FINAL EXAM STUDY GUIDE
Topics going over in this Study Guide:
➔ Unit 1(Logic and Euclidean Geometry)
➔ Unit 2(Coordinate Geometry and Transformations)
➔ Unit 3(Relationships of lines and Transversals)
➔ Unit 4(Relationships of Triangles, including Congruence and Similarity)
➔ Unit 5(Relationships of Right Triangles, including Trigonometry)
➔ Unit 6(Relationships of Circle, including Radian Measure and Equations of Circles)
➔ Unit 7(Relationships of Two - and Three Dimensional Figures)
➔ Unit 8(Measurement of Two-Dimensional Figures)
➔ Unit 9(Measurement of Three-Dimensional Figures)
UNIT 1(Logic and Euclidean Geometry)
Shapes
Equilateral Triangle
Characteristics of shapes
Equilateral triangles have all angles equal to
60° and all sides equal length.
All equilateral triangles have 3 lines of
symmetry.
Isosceles Triangle
Isosceles triangles have 2 angles equal and 2
sides of equal length.
All isosceles triangles have a line of
symmetry.
Scalene Triangle
Scalene triangles have no angles equal, and
no sides of equal length.
Right Triangle
Right triangles (or right angled triangles) have
one right angle (equal to 90° ).
Obtuse Triangle
Obtuse triangles have one obtuse angle (an
angle greater than 90° ). The other two angles
are acute (less than 90° ).
Acute Triangle
Acute triangles have all angles acute.
Geometric Relationships
Triangles:
Interior Angle: angle formed on the inside of a polygon by two sides meeting at a vertex.
Exterior Angle: angle formed on the outside of a geometric shape by extending one side past a
vertex.
Quadrilaterals:
The sum of interior angles of a quadrilateral is 360 degrees.
The sum of exterior angles of a quadrilateral is also 360 degrees.
Parallelograms:
Adjacent angles in a parallelogram are supplementary(add to 180)
Opposite angles in a parallelogram are equal.
Polygons:
Convex Polygon: All interior angles measure less than 180 degrees.
Concave Polygon: Can have interior angles greater than 180 degrees.
Regular Polygon: All sides are equal and all interior angles are equal.
Geometric Representations
Vectors
Logical Arguments
Name
Definition
Example
Direct Argument
If p is true, q is true
If a shape is a square, then it
is a rectangle.
P is true
Therefore, q must also be
true.
Indirect Argument
If p is true, q is true.
Q is not true.
Therefore p cannot be true.
Chain Rule
If p is true, then q is true.
If q is true, then r is true.
Therefore, if p is true, then r
is true.
Or Rule
Either p is true, or q is true.
P is not true.
So q must be true.
And Rule
P and q are not both true.
Q is true.
So p must be false.
HIJK is a square
Therefore HIJK must also be
a rectangle.
If a shape is a square, then it
is a rectangle.
HIJK is not a rectangle.
Therefore HIJK can’t be a
square.
If a shape is a square, then it
is a rectangle.
If a shape is a rectangle, then
it is a parallelogram.
Therefore, if a shape is a
square, then it is a
parallelogram.
Figure A is a circle or a
square.
Figure A is not a circle
So Figure A must be a square.
Figure A is not both a circle
or a square.
Figure A is a square.
So Figure A is not a circle.
Unit 2(Coordinate Geometry and Transformations)
There are 4 types of transformations:
- Reflection
- Rotation
- Translation
- Dilation
Reflection:
- Is a flip
- Equation of a line
Rotation
- Is a turn
- direction(counterclockwise, clockwise)
- Degree
- Center point of rotation
Translation
- Shift or slide
- direction(left, right, up, down)
- magnitude(number of units)
Dilation
- Enlargement or reduction
- Center point of Dilation
- Scale factor
One Dimensional Coordinate:
A one-dimensional coordinate system is defined by its origin and a single basis vector that
defines the positive direction of the coordinate axis (x-axis).
UNIT 3(Relationships of lines and Transversals)
Angles, lines, and transversals
Parallel Lines are lines that will never intersect.
Perpendicular Lines are lines that intersect to form 90 degrees.
Vertical Angles - When two lines intersect, they form four angles. The angles
that are across from each other are vertical angles.
Angles
Definition
Interior Angles
Angles that are between the two lines that are
intersected by the transversal. In the diagram
above, they are angles 3, 4, 5, and 6.
Exterior Angles
Angles that are NOT between the two lines,
that are intersected by the transversal. In the
diagram above, they are angles 3, 4, 5, and 6.
Alternate Interior Angles
Interior angles that are the opposite sides of
the transversal. In the diagram above, they
are angles 3 and 6 as well as angles 4 and 5.
Alternate Exterior Angles
Exterior angles that are the opposite sides of
the transversal. In the diagram above, they
are angles 1 and 8, as well as 2 and 7.
Corresponding Angles
Angles that are in the same relative position.
In the diagram above, they are angles 1 and 5;
angles 2 and 6; and 3 and 7; and 4 and 8.
UNIT 4(Relationships of Triangles, including Congruence and Similarity)
Acute
< 90°
Obtuse
> 90°
Equilateral Triangle = 3 sides
Isosceles Triangle = 2 sides
Scalene Triangle = No sides
Right
= 90°
Straight
= 180°
Right Triangle
Obtuse Triangle
Acute Triangle
UNIT 5(Relationships of Right Triangles, including Trigonometry)
A right triangle has a value of 90 degrees. A right triangle is a
triangle in which one angle is a right angle. The relation between the sides and angles of a right
triangle is the basis for trigonometry.
Pythagorean Theorem
a^2 + b^2 = c^2
In this equation, c represents the length of the hypotenuse and a and b the lengths of triangles
two sides.
Pythagorean Theorem
UNIT 6(Relationships of Circle, including Radian Measure and Equations of Circles)
Inscribed Angle
Angle whose vertex is on a circle and whose
sides contain chords of the circle.
Intercepted Arc
The arc that lies in the interior of the
inscribed angle and has endpoints on the
angle.
Inscribed Polygon
Polygon whose vertices lie on a circle
Circumscribed Circle
A circle that contains the vertices of an
inscribed polygon.
Theorem
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.
Measure of an Inscribed Angle Theorem
The measure of an inscribed angle is one half the measure of its intercepted arc.
UNIT 7(Relationships of Two - and Three Dimensional Figures)
Three-Dimensional Figures
Type
Triangular Prism
Examples
Properties
●
●
●
Rectangular Prism
●
●
●
5 faces
Two triangular bases
3 rectangular faces
9 edges
6 vertices
6 faces
2 rectangular bases
4 rectangular faces
12 edges
8 vertices
Cube
●
●
●
Square Pyramid
●
●
●
Triangular Pyramid
●
●
●
6 faces
2 square bases
4 square faces
12 edges
8 vertices
5 faces
1 square base
4 triangular faces
8 edges
5 vertices
4 faces
1 triangular base
3 triangular faces
6 edges
4 vertices
Two Dimensional Figures
N- gon A polygon with n sides
Equilateral polygon
A polygon in which all sides are congruent
Equiangular polygon
A polygon in which all angles are congruent
Regular polygon
A convex polygon that is both equilateral and equiangular
UNIT 8(Measurement of Two-Dimensional Figures)
Triangle
Square
Rectangle
Circle
P=c+b+d
P = 4s
P = 2l + 2w
C = 2πr OR πd
A = ½ (bh) = bh/2
A = s^2
A = lw
A = πr^2
P = Perimeter of polygon
B = base, h = height
A = area of figure
l = length, w = width
c = circumference
r = radius, d = diameter
UNIT 9(Measurement of Three-Dimensional Figures)
●
-
Mathematical arguments
when making an argument you must be able to back yourself.
To this you may use pictures, word problems, questions, etc. When
doing this, and to keep the conversation going use mathematical
terminology to communicate.
½ x perimeter x apothem
s= side of length
s= number of sides
a=apothem length
Formula
How to find Apothem?
S
_______
2tan(180/n)
-
FORMULAS
The volume of a Cuboid is given by(area of base x height) i.e. height x
(length x breadth.)
The volume of a Cube is given by edge^3.
The volume of a Cylinder is πr^2h.
The volume of a Cone is ⅓ πr^2h
The volume of a Sphere is 4/3πr^3
Sphere
- Volume is 4/3πr^3
- Area is A= 4πr^2
Cone
- Surface Area is πrs + πr^2
- Volume is v=⅓ πr^2h
Cylinder
- Area is 2πr^2 + 2πrh
- Volume is πr^2h
Rectangular Prism
- Area is 2(wh + lw + lh)
- Volume is V= lwh
Lesson:
Pyramid
- Area is 2bs + b^2
- Volume is V=⅓ b^2h