Geometry SOL Blitz
Congruent segments
– set lengths equal
–
A
AB  CD
B
C
D
Congruent angles
– set measures equal
–
1
1  2
2
Complementary Angles – 2 angles (adjacent or non-adjacent) that add to 90°
Supplementary Angles – 2 angles (adjacent or non-adjacent) that add to 180°
Linear pair
– 2 adjacent angles that make a line
– angles add to 180°
2
1
Vertical angles
– 2 angles on opposite sides of an “X” (see
1 and 2 )
2
– angles are congruent (set measures equal)
1
2 Column Proofs
– every statement is numbered.
– each reason matches each corresponding statement.
– can use definitions, theorems, postulates, properties, and given information to justify each step
Statements
Reasons
1.
1.
2.
2.
Conditional Statement: If hypothesis, then conclusion.
p→q
Converse: If conclusion, then hypothesis.
q→p
Inverse: If not hypothesis, then not conclusion.
~p →~q
Contrapositive: If not conclusion, then not hypothesis.
~q → ~p
p↔q
Biconditional: Hypothesis if and only if conclusion.
Law of Syllogism:
Law of Detachment
p→q
p→q
q→r
p is true
p→r
q is true
New Properties:
Reflexive Property of (Equality or Congruence)
Example: 5 = 5 (1 thing)
Symmetric Property of (Equality or Congruence)
Example: If 5 = x, then x = 5. (2 things)
Transitive Property of (Equality or Congruence)
Example: If x = y and y = z, then x = z (3 things)
2
1
3
4
6
5
7
8
Lines Cut by a Transversal
Corresponding – same relative position at points of intersection (∠1 and ∠5 or ∠3 and ∠7 for example). One angle
is between the lines, the other outside the lines.
Alternate exterior – both angles are outside the lines and on opposite sides of the transversal (∠2 and ∠8 or ∠1
and ∠7 for example)
Alternate interior – both angles are inside the lines and on opposite sides of the transversal (∠3 and ∠5 or ∠4
and ∠6 for example)
Consecutive interior – both angles are inside the lines and on same side of the transversal (∠4 and ∠5 or ∠ 3 and
∠6 for example)
If the lines are parallel, the corresponding, alternate interior, and alternate exterior angle pairs are congruent and
the consecutive (or same side) interior angles are supplementary.
Parallel lines have the same slope.
m1  m2
Perpendicular lines have slopes that are negative reciprocals.
m2  
1
m1
If you have lines on a graph grid, count blocks to determine slope and to graph lines that are parallel or
perpendicular to them.
Congruent Triangles – SAME SHAPE, SAME SIZE
5 ways to prove Triangles are Congruent: SSS, SAS, ASA, AAS, HL (right triangles only)
Once two or more triangles are proved to be congruent, all other corresponding parts of the triangles can be shown
congruent because Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
Triangle Inequalities
The lengths of any 2 sides of a triangle must add to be greater than the 3rd side of the triangle.
To determine if 3 side lengths form a triangle, add the 2 shortest and compare to the longest. If the sum of the 2 shortest
is more than the longest, it forms a triangle.
Similar figures – SAME SHAPE, DIFFERENT SIZE
3 ways to prove triangles similar: SSS, SAS, AA
Similar triangles must have congruent angles and sides must be proportional (called the scale factor). Write
proportionality statements to solve for missing sides. Set corresponding angles equal to each to find missing angles.
Perimeters of similar figures are proportional by the same scale factor as the individual pairs of
corresponding sides.
Areas of similar figures are proportional by the square of the scale factor.
Volumes of similar objects are proportional by the cube of the scale factor.
A segment in a triangle that is parallel to one of the sides of the triangle divides the sides into proportional pieces.
Right Triangles
Pythagorean Theorem:
c 2  a 2  b2
(hypotenuse (side opposite the right angle) is always c
Classifying triangles using Pythagorean Theorem (the longest side is always c)
If
If
If
c 2  a 2  b2 , the longest side is too short – acute triangle
c 2  a 2  b2 , the longest side is too long – obtuse triangle
c 2  a 2  b2 , the longest side is exactly right – right triangle
Trig Ratios: SOH CAH TOA
SOH: sine of an angle =
length of the side opposite the angle
length of the hypotenuse
CAH: cosine of an angle = length of the side adjacent to the angle
length of the hypotenuse
TOA: tangent of an angle = length of the side opposite the angle
length of the side adjacent to the angle
To find an angle, use the inverse of the trig ratios.
For example: If sin A 
opp
opp
, then A  sin 1
.
hyp
hyp
Special Right Triangles
30 -60 -90
2
hypotenuse = 2 * short leg
3
long leg = short leg *
short leg
 3
30
long leg
45 -45 -90
hypotenuse = leg *
hypotenuse
2
leg
hypotenuse
45
leg  2
If you want to go to a longer side, multiply by something; a shorter side, divide by something.
Angle of elevation – how much you have to lift your eyes from the horizontal
angle of depression
Angle of depression – how much you have to lower your eyes from the
horizontal
angle of elevation
Polygons
Sum of Interior Angles: (n  2)180
Sum of Exterior Angles: 360
For a Regular Polygon,
each interior angle is
(n  2)180
n
each exterior angle is
360
n
Parallelograms:
Opposite sides are congruent (distance formula)
Opposite sides are parallel (same slope)

Opposite angles congruent
Diagonals bisect each other (same midpoint)
Adjacent angles are supplementary
Rectangles – all the properties of parallelograms, plus
Rhombuses – all the properties of parallelograms, plus
4 congruent angles (right angles)
4 congruent sides
diagonals are congruent
diagonals are perpendicular
diagonals bisect opposite angles
Squares – all the properties of parallelograms, plus
all the properties of rectangles and rhombuses
Trapezoids – only 1 pair of parallel sides
Isosceles trapezoids – legs are congruent and each pair of base angles are congruent
Kites – two pairs of consecutive congruent sides
diagonals are perpendicular
1 pair of opposite angles congruent
Transformations
Translations – slides an object with no change in orientation, size or shape
Rotation – spins an object, no change in size or shape
Counterclockwise rotation rules:
Reflection – “mirrors” an object across some line,
no change in size or shape
90 ( x, y )  ( y,  x)
Dilation – makes an object bigger or smaller by some scaling
factor, no change in shape. Multiply every coordinate (both x and y)
180 ( x, y )  ( x,  y )
by the scale factor to get new coordinates.
270
( x, y)  ( y, x)
Vertex Location
Circles – Summary of Angles and Arcs
Measure relationships
Example
C
Inside (at center –
central angle)
»
mÐ1 = mCB
1
A
B
mÐ1 =
Inside (not at center)
1 »
»
mAB + mCD
2
(
A
)
1 »
»
mÐ2 =
mBC + mAD
2
(
D
E
2
B
1
)
C
B
On circle (both sides
inside circle – inscribed
angle)
mÐA =
1 »
mBC
2
C
A
P
On circle (one side inside
and one tangent)
mÐ1 =
1 »
mPM
2
1
M
N
T
Two secants
mÐS =
1 »
¼
mTU - mVW
2
(
V
)
S
U
W
O
T
V
Outside
One secants
& one
tangent
mÐS =
1 ¼
¼
mTUW - m VW
2
(
)
S
U
W
T
Two tangents
1 ¼
¼
mÐS =
mTUW - mTW
2
(
U
)
S
W
Summary of Tangents, Segments, and Chords
B
C
Tangent at end
of radius on a
circle
Two tangents
from same
point outside
circle
D
mÐABC = mÐABD = 90°
A
J
KJ = KI
H
K
I
DB = CB
if and only if
» = mCB
»
mDB
D
C
A
B
G
GF ^ JK
if and only if
JH = HK and
Chords – inside
circle
O
H
º = mFK
»
mJF
K
J
F
LM @ HK
if and only if
ON = OJ
H
J
K
O
M
N
L
D
A
E
(POP)1 = (POP)2
B
C
T
2 Secants from
outside circle
(WE)1 = (WE)2
U
V
S
W
S
Secant and
Tangent from
outside circle
V
(WE) = T2
T
W
U
( x  h)  ( y  k )  r
2
Equation of a circle:
2
2
Center (h, k), radius = r
X
Quadrilateral inscribed in a circle – opposite angles must be supplementary
V
S
T
W
Tessellations – a collection of figures that cover a plane with no gaps or overlaps
– sum of angles at vertices must be 360
tessellation
so no regular polygon with greater than 6 sides can used in a regular
Symmetry
– try to picture the figure “folded” over a line running through the figure. If the sides can lie right on top of each
other, it has line symmetry
– rotational symmetry happens if you can spin the object no more than 180
and it maps back onto itself.
Areas of Quadrilaterals – use SOL formula sheets
only major formula not given is area of rhombus and kites:
1
Area  d1d2
2
where d1 and d2 are the lengths of the diagonals
Similar polygons
– perimeters that are proportional by the same scale factor as the sides (sf)
– areas are proportional by the square of the scale factor (sf2)
Arc length – the length of the piece of the circle if you stretched it out straight (don’t confuse with arc measure which is
the degree measure)
– set up a proportion
length of AB Circumference of Circle

mACB
360
r
C
B
 mACB 
length of AB  
 (2 r )
 360 
A
Sector Area – the portion of the entire circle that is inside a central angle. Set up a proportion:
area of shaded sector Area of Circle

mACB
360
C
r
A
Solids – use the formula sheet
B
 mACB 
2
area of shaded sector  
 ( r )
 360 
Similar solids
Similar polygons
– straight line segments and perimeters are proportional by the scale factor (sf)
– surface areas and lateral areas are proportional by the square of the scale factor (sf2)
– volumes are proportional by the cube of the scale factor (sf3)
Use Pythagorean Theorem to find slant height or actual height of a pyramid or cone.
Remember that B in the formulas is area of the base – use other formulas to find the area