Download Geometry EOC

Geometry EOC
Content Review Packet
Name:
Teacher:
Period:
Page 2
Review resources
Review Resources
Videos:
 Pearson Online
www.pearsonsuccessnet.com
(Covers Geometry topics in tutorials & videos. Aligns with the
curriculum used in Geometry)
 Khan Academy
https://www.khanacademy.com
(Covers all these topics in 10-20 minute videos with some good
visual representations.)
 Brightstorm Videos Online
www.brightstorm.com
(Search any topic you are interested in exploring further to see
video clips)
Page 31
FORMULA SHEET:
Page 30
FORMULA SHEET Cont…
Page 3
G.1.C Use deductive reasoning to prove that a valid geometric
statement is true.
Deductive Reasoning- Reasoning accepted as logical from agreed-upon
assumptions and proven facts.
Example Theorem: All right angles are congruent.
If ∠1 and ∠2 are right angles, then ∠1≅∠2.
Proof: ___________ are right angles because it is given. By the
definition of _______ m∠1= 90 and ∠2 = 90. By the transitive property
of equality, ______ =_______.
Examples:
1. The measure of an angle is three more than twice its supplement.
What is the measure of the angle?
2.
Resources:
 2.4 (Pearson Geometry Textbook)
 http://www.brightstorm.com/math/geometry/reasoning-diagonals-angles-andparallel-lines/deductive-reasoning/all
Page 4
Page 29
G.1.D Write the converse, inverse, and contrapositive
of a valid proposition and determine their validity.
Original Statement: ________________________________________
Converse Statement: _______________________________________
Inverse Statement: _________________________________________
Contrapositive Statement: ___________________________________
Tips for doing well on the EOC



Don’t skip any questions – answer them!
Read the question carefully to get to what they are truly asking
you.
Re-read what you wrote and make sure it answers the prompt
Valid Statement: ____________________________________
Invalid Statement: ___________________________________
Just think back to what you have learned, do your best, and you will be
fine! Even if your answer feels a bit silly, write it down, it’s better than
nothing!
Example: Write the converse, inverse, and contrapositive of
the following statement:
If the weather is rainy, then the sidewalks will be wet.
Do well! Take your time, be thorough and careful, and you will do well!
You can do it!
Calculators will be allowed on the Geometry EOC…
…BRING YOUR OWN CALCULATOR!
Examples:
1. If m and n are odd integers, then the sum of m and n is an even
integer. State the converse and determine whether it is valid.
2. If a quadrilateral is a rectangle, the diagonals have the same
length. State the contrapositive and determine whether it is valid.
Resources:
 2.2, 2.3,2.5 (Pearson Geometry Textbook)
 http://hotmath.com/hotmath_help/topics/converse-inversecontrapositive.html
Page 28
Page 5
What will the Geometry EOC be like?
Logical arguments & proofs: 6-8 points
– 5-8 multiple choice
– 0-1 short answer
Proving & applying properties of 2D shapes: 24-26 points
– 15-19 multiple choice
– 2-4 completion
– 1-3 short answer
Coordinate planes & measurement: 7-9 points
– 5-8 multiple choice
– 1-3 completion problems
– 0-1 short answer
G.1.E Identify errors or gaps in a mathematical argument and
develop counterexamples to refute invalid statements about
geometric relationships.
A Counterexample is: ______________________________________
_________________________________________________________
An Invalid statement is: _____________________________________
_________________________________________________________
Examples:
1. Give a counterexample that disproves each conjecture below.
a. No triangles have two sides of the same length.
b. No women have been elected U.S. senators.
Details for scoring please!?!?!?!
c. All basketball players are more than 6 feet tall.
2. Error Analysis:
Identify errors in reasoning in the following proof:
Given ∠ABC ≅ ∠PRQ, AB ≅ PQ, and BC ≅ QR,
then ∆ABC ≅ ∆PQR by SAS.
Resource: Look at Error Analysis problems in Pearson textbook
Page 27
Page 6
G.1.F Explain the role of definitions, undefined terms, postulates
(axioms), and theorems.
Undefined terms: Can be described but cannot be given precise
definitions. List the 3 undefined Geometric Terms:
__________________, ________________, _________________
Postulates: Statements that we assume to be _____ without proof.
Theorems: Statements that ______________________ within a
deductive system.
Examples:
1.
2.
3.
Resource:
http://www.brightstorm.com/math/geometry/geometry-building-blocks/postulateaxiom-conjecture/all
Additional Notes/Practice:
Page 26
Additional Notes/Practice:
Page 7
G.3.A Know and apply basic postulates and theorems about
triangles and the special lines, line segments, and rays associated
with a triangle.
Examples:
• Prove that the sum of the angles of a triangle is 180°.
Triangle Sum Theorem
Definition:
Picture/Sketch:
• Prove and explain theorems about the incenter,
circumcenter, orthocenter, and centroid.
Incenter
Definition:
Picture:
Circumcenter
Definition:
Picture:
Page 25
Page 8
Additional Notes/Practice:
G.3.A Continued
Orthocenter
Definition:
Picture:
Centroid
Definition:
Picture:
Example:
1. The rural towns of Atwood, Bridgeville, and Carnegie are building a
communications tower to serve the needs of all three towns. They want
to position the tower so that the distance from each town to the tower is
equal.
a. Where should they locate the tower?
b. How far will it be from each town?
2. If the measures of three angles of a triangle are represented by
(y + 30)°, (4y +30)°, and (10y – 30)°, then the triangle must be:
A. obtuse
B. isosceles C. scalene D. right
Resources:
3.1, 3.2, 3.3, 3.4, 3.5 (Pearson Geometry Textbook)
http://www.sparknotes.com/math/geometry2/theorems/section2.rhtml
Page 24
Additional Notes/Practice:
Page 9
G.3.B Determine and prove triangle congruence and other
properties of triangles.
Postulate: Side-Side-Side (SSS)
Example:
Postulate: Side-Angle-Side (SAS)
Example:
Postulate: Angle-Side-Angle (ASA)
Example:
Theorem: Angle-Angle-Side (AAS)
Example:
Using Corresponding Parts of Congruent Triangles (CPCTC)
Example:
Resources:
 4.2, 4.3, 4.4, 4.5 (Pearson Geometry Textbook)
 http://www.mathopenref.com/congruenttriangles.html
Page 10
G.3.D Know, prove, and apply the Pythagorean Theorem and its
converse.
Then:
Page 23
G.6.F Solve problems involving measurement conversions within
and between systems, including those involving derived units, and
analyze solutions in terms of reasonableness of solutions and
appropriate units.
_____________________
Examples:
1. You want to make a triangular sail that has a base of 13 ft. 4 in.
and a height of 12 ft. 2 in. How many square feet of material do
you need?
Examples:
1)
2. An airplane is flying at an altitude of 10,000 ft. The airport at
which it is scheduled to land is 50 mi away. Find the average
angle at which the airplane must descend for landing. Round
your answer to the nearest degree.
2)
3) Classifying a Triangle
Resources:
 8.1 (Pearson Geometry Textbook)
 http://mathworld.wolfram.com/PythagoreanTheorem.html
Resource: Throughout Pearson Geometry Textbook
Page 22
G.6.E Use different degrees of precision in measurement, explain the
reason for using a certain degree of precision, and apply estimation
strategies to obtain reasonable measurements with appropriate
precision for a given purpose.
Page 11
G.3.E Solve problems involving the basic trigonometric ratios of
sine, cosine, and tangent.
Examples:
1. Find the value of each variable.
Sin(A) =
Cos(A) =
Tan(A) =
Examples:
1) Write each ratio for the following:
2. The International Space Station orbits 350 km above
Earth’s surface. Earth’s radius is about 6370 km. Use the
Pythagorean Theorem to find the distance from the space
station to Earth’s horizon. Round your answer to the nearest
10 kilometers. (Diagram is not to scale.)
Sin(A) =
Cos(B) =
2) Find the value of x for the following:
Resource: Throughout Pearson Geometry Textbook
Tan(A) =
Page 12
Page 21
G.3.F Know, prove, and apply basic theorems about parallelograms.
G.3.C Continued
Properties of Parallelograms:
Examples:
3. What is the value of z?
A. 2 2
B. 2 3
C. 4 2
D. 8 2
If….
Then….
Examples:
1) Find the value of the variables for each:
2) Find the value of the variables:
3) Find the measures of the numbered angles:
Resource:
 6.2, 6.3, 6.4, 6.5 (Pearson Geometry Textbook)
4. Determine the length of the altitude of An equilateral triangle whose
side lengths measure 8 units. Keep your answer written as an exact
value.
Page 13
Page 20
G.3.C Use the properties of special right triangles
(30°–60°–90° and 45°–45°–90°) to solve problems.
Rules:
45-45-90 Triangle:
Examples:
1) What are the values of x and y?
2) What are the values of x?
30-60-90 Triangle:
G.3.F Continued
Page 14
G.4.B Determine the coordinates of a point that is described
geometrically.
Midpoint Formula: ________________________________
Distance Formula: _________________________________
Examples:
1. b. Determine the coordinates for the midpoint of a line segment
whose endpoints are (6, 8) and (-4, -5).
b. Determine the coordinates for the midpoint of a line segment
whose endpoints are (-5, 8) and (-2, -9).
2. a. Points X, Y, Z are collinear. Y is the midpoint of XZ. The
coordinates of point X are (-4, 5). The coordinates of point Y are
(2, 4). Determine the coordinates of point Z.
b. Points A, B, C are collinear. B is the midpoint of AC. The
coordinates of point Z are (-3, 9). The coordinates of point B are
(2, -2). Determine the coordinates of point C.
3. Three vertices of parallelogram ABCD are ( 4, 9 ), ( 7, 7 ), and
( 1, 1 ). Which of the following coordinates would be possible
for the fourth vertex?
A. ( 4, 0 )
B. ( 4, 1 ) C. ( -3, 2 ) D. ( -2, 3 )
Resources:
 6.8 (Pearson Geometry Textbook)
 http://www.mathopenref.com/coordintro.html
 http://www.onlinemathlearning.com/coordinate-geometry.html
Page 19
G.4.C Continued
Example:
3. Classify the quadrilateral ABCD whose vertices lie on A (-11,12),
B(4,15), C(7,7), and D(-3,5).
Classify the quadrilateral and justify your answer.
Page 15
Page 18
G.4.C Verify and apply properties of triangles and
quadrilaterals in the coordinate plane.
Triangle Properties:
Obtuse:
Acute:
Scalene:
Right:
Isosceles:
Equilateral:
G.4.B Cont.
Examples:
4. A kite is designed for an upcoming craft show. In order to get the
design lined up properly, point S must be located. Point S is the
midpoint of segment EI. What are the coordinates of point S?
A. (2, 2)
B. (5,10)
C. (5.5, 9.5) D.(7.5, 7.5)
5. In parallelogram WXYZ, what are the coordinates of the point of
intersection of WY and ZX?
Examples:
1. The vertices of ΔABC are A( –1, –2), B( –1, 2), and C( 6, 0). Which
conclusion can be made about the angles of ΔABC?
mACB  90
A. mA  mB
B.
C. mA  mC
D. mABC  60
2. Classify triangle FGH by its sides and angles. F(-12,3), G(0,0),
H(-10,11)
A. Scalene, Right
B. Scalene, Obtuse
C. Isosceles, Right
D. Equilateral, Acute
Resource:
 6.8, 6.9 (Pearson Geometry Textbook)
Page 16
G.3.G Know, prove, and apply theorems about properties of
quadrilaterals and other polygons.
Special Parallelograms:
Page 17
G.3.G Cont.
Examples:
1. LMNP is a rectangle. Find the value of x and the length of each
diagonal:
LN = 9x – 14 and MP = 7x + 4
1) A ___________ is a parallelogram with four congruent
sides.
Other Properties:
2. a. What is the length of the apothem of a regular hexagon with a
side length 9in?
2) A _______________ is a parallelogram with four right
angles.
Other Properties:
3) A ______________ is a parallelogram with four congruent
sides and four right angles.
Other Properties:
b. What is the length of the apothem of a regular hexagon with a
side length 12in?
3. a. What is the sum of the interior angles of a regular nonagon?
b. What is the sum of the interior angles of a regular pentagon?
Geometric Properties of polygons:
1. If a figure is a trapezoid, then consecutive angles between a pair
of parallel lines are ___________________________
2. If a figure is a kite, then the diagonals are __________________
3. The sum of the exterior angles of a polygon is ______________
4. The sum of the interior angles of a polygon is _______________
when n is the _________________________________________
5. The area of a regular polygon is found by: __________________
6. The diagonals of a rectangle ______________ each other.
7. The diagonals of a rhombus are ______ and intersect at _______
4. Solve for x, using the information from the trapezoid below.
5x + 5
3x + 7
Resources:
 6.4, 6.5 (Pearson Geometry Textbook)