Download Geometry—Segment 1 Reference Sheet

Geometry—Segment 1 Reference Sheet
Module 1
 You will need many tools for geometry such as a compass and straightedge,
writing utensil, geometric drawing software, and even paper.
A postulate is considered a known fact. Theorems must be
proven to be true.
° Negation: stating the opposite of a statement
° Bi-conditional: statement in the form “if and only if”
Constructions are the most accurate and detailed depiction,
followed by drawings and then sketches.
 Undefined terms will be used as foundational elements in
defining other “defined” terms. The undefined terms include
point, line, and plane.
Algebraic Properties
Example
Sample Conditional Statement
If it is cloudy outside, then it is raining.
Commutative (of addition and
mult.)
3+2+5=5+2+3
Associative (of addition and mult.)
6 x 5 x 2 = (6 x 5) x 2
Distributive
4(x - 2) = 4x - 8
Symmetric
x = y and y = x
Transitive
If a = b and b = c, then a = c
Reflexive
x=x
Module 2
Transformations Rules
Translations
(x, y) -> (x+1, y-2) sample
Reflections
x-axis: (x, y) -> (x, -y)
y-axis: (x, y) -> (-x, y)
line y=x: (x, y) -> (y, x)
Rotations
90 clockwise: (x, y) -> (y, -x)
90 counterclockwise: (x, y) -> (-y, x)
180 : (x, y) -> (-x, -y)
Inverse
Opposite of the original but in the same order
If it is not cloudy outside, then it is not raining.
Converse
Same as the original, but in a different order
If it is raining, then it is cloudy outside.
Contrapositive
(logically equivalent)
Opposite of the original AND in a different order
If it is not raining, then it is not cloudy outside.
° Translational: pattern is moved
˄, ˅, ˂, or ˃ without changing.
° Reflectional: pattern may be
flipped over a line without
changing.
° Rotational: pattern may be
turned about a fixed point without
changing.
Corresponding parts of triangles can be labeled for all
six parts of two triangles. You
need a minimum of 3 specific
parts to declare two triangles
congruent using SSS, SAS, or
ASA.
° Side-Side-Side, SSS: If the corresponding sides of one triangle are congruent to the corresponding sides of a second triangle, then the triangles are congruent.
° Side-Angle-Side, SAS: If two corresponding sides and the included angle of one triangle are congruent to two corresponding sides and the included a ngle of
another triangle, then the triangles are congruent.
° Angle-Side-Angle, ASA: If two corresponding angles and the included side of one triangle are congruent to two corresponding angles and the included side of
another triangle, the triangles are congruent.
Module 3
1 & 2 are adjacent angles (supplementary)
1 & 3 are vertical angles (congruent)
1 & 5 are corresponding angles (congruent)
4 & 5 are same-side interior angles (supplementary)
1 & 8 are same-side exterior angles (supplementary)
4 & 6 are alternate interior angles (congruent)
Properties of a parallelogram:
1.
The opposite sides are congruent and parallel
2.
The opposite angles are congruent
3.
The diagonals bisect each other
4.
Consecutive angles are supplementary
Properties of a Rectangle:
Properties 1 through 4 of a parallelogram and
5. Contains four right angles
6. The diagonals are congruent
Steps for creating an Indirect Proof:
1.
Assume the opposite of the conclusion (or prove statement).
2.
Reason logically to show the assumption leads to a contradiction of a
known fact. Be sure to explain your whys
3.
Conclude the assumption is false, which in turn proves the conclusion is
true.
Triangle sum theorem: The sum of the
measures of the angles in a triangle will
always add up to 180 degrees
° Isosceles Triangle Theorem: If two sides
of a triangle are congruent, the angles
opposite them are congruent 
Converse: If two angles of a triangle are
congruent, the sides opposite them are
congruent.
Equidistance of a point on a perpendicular bisector of
a Segment: A point that lies on the perpendicular bisector
of a line segment is equidistant (or the same distance
away) from the endpoints of the line segment.
Mid-segment of a Triangle Theorem: A line segment
connecting the midpoints of two sides of any triangle is
parallel to the third side and half its length.
 Concurrency of Medians Theorem: The three medians
of any triangle intersect at one point called the centroid.
Geometry—Segment 1 Review
Need to dilate the image?
Module 4
Multiply the scale factor by the original coordinates
(x,y) to find the new coordinates for the dilation image.
° Dilation: When a figure is dilated from the origin, each ordered pair of the
image may be found according to the rule (x, y) → (kx, ky) where k is the scale
factor.
Ex: B (-1,2) Scale Factor is 2.
x(-1*2) = -2 and y(2*2) = 4 —-> B’ (-2,4)
° Scale Factor: the constant by which a figure (or the dimensions of a figure)
Need to find the scale factor?
Scale Factor > 1 = Enlargement
Divide the coordinates (x,y) of the original figure by the
new coordinates of the dilated.
0 < Scale Factor < 1 = Reduction
**When dilating a figure, the rule (x, y) → (kx, ky), where k is the scale factor,
can be seen as a function machine. Remember that (x, y) is the input and (kx,
ky) is the output.
Ex: B (-1,2) / B’ (-2,4) ——> -1/-2 = 1/2 and 2/4 = 1/2
° Similar polygons are polygons that have congruent angles and corresponding sides that are proportional to one another.
° Corresponding sides of similar triangles are proportional and will have
equal ratios
° Similar polygons have the same shape but different sizes.
° Corresponding angles of similar triangles are congruent
WAYS TO WRITE RATIOS
° A ratio shows the relative sizes of
two or more values and can be
written as a fraction or with a colon
symbol separating the values.
a/b = 1/2 or a:b = 1:2
Is Rectangle ABCD SIMILAR to EFGH?
Take a look at the corresponding sides. Are
they in proportion?
° Angle-Angle (AA) Similarity Postulate:
If two corresponding angles of two or
more triangles are congruent, the triangles are similar
° Similar triangles: have corresponding parts that form a proportion with
their corresponding sides.
Module 5
Triangle Proportionality Theorem
° Congruent triangles - have the same shape AND size
° CPCTC: Corresponding Parts of Congruent Triangles are Congruent
° CPSTP: Corresponding Parts Similar Triangles are Proportional
° Congruency postulates are: SSS, SAS, ASA, AAS
° Similar triangles - have the same shape but sides are proportional
° Similarity Postulates are: AA Similarity Postulate, SSS Similarity Postulate,
AD
12
4
AB
4
12
=
Pythagorean Theorem
AE
AC
= 6
18
6
18
SAS Similarity Postulate
° Hypotenuse Leg: Two right triangles are congruent if the hypotenuse and
Let a= 6 and c = 10 find b
a2 + b2 = c2 62 + b2 = 102
36 + b2 = 100
b2 = 100-36
b2 = 64 ; b2 = 64 ; b= 8
Pieces of Right Triangles Similarity Theorem
One corresponding leg are equal in both triangles.
Remember if an altitude is drawn from the right angle of
a right triangle, the two smaller triangles created are
similar to one another and to the larger triangle..