Download 001-1st-Semester-Review-Geometry

1st Semester Geometry Notes page 1
1-3 Points, Lines, Planes
A
point A
Points A, B are collinear
Points A, B, and C are
coplanar
plane M or
plane ABC
(name with
3 pts)
•Intersection of two distinct lines is a point
•Intersection of two distinct planes is a line
1-4 Segments, Rays, Parallel Lines, Planes
Segment AB or
•Opposite rays share same endpoint
•Opposite rays are collinear
Parallel Lines
-Never intersect
-Extend in the same directions
-Coplanar
Skew Lines
-Never intersect
-Extend in different directions
-Noncoplanar
Parallel Planes
-Can never intersect
1-5 Measuring Segments
AB is the abbreviation for the distance between points A and B.
Segment Addition Postulate
Midpoint
•B is exactly halfway between A and C
•B is the average coordinate of A and C
1-6 Measuring Angles
vertex
Congruent Angles
m 1 = m 2 (the measure of angle 1 equals the measure of angle 2)
 1 ≅  2 (Angle 1 is congruent to angle 2)
(May also be indicated by arc on both angles)
1st Semester Geometry Notes page 2
Pairs of Angles
1 and 2 are adjacent angles
-No interior points in common
-Share the same vertex R
-Share common side
Vertical Angles
-Non-adjacent;
-Formed by two intersecting lines
-Are congruent
1 and 3
Angle Addition Postulate
mAOB + mBOC = mAOC
2 and 4
Linear Pairs
-Form a straight angle
-Are supplementary
(sum = 180)
Compl.
Corner
Suppl.
Straight
1-1 Inductive Reasoning
Law of Detachment:
If a, then b. (True)
Given: a is True
b is therefore True.
Law of Syllogism
If a, then b. (True)
If b, then c. (True)
If a, then c must be True.
If A is falls to the
right, then B falls to
the right
A B
Given: A falls to the right is
True
Then: B falls to the right.
2-1 2-2 2-3 5-4 Deductive Reasoning
Conditional: If a, then b statement
a is the hypothesis; b is the conclusion
Converse: Switch the hypothesis and conclusion.
If b, then a.
Truth value of a statement: Either True or False,
where True means always true
Biconditional: Both the conditional and its
converse are true. You can combine both
statements with if and only if.
a if and only if b.
2-4 Algebraic Properties
A B C
If B is falls to the right, then C falls to
the right.
If A falls to the right, then C falls to
the right.
Addition
Property
If a = b, then a + c = b + c
Subtraction
Property
If a = b, then a – c = b – c
Multiplication
Property
If a = b, then a * c = b * c
Division
Property
If a = b and c ≠ 0, then a/c = b/c
Reflexive
Property
a=a
Symmetric
Property
If a = b, then b = a
Transitive
Property
If a = b and b = c, then a = c
Substitution
Property
If a = b, then b can replace a in
any expression
Distributive
Property
a(b + c) = ab + ac
1st Semester Geometry Notes page 3
1-8 8-1 Pythagorean Theorem, Midpoint, Distance Formula
Pythagorean Theorem
Classifying Triangles
(leg1)2 + (leg2)2 = hypotenuse2
True only for right triangles
Distance between 2 points
-Use the Pythag. Thm
Let a, b, c be the lengths of
the sides of a triangle,
where c is the longest
Acute: c2 < a2 + b2
Right: c2 = a2 + b2
Obtuse: c2 > a2 + b2
Midpoint
(average x, average y)
3-1 3-2 3-3 Parallel Lines and Angles
Transversal: line that
cuts across two or more
lines
Congruent if and only if l and m are parallel
Vertical angles are congruent
Linear pairs are supplementary
3-4 Triangle Sum Thm, Exterior Angle Thm
4-5 Isosceles and Equilateral Triangles
A triangle is isosceles if and only if the
base angles are congruent.
A triangle is equilateral if and only if the
triangle is equiangular
Same-side exterior:
∠1 and ∠4
∠5 and ∠8
Supplementary if and only if l and m are parallel
3-5 Polygon Angle
Sum Thms
n = number of sides
1st Semester Geometry Notes page 4
Interior
angle
Exterior
angle
360
for regular polygons
180 – n
3-6 3-7 Graphing Equations of Lines
•Any point on the line must
satisfy the equation of the
line (y = mx + b)
•Parallel lines have equal
slopes (same steepness)
•Perpendicular lines have
slopes
that are negative reciprocals
of each other
Standard Form:
Ax + By = C
Point-Slope Form:
y – y1 = m (x – x1)
9-2 Reflections:
Preimage and image are
-on opposite sides of line of reflect.
-equidistant from line of reflection
Reflect about x-axis
(x, y) → (x, -y)
Reflect about y-axis
(x, y) → (-x, y)
Reflect about y = x
(x, y) → (y, x)
9-1 Translations:
(x, y) → (x+a, y+b)
9-3 Rotations about origin:
For each 90° of rotation,
switch the x and y coordinates;
then determine signs based on
the quadrant after rotation
Preimage: before the transformation
Image: after the transformation
Isometry: size and shape stay the same
Reflections, Translations, and Rotations are isometries
9-5 Dilations
9-4 Symmetry
Enlargement:
Multiply both x and y by a scale factor k greater than 1
(x, y) → (kx, ky)
Reduction:
Multiply both x and y by a scale factor k between 0 and 1
(x, y) → (kx, ky)
Vertical stretch
Multiply the y only by a scale factor k greater than 1
(x, y) → (x, ky)
Horizontal shrinkage
Multiply the x only by a scale factor k between 0 and 1
(x, y) → (kx, y)
1st Semester Geometry Notes page 5
7-1 Ratios and Proportions
7-2 Similarity
2
4
7-3 Proving Triangles Similarity SSS, SAS, AA
=
3
6
1st Semester Geometry Notes page 6
7-4 Similarity in Right Triangles
1)
2)
3)
Redraw and label triangles.
Fill in the table with given information
Use proportions or Pythag. Thm to
solve for missing lengths
Small leg
Medium leg
Hypotenuse
∆1
a
b
c
∆2
r
h
a
∆3
h
s
b
7-5 Proportions in Triangles
in a triangle
4-1 Congruent Polygons
Two polygons are congruent if they have the same
size and shape.
Two polygons are congruent if and only if all
corresponding sides and corresponding angles are
congruent.
1st Semester Geometry Notes page 7
4-2 4-3 4-6 Proving Triangles Congruent SSS SAS ASA AAS HL
4-4 4-7 Using CPCTC in Proofs
CPCTC is an abbreviation of the phrase “Corresponding Parts of
Congruent Triangles are Congruent.”
1st Semester Geometry Notes page 8
5-5 Triangle Inequalities
Given two sides a
and b, the third side
the triangle with
length c must satisfy
|a–b|<c<a+b
5-1 5-2 5-3 Special Segments in Triangles
Altitudes (vertex to opposite side at right angles
– Orthocenter
Perpendicular Bisectors (90 degrees through
midpoint of a side) – Circumcenter
Medians (vertex to midpoint of opposite side) –
Centroid
Angle Bisectors (vertex to opposite
side through line splitting the vertex
angle in half– Incenter