Download Geometry College Prep B Final Exam 2012 Study Guide Mrs

Geometry College Prep B
Final Exam 2012 Study Guide
Mrs. Mutovic & Ms. Errico
Unit 3: Triangle Properties & Theorems
Classifying a triangle by
side length
Equilateral
Description
All sides are equal
Isosceles
Two sides are equal
Scalene
No sides are equal
Classifying a triangle by
angle measure
Equiangular
All angles are equal
Acute triangle
All angles are acute
c < a +b
2
2
c = a +b
2
c > a +b
2
One angle is = 90°
2
Obtuse triangle
2
Image
2
Right triangle
2
Description
Image
One angle is obtuse
2
1
Triangle
Relationships
Median
Intersections of
Medians of a
Triangle
Description
Image
A segment from a
vertex to the midpoint
of the opposite side.
The medians of a
triangle intersect at the
centroid, a point that is
two thirds of the
distance from each
vertex to the midpoint
of the opposite side.
If P is the centroid of
Centroid
The point at which the
three medians of a
triangle intersect.
, then AP =
2
CP = CE
3
2
2
AD, BP = BF , and
3
3
point P is the centroid
Circumcenter
The point at which the
three perpendicular
bisectors intersect.
Theorem 4.10 &
Theorem 4.11
The shortest side is
opposite the smallest
angle and the longest
side is opposite the
largest angle and visa
versa.
The sum of the lengths
of any two sides of a
triangle is greater than
the length of the third
side.
Triangle
Inequality
Theorem
2
Triangle Theorems
Triangle Sum Theorem
Description
The sum of the measures
of the angles in a triangle is 180°.
Exterior Angles Theorem
The measure of an exterior angle
of a triangle is equal to the sum of
the measure of the two
nonadjacent interior angles.
Isosceles Triangle Theorem
1
If two sides of a triangle are
congruent, then the angles
opposite them are congruent.
Image
If
Isosceles Triangle Theorem
2
AB @ AC, then ÐC @ ÐB
If two angles of a triangle are
congruent, then the sides opposite
them are congruent.
If ÐB @ ÐC, then
Equilateral Theorem 1
AC @ AB
If a triangle is equilateral, then it is
equiangular.
AB @ AC @ BC , then
ÐA @ ÐB @ ÐC
If
Equilateral Theorem 2
If a triangle is equiangular then its
equilateral.
If ÐB @ ÐC @ ÐA, then
AB @ AC @ BC
Triangle Congruence
3
Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
 Figures are congruent if all pairs of corresponding angles are congruent and all pairs
of corresponding sides are congruent.
J KL  RS T

SIDE-SIDE-SIDE (SSS)
 If three sides of one triangle are congruent to three sides of a second triangle, then
the two triangles are congruent.
o For example, in the diagram below ∆MNP  ∆QRS, by SSS.

SIDE-ANGLE-SIDE (SAS)
 If two sides and the included angle of one triangle are congruent to two sides and
the included angle of a second triangle, then the two triangles are congruent.
o For example, in the diagram below ∆PQS  ∆WXY, by SAS.

ANGLE-SIDE-ANGLE CONGRUENCE (ASA)
 If two angles and the included side of one triangle are congruent to two angles and
the included side of another triangle, then the triangles are congruent.
o For example, in the diagram below ∆ABC  ∆XYZ, by ASA.

4
ANGLE-ANGLE-SIDE CONGRUENCE (AAS)
 If two angles and a NON-included side of one triangle are congruent to the
corresponding two angles and NON-included side of a second triangle then the two
triangles are congruent.
o For example, in the diagram below ∆ABC  ∆XYZ, by AAS.

HYPOTENUSE-LEG CONGRUENCE (HL)
 If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and
leg of another right triangle, then the triangles are congruent.
o For example, in the diagram below ∆ABC  ∆DEF, by HL.

Angle Bisectors &
Perpendicular
Bisectors
Angle Bisector Theorem
Perpendicular Bisector
Theorem
Description
Image
If a point is on
the bisector of
an angle, then it
is equidistant
from the two
sides of the
angle.
If a point is on
the
perpendicular
bisector of a
segment, then it
is equidistant
from the end
points of the
segment.
5
Polygons & Quadrilaterals

Polygon: A figure that is formed by three or more segments.

Diagonal: A segment that joins two non-consecutive vertices of a polygon.

Sum of interior angles: 180(n - 2) where n = number of sides in a polygon

Sum of exterior angles: always add to 360°
Properties of Parallelograms
Parallelogram
Description
If a quadrilateral is a
parallelogram, then both
pairs of opposite sides are
parallel.
Theorem 6.2
If a quadrilateral is a
parallelogram, then its
opposite sides are
congruent.
Theorem 6.3
If a quadrilateral is a
parallelogram, then its
opposite angles are
congruent.
Theorem 6.4
If a quadrilateral is a
parallelogram, then its
consecutive angles are
supplementary.
Theorem 6.5
If a quadrilateral is a
parallelogram, then its
diagonals bisect each other.
Image
6
Showing Quadrilaterals are
Parallelograms
Theorem 6.6
Description
Image
If both pairs of opposite
sides of a quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.
Theorem 6.7
If both pairs of opposite
angles of a quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.
Theorem 6.8
If an angle of a
quadrilateral is
supplementary to both of
its consecutive angles, then
the quadrilateral is a
parallelogram.
Theorem 6.9
If the diagonals of a
quadrilateral bisect each
other, then the
quadrilateral is a
parallelogram.
Properties of Trapezoids
Theorem 6.12
Description
If a trapezoid is isosceles,
then each pair of base
angles are congruent.
Theorem 6.13
If a trapezoid has a pair of
congruent base angles, then
it is isosceles.
Midsegments of a trapezoid
The segment that connects
the midpoints of its legs.
Image
1
MN = (AD + BC)
2
****Special Quadrilaterals (see chart in study guide 2)****
7
Unit 4: Similarity

Simplifying ratios: be sure to convert to the same units of measurement first

Solving proportions for a variable: cross multiply then solve for the value of the variable

Proportions and Similar Triangles: Triangle Proportionality Theorem, Converse of the
Triangle Proportionality Theorem, and the Midsegment Theorem

Scale factor: ratio of the lengths of two corresponding sides of two similar polygons
Theorem
Similar polygons
Description
Two polygons are
similar if
corresponding
angles are
congruent and
corresponding
side lengths are
congruent
Angle-Angle
Similarity
Postulate (AA)
If two angles of
one triangle are
congruent to two
angles of another
triangle, then the
two triangles are
similar
Image
8
Unit 5: Right Triangle Trigonometry
Special Right Triangles:

A right triangle with angle measures of 45°, 45°, 90° is called a 45°-45°-90°
triangle.
hypotenuse = leg· 2

A right triangle with angle measures of 30°, 60°, 90° is called a 30°-60°-90°
triangle.
hypotenuse = 2 · shorter leg
longer leg = shorter leg · 3
Trigonometric Ratios
SOHCOATOA
sin =
opposite
hypotenuse
cos =
adjacent
hypotenuse
tan =
opposite
adjacent


-1
-1
-1
To find missing angle measures use the inverse keys ( sin ,cos ,tan )
o You can use sin, cos, or tan if you are given all three side lengths because you
will still get the same answer
To find the missing side lengths us the trigonometric ratios and solve for the variable
o When the variable is in the numerator, multiply the two known numbers
(example: cos54 =
o
x
 x = (10)(cos54))
10
When the variable is in the denominator divide the two known numbers
(example: cos54 =
10
10
 x=
)
x
cos54
9
Unit 6: 3D Calculations
Polyhedra
 Rectangular prism
 Triangular pyramid
 Triangular prism
 Square pyramid
o Faces: the plane surfaces
o Edges: the segments joining the vertices
o Vertex: point that joins two sides of a figure
Net: a flat representation of all the faces of a polyhedron
Not a Polyhedra
 Sphere
 Cone
 Cylinder
Volume Formulas:

V = lwh

Rectangular Prism:

Triangular Prism: V =

Cylinder: V =π r 2 h

Square Pyramid: V =


Sphere: V =
1
lwh
2
1
lwh
3
1
Triangular Pyramid: V = lwh
6
1
Cone: V = π r 2 h
3
4 3
πr
3
Surface Area: the sum of the areas
of all sides

Prisms: add up areas of all
faces

Pyramids: Area of base + 4
1
1
( s (h2 + ( ) 2 )
2
2s

Cylinders: 2π r 2 + 2π rh

Cone: π r 2 + π r (h2 + r 2 )
10

Sphere: 4π r 2
Unit 8: Circles
Parts of a circle
Chord
Description
A segment that has
endpoints on the circle.
Image
Diameter
A chord that goes through
the center of a circle.
Radius
Secant
½ the diameter
A line that has endpoints
on the circle.
See above image
Tangent
A line that hits the circle
once.
See above image
Properties of Tangents
Theorem 11.1
Description
If a line is tangent to a circle,
then it is perpendicular to
the radius drawn at the
point of tangency.
Theorem 11.2
In a plane, if a line is
perpendicular to a radius of
a circle at its endpoint on the
circle, then the line is
tangent to the circle.
Theorem 11.3
If two segments from the
same point outside a circle
are tangent to the circle,
then they are congruent.
Image
11
*use the Pythagorean Theorem ( a2 + b2 = c2) to find the radius of a circle
Arcs and Central Angles
Minor arc and major arc
Description
Any two points A and B on a
circle C determine a minor
arc and a major arc (unless
the points lie on a diameter)
Measure of a minor arc
The measure of its central
angle.
Measure of a major arc
The difference of 360° and
the measure of the related
minor arc.
An arc whose central angle
measures 180°.
Semicircle
Congruent circles
Two circles are congruent if
they have the same radius
Congruent arcs
Two arcs of the same circle or
of congruent circles are
congruent arcs if they have
the same measure.
Arc length
Portion of a circumference of
a circle.
Image
See above image.
12