and their angles are __________________________.
Medians are concurrent at the ____________________________________
Altitudes are concurrent at the ____________________________________
Angle bisectors are concurrent at the ________________________________
Perpendicular bisectors are concurrent at the __________________________
Name __________________________ Support Exam Review #2: Transformations
Translations (_________________)
Notation: a translation of left 3 up 4 can also be written:
Vector notation: 〈
〉
Arrow notation/rule: (x, y)
(x – 3, y + 4)
1. ______ Which is the same as a translation of right 5 and down 7?
A. 〈
〉
B. 〈
〉
C. 〈
〉
D. 〈
〉
2. ______ Which is the same as a translation of left 6?
A. 〈
〉
B. 〈
〉
C. 〈
〉
D. 〈
〉
3. ______ Which is the same as a translation of left 3 down 8?
A. (x, y)
(x + 3, y + 8)
B. (x, y)
(x – 3, y + 8)
C. (x, y)
(x + 3, y – 8)
D. (x, y)
(x – 3, y – 8)
4.
5.
Reflections (_________________):
Image should be equidistant from the reflection line
6.
9.
7.
8.
10.
11. Reflect over y = x
Rotations (____________________):
Clockwise:
counterclockwise:
12.
13.
14.
270
Dilations:
Sides are proportional, angles are congruent, shapes are similar
What is the ratio of FG to F’G’?
What is the ratio of F’G’ to FG?
The center of dilation:
This picture is modeling that the new image is twice the
distance from the origin (center (0, 0))
15. find the scale factor:
16. Dilate the triangle with scale factor 2, center (0, 0)
Image Rules:
V (2, -3) V’ ________
R (5, -3) R’ ________
Z (4, 2) Z’ _________
Rule for reflection over x-axis
(x, y)
A (0, 2) A’ _______
L (2, 0) L’ ________
V (5, 4) V’ ________
Rule for reflection over y-axis
(x, y)
T (-4, -2) T’ _______
G (-2, -1) G’ _______
W (-2, -4) W’ _______
Rule for rotation 90° clockwise
OR 270° counterclockwise
(x, y)
E (2, 1) E’ ________
N (4, -1) N’ _______
F (4, 4) F’ ________
Rule for rotation 180°
(x, y)
Rotate 270° clockwise about (0,0)
T (-3, 5) T’ _______
H (-3, 2) H’ _______
D (-4, 3) D’ ________
Rule for rotation 270° clockwise
OR 90° counterclockwise
(x, y)
Practice: 1. What is the scale factor? What is the center of the dilation?
Name _____________________________ Support Exam Review 3
Midpoint Formula:
Find the midpoint of each side of the triangle:
Distance formula:
Find the perimeter of the triangle:
(find the length of each side, add)
Find the perimeter of the triangle:
(hint – because it is a right triangle you can
use Pythagorean theorem)
Find the area of the triangles: (A = ½ bh)
Practice:
Midpoint of RC _____
Midpoint of CD _____
Midpoint of RD _____
Length RC ________
Length CD ________
Length RD ________
Perimeter of triangle ____________
Area of triangle ________________
Name ____________________________ Support Exam Review 4
Congruent Angles
Vertical angles
1 = 74° 4 = ________
6 = (9x + 5)°, 7 = (3x + 29)°
X = _____
6 = ____
7 = ____
Alternate interior angles
3 = 88° 6 = ________
4 = (2x – 7)° 5 = (x + 30)°
X = _____
4 = ____
5 = ____
Corresponding angles
3 = 72° 7 = ________
3 = (5x + 20)° 7 = (-3x + 52)°
X = _____
3 = ____
7 = ____
Alternate exterior angles
Supplementary Angles
Linear Pair
1 = 74° 2 = ______
5 = (5x + 25)° 7 = (5x + 15)°
X = _____
5 = ____
7 = ____
Same Side Interior (Consecutive interior)
4 = 110° 6 = ______
3 = (15x- 10)° 5 = (15x + 10)°
X = _____
3 = ____
5 = ____
Same Side Exterior
1 = 123° 7 = ______
1 =(20x + 6) ° 8 = (5x + 24)°
X = _____
1 = _____
8 = _____
Triangle Angle Sum Theorem: the sum of the angles in a
triangle = 180° (X + Y + Z = 180)
Exterior Angle Theorem: The exterior angle of a triangle =
sum of two remote interior angles (W = X + Y)
Examples:
1.
2.
3.
4.
5.
6.
Name _____________________________ Support Exam Review 5
Parallel lines have the same slope and different y-intercepts
Write an equation of the line parallel to the given line passing through the given
point:
1. y = 2x + 4 (-3, 1)
ଶ
ଵ
2. y = - x + 1 (6, -2)
3. y = x – 4 (-4, 6)
4. 4y = 8x + 10
5. 3x + 6y = 12 (-2, 5)
ଷ
(-1, 3)
ଶ
Perpendicular Lines have opposite reciprocal slopes
Write an equation of the line perpendicular to the given line through the point
ଶ
6. y = - x + 5 (2, 3)
ଷ
7. y = 2x – 3 (-4, 1)
9. 3y + 6 = 4x
(4, 8)
8. Y = -3x + 1 (3, 6)
10. 2x – 4y = -12 (-2, 5)
Practice:
Write an equation of the line parallel to the given line through the given point:
ସ
1. y = x + 3 (5, 2)
2. Y = -3x – 7
3. 3y + x = 6 (3, 1)
4. 2x + y = 4 (-3, 5)
5. 4y – 8 = 3x (-4, 4)
6. -4x + 2y = 10
ହ
(1, 1)
(3, 5)
Write an equation of the line perpendicular to the given line through the point:
ଵ
7. y = - x + 5 (-2, 6)
8. Y = 3x – 2 (-6, 8)
9. 4y = 5x + 12
10. 3x – 2y = 6
ସ
ଶ
(-5, 2)
11. y + 5 = x (-8, 2)
ଷ
12. Y + 2x = 6
(6, 4)
(2, 1)
Your exam has 40 questions. If you google the standard, look for the Shmoop.com
entry. It will have additional practice problems (sample assignments)
G.CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line,
and line segment, based on the undefined notions of point, line, distance along a
line, and distance around a circular arc.
1.
3.
2.
4. G.CO.2: Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in the
plane as inputs and give other points as outputs. Compare transformations that
preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
5.
7.
6.
8.
G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe
the rotations and reflections that carry it onto itself.
9.
11.
10.
12.
13.
G.CO.4 Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
14.
15.
16.
G.CO.5:Given a geometric figure and a rotation, reflection, or translation, draw
the transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure onto
another.
17.
G.CO.6: Use geometric descriptions of rigid motions to transform figures and to
predict the effect of a given rigid motion on a given figure; given two figures, use
the definition of congruence in terms of rigid motions to decide if they are
congruent.
18.
19.
20.
21.
22.
A triangle was rotated 90 degrees
counterclockwise and then translated 2
down and 4 left. The final coordinates of
the triangle are: (1, -3), (-2, 0), and (3,
2). What were the original coordinates?
A (1, 1) (4, 4) (6, -1)
B (0, 2) (2, -3) (-3, -1)
C (-1, 3) (2, 0) (-3, -2)
D (3, 7) (8, -2) (6, 4)
G.CO.7 Use the definitions of congruence in terms of rigid motions to show that
two triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
23.
24.
25.
26.
G.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS)
follow the definition of congruence in terms of rigid motions.
27.
28.
G.CO.9: Prove theorems about lines and angles. Theorems include: vertical
angles are congruent; when a transversal crosses parallel lines, alternate interior
angles are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from the
segment's endpoints.
29.
31. 30.
32. 33.
G.CO.10: Prove theorems about triangles. Theorems include: measures of
interior angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel to
the third side and half the length; the medians of a triangle meet at a point.
34.
35.
36.
G.CO.11: Prove theorems about parallelograms. Theorems include: opposite
sides are congruent, opposite angles are congruent, the diagonals of a
parallelogram bisect each other, and conversely, rectangles are parallelograms
with congruent diagonals.
37.
38.
G.CO.12: Make formal geometric constructions with a variety of tools and
methods (compass and straightedge, string, reflective devices, paper folding,
dynamic geometric software, etc.). Copying a segment; copying an angle;
bisecting a segment; bisecting an angle; constructing perpendicular lines,
including the perpendicular bisector of a line segment; and constructing a line
parallel to a given line through a point not on the line.
39. 40.
41.
G.CO.13: Construct an equilateral triangle, a square, and a regular hexagon
inscribed in a circle.
G.SRT.1: Verify experimentally the properties of dilations given by a center and a
scale factor:
42.
43.
44.
G.SRT.2: Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs of
sides.
G.SRT.3: Use the properties of similarity transformations to establish the AA
criterion for two triangles to be similar.
45.
46.
47.
G.SRT.4: Prove theorems about triangles. Theorems include: a line parallel to
one side of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
48.
G.SRT.5: Use congruence and similarity criteria for triangles to solve problems
and to prove relationships in geometric figures.
G.C.3: Construct the inscribed and circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle.
49.
50.
51.
A (4, 6)
B (6, 4) C (3, 3) D (1, 2)
G.GPE.5: Prove the slope criteria for parallel and perpendicular lines and use
them to solve geometric problems (e.g., find the equation of a line parallel or
perpendicular to a given line that passes through a given point).
G.GPE.6: Find the point on a directed line segment between two given points
that partitions the segment in a given ratio.
52.
53.
54.
55.
Where should you plot point X so that
PX is of the length of PQ?
A. (4, 0)
B (6, 0)
C (7, 0)
D (8, 0)
Where should you plot X so it divides PQ in a
ratio of 3:4?
A. (5 , 0)
B (4 , 0)
C 4 , 0)
D (3 , 0)
G.GPE.7: Use coordinates to compute perimeters of polygons and areas of
triangles and rectangles, e.g., using the distance formula.
56.
57.
58.
59.
60 .
Additional midterm review key 1. A 2. D 3. A 4. B 5. B 6. B 7. A 8. C 9. D 10. D 11. A 12. C 13. B 14. A 15. B 16. B 17. A 18. D 19. C 20. A 21. C 22. A 23. D 24. D 25. C 50. C 26. C 51. A 27. C 52. B 28. B 53. C 29. B 54. A 30. B 55. D, A 31. C 56. B 32. D 57. D 33. D 58. C 34. A 59. B 35. C 60. C 36. B 37. D 38. A 39. A 40. A 41. B 42. B 43. D 44. C 45. C 46. B 47. B 48. C 49. C