Geometry Midterm Review (Chapters: 1, 2, 3, 4, 5, 6) Algebra Review Chapter 2 Topics 2 ·...
Transcript of Geometry Midterm Review (Chapters: 1, 2, 3, 4, 5, 6) Algebra Review Chapter 2 Topics 2 ·...
Geometry Midterm Review (Chapters: 1, 2, 3, 4, 5, 6)
Algebra Review
Systems of equations
Simplifying radicals
Equations of Lines: slope-intercept, point-slope
Writing and solving equations: linear and quadratic
Parallel lines
Chapter 1 Topics
1.1
Undefined Terms—point, line, plane
Collinear, Coplanar
Segment
Endpoint
Ray
Opposite Rays
Postulates—Points, Lines, and Planes
Intersections
1.2
Coordinate
Ruler Postulate
Distance
Congruent Segments
Segment Addition Postulate
Midpoint
Segment Bisector
1.3
Angle
Interior of an Angle/Exterior of an Angle
Measure of an angle/Degree
Protractor Postulate
Types of Angles
Congruent Angles
Angle Addition Postulate
Angle Bisector
1.4
Adjacent Angles
Linear Pair
Complementary Angles
Supplementary Angles
Vertical Angles
1.5
Perimeter (P) and Area (A) of:
Rectangle, Triangle, Square
Parallelogram
Circumference (C) and Area (A) of:
Circles
1.6
Midpoint Formula
Parts of a right triangle
Finding distance using:
Distance Formula
Pythagorean Theorem
Chapter 2 Topics
2.1
Inductive Reasoning
Finding and describing a pattern
Writing conjecture
Counterexamples
2.2
Conditional Statement
Hypothesis
Conclusion
Writing Conditional Statements
Truth Value
Negation
Related Conditionals
-Converse
-Inverse
-Contrapositive
Logically Equivalent Statements
2.3
Deductive Reasoning
Law of Detachment
Law of Syllogism
Making Conclusions
2.4
Biconditional Statement
Truth Value of a Biconditional Statement
Definition
Polygon
Triangle
Quadrilaterals
Definition—Biconditional
2.5
Properties of Equality
Distributive Property
Algebraic Proof
D=rt
Properties of Congruence
Difference between Congruence and Equivalencies
2.6
Geometric Proofs (2-column)
Theorem
Linear Pair Theorem
Congruent Supplements Theorem
Right Angle Congruence Theorem
Congruent Complements Theorem
Vertical Angle Theorem
2.7
Common Segments Theorem
If two congruent angles are supplementary, then each
angle is a right angle.
Chapter 3 Topics
3.1
Parallel Lines/Planes
Perpendicular Lines
Skew Lines
Transversal
Corresponding Angles
Alternate Interior Angles
Same-Side Interior Angles
Alternate Exterior Angles
3.2
Corresponding Angles Postulate
Alternate Interior Angles Theorem
Alternate Exterior Angles Theorem
Same-Side Interior Angles Theorem
If transversal is perpendicular to parallel lines, then all
angles are right angles.
3.3
Converse of the Corresponding Angles Postulate
Converse of the Alternate Interior Angles Theorem
Converse of the Alternate Exterior Angles Theorem
Converse of the Same-Side Interior Angles Theorem
Parallel Postulate
3.4
Perpendicular Bisector
Distance from a point to a line
If 2 intersecting lines form a linear pair of congruent
angles, then the lines are perpendicular.
Perpendicular Transversal Theorem
If 2 coplanar lines are perpendicular to the same line,
then the 2 lines are parallel to each other.
3.5
Slope from a graph/coordinates
Positive/Negative/Zero/Undefined Slopes
Parallel Lines Theorem
Perpendicular Lines Theorem
3.6
Point-Slope Form
Slope-Intercept Form
Vertical Lines
Horizontal Line
Transform between both equations
Graphing Lines
Pairs of Lines:
-Parallel Lines
-Intersecting Lines/Perpendicular Lines
-Coinciding Lines
Chapter 4 Topics
4.1
Classifying Triangles by Angles and Sides
Using Triangle Classification
4.2
Triangle Sum Theorem
Auxiliary Line
Corollary
The acute angles of a right triangle are complementary.
The measure of each equiangular triangle is 60 degrees.
Interior/Exterior
Interior Angles/Exterior Angles
Remote Interior Angles
Exterior Angle Theorem
3rd Angles Theorem
4.3
Congruent
Corresponding Angles and Corresponding Sides
2 polygons are congruent iff their corresponding sides
and angles are congruent
CPCT-Corresponding Parts of Congruent Triangles
Proving Triangles Congruent
4.4
Triangle Rigidity
SSS
Included Angle
SAS
AAS
Verifying Triangle Congruence
4.5
Included Side
ASA
HL/HA/LA
4.6
CPCTC—Corresponding Parts of Congruent Triangles are
Congruent
Remember: SSS, SAS, ASA, AAS, HL, HA, LA use
corresponding parts to prove triangles congruent
CPCTC uses congruent triangles to prove corresponding
parts are congruent
4.8
Isosceles Triangle
Legs, Vertex Angle, Base, Base Angles
Isosceles Triangle Theorem (ITT)
Converse of Isosceles Triangle Theorem
If a triangle is equilateral, then it is equiangular.
Chapter 5 Topics
5.1
Equidistant
Perpendicular Bisector Theorem
Converse of Perpendicular Bisector Theorem
Angle Bisector Theorem
Converse of Angle Bisector Theorem
Applying Angle Bisector Theorem
5.2
Concurrent
Circumcenter Theorem
Incenter
Incenter Theorem
Inscribed
5.3
Median of a Triangle
Centroid of a Triangle
Centroid Theorem
Altitude of a Triangle
Orthocenter of a triangle
5.4
Midsegment of a Triangle
Triangle Midsegment Theorem
5.5
Indirect Proof
In a triangle, the longer side is opposite the larger angle
In a triangle, the larger angle is opposite the longer side
Inequality Properties:
-Addition Property of Inequality
-Subtraction Property of Inequality
-Multiplication Property of Inequality
-Division Property of Inequality
-Transitive Property of Inequality
-Comparison Property
Triangle Inequality Theorem
5.6
Hinge Theorem
Converse of Hinge Theorem
Simplifying Radicals
5.7
Pythagorean Triples
Pythagorean Inequality Theorems
5.8
45-45-90 Triangle Theorem
30-60-90 Triangle Theorem
Chapter 6 Topics
6.1
Side of a Polygon, Vertex of a Polygon, Diagonal of a
Polygon
Names of Polygons
Definition of Polygon
Regular Polygon
Concave
Convex
Polygon Angle-Sum Theorem
Polygon Exterior Angle Sum Theorem
6.2
Properties of a parallelogram
If quad is a parallelogram, then its opposite sides are
congruent
If a quad is a parallelogram, then its opposite angles are
congruent
If a quad is a parallelogram, then its consecutive angles
are supplementary.
If a quad is a parallelogram, then its diagonals bisect each
other.
6.3
Conditions for Parallelograms
Quad with 1 pair of opposite sides parallel and congruent
is a parallelogram.
Quad with opposite sides congruent is a parallelogram
Quad with opposite angles congruent is a parallelogram
Quad with angles supp. to consecutive angles is a
parallelogram
Quad with diagonals bisecting each other is a
parallelogram
Algebra Review
Simplify completely.
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
9.)
10.)
Solve the following equations.
11.)
12.)
13.)
14.)
15.)
16.)
17.) Write an equation for the line that goes through the points
18.) What is the equation of the perpendicular bisect of if ?
19.) What is the equation for a line parallel to through point
Review Chapter 1
Use the following diagram for problems 20-23.
20.) Two opposite rays. _____________________________________
21.) A point on . _____________________________________
22.) The intersection of the two planes. ________________________________
23.) A plane containing X, V, and P. _____________________________________
24.) The intersection of two planes is a(n) ______________________________.
25.) M is between N and R. MR=15 m. NR=17.6 m. Write an equation and solve for MN. Draw a diagram.
26.) bisects at M. GM=(2x+6) m and GK = 24 m. Write an equation and solve for x. Draw a diagram.
Use the following diagram for problems 27-30. is a right angle.
27.) Name two acute angles.
28.) Name two obtuse angles.
29.) Name two angles that form a linear pair.
30.) Name a pair of angles that are supplementary and
congruent.
31.)
Use the following diagram for problem 32.
32.) . Solve for x.
If find the measure of the following. Write an equation to solve.
33.) Complement of A 34.) Supplement of B
Find the perimeter and area in problems 35 and 36. Draw a diagram. All units are in inches.
35.) Rectangle with 36.) Triangle: with a=3x, b=10, c=x+6,
height=2x where b is the base.
37.) Find the circumference of a circle with radius 4cm. Leave circumference in terms of pi.
38.) Find the area of a circle with diameter 12 ft. Leave area in terms of pi.
39.) Find the area of a circle with circumference Leave area in terms of pi.
40.) Find the coordinates of the midpoint of with endpoints .
41.) K is the midpoint of . H has coordinates Solve for the coordinates of L.
Review Chapter 2
42.) Find the next two items in the pattern, then write a conjecture about the pattern: 0.7, 0.07, 0.007,…
43.) Find the next two items in the pattern, then write a conjecture about the pattern:
44.) Show that the conjecture is false by providing a counterexample: For every integer n, n5 is positive.
45.) Show that the conjecture is false by providing a counterexample: Two complementary angles are not congruent.
46.) Write the inverse and determine the truth value for the inverse: All even numbers are divisible by 2.
47.) Write the converse and determine the truth value for the converse: A triangle with one right angle is a right triangle.
48.) Write the contrapositive and determine the truth value for the contrapositive: If n2 = 144, then n = 12.
Determine if each conjecture in 49 and 50 is valid and by which law of deductive reasoning.
49.) If n is a natural number, then n is an integer. n is a rational number, if n is an integer. If n is 0.875 is a rational
number, then n is a natural number.
50.) If you do your homework, then your grade will be at least a C. You have a C-.
51.) For the conditional, “If an angle is a right, then its measure is 90 degrees,” write the converse and a biconditional
statement.
Converse:
Biconditional:
52.) Write the definition as a biconditional: An acute triangle is a triangle with three acute angles.
53.) Write, solve, and justify each step of an equation. J is a point on segment GH.
Draw a diagram. All lengths are in meters.
Review Chapter 3
For problems 54-61, identify the following using the diagram on the right.
54.) One pair of parallel segments
55.) One pair of skew segments
56.) One pair of perpendicular segments
57.) One pair of parallel planes
58.) One pair of alternate interior angles
59.) One pair of corresponding angles
60.) One pair of alternate exterior angles
61.) One pair of same-side interior angles
62.) Use diagram below to solve for x and y.
Use the diagram to the right to answer questions 63-70.
In problems 63-66, state the theorem/postulate that is related to the
measures of the angles in each pair. Then, write an solve an equation to
find the unknown angle measure.
63.)
64.)
65.)
66.)
1 2
4 3
6 5
8
m
7
l
In problems 67-70, name the theorem/postulate that proves 𝓂.
67.) ________________________________________________
68.) ________________________________________________
69.) ________________________________________________
70.) are supplementary _________________________________
71.) Write and solve an inequality for x.
72.) Solve to find x and y:
73.) Determine if XY ⃡ and AB ⃡ are parallel, perpendicular, or neither.
1 2
4 3
6 5
8
m
7
l
74.) Write the equation of the line through in slope-intercept form.
Review Chapter 4
Use the following diagram for problems 75-77 to classify each triangle by its angles and sides.
75.) ∆MNQ: ____________________________________________________
76.) ∆NQP: ____________________________________________________
77.) ∆MNP: ____________________________________________________
78.) Write and solve an equation to find the side lengths of the triangle.
79.) Solve for
80.) Write and solve an equation to find
81.) Given write and solve an equation to find x and AB. Draw a
diagram.
82.) Write and solve an equation to find y. All lengths are in centimeters.
83.) Write and solve an equation to find x.
Review Chapter 5
Use the diagram to the right for problems 84 and 85.
84.) Given that find the
85.) Given that
Use the diagram to the right for problems 86 and 87.
86.) Given that write and solve an
equation for FG.
87.) Given that EF=10.6 ft, EH=4.3 ft, and FG=10.6 ft. Write and solve an equation for EG.
88.) Write an equation for the perpendicular bisector of the segment with endpoints
89.) are the perpendicular bisectors of triangle ABC. ED=8 ft, DC=17 ft. What is BD?
90.) are angle bisectors of . PL=3 in, LK=4 in, and JP=5 in. Find the distance from P to HK.
Use the figure to the right for items 91-93. In , AE=12 ft, DG=7 ft, and BG=9 ft. Find each length.
91.) AG
92.) GC
93.) BF
For items 94-96, use with vertices Find the coordinates of each point.
94.) The centroid
95.) The orthocenter
96.) The circumcenter
Use the diagram to the right for items 97-99. Find each measure. All lengths are in miles.
97.) ED
98.) AB
99.)
100.) Write and solve an equation for n. All lengths are in meters.
101.) is the midsegment triangle of . What is the perimeter of ? All lengths are in feet.
102.) Draw , then write the angles in order from smallest to largest. AB=7 m, BC=9 m, and AC=8 m.
103.) Draw , then write the sides in order from shortest to longest .
104.) The lengths of two sides of a triangle are 17cm and 12cm. Find the range of possible lengths for the third side.
105.) Can a triangle have sides with lengths 2.7 m, 3.5 m, and 5.8 m? Would the triangle be acute, obtuse, or acute?
Show your work.
106.) Ray wants to place a chair 10 feet from his television set. Can the distance from the chair to his fireplace be 6 ft and
the distance between the fireplace and the television is 8 feet? Draw a diagram. Show your work or explain your
answer.
107.) Compare
108.) Compare EF and GF below.
109.) Find the range of values for k.
110.) Find the range of values for z.
111.) Compare
112.) Compare PS and QR below.
113.) Find the value of x. All lengths are in yards. If necessary, write your answer as a simplified radical.
114.) An entertainment center is 52 inches wide and 40 inches high. Will a TV with a 60 inch diagonal fit in it?
115.) Find the missing side length, if a=5 cm and b=12 cm. Do the side lengths form a Pythagorean triple? Explain.
116.) Can the sides of a triangle measure 7 m, 11 m, and 15 m? If so, classify the triangle as acute, obtuse, or right.
Find the values of the variables. Give your answers in simplest radical form. All sides lengths are in inches.
117.)
118.)
119.)
120.)
Find the perimeter and area of each figure. Give your answers in simplest radical form.
121.) A square with a diagonal length of 20 cm.
122.) An equilateral triangle with height 24 in.
123.) In a right triangle, the hypotenuse has length 18 – 3z and one leg has length z + 4. Draw a diagram and write an
inequality to show all possible values for z. All lengths are in feet.
Review Chapter 6
124.) Find the sum of the interior angle measures of a convex 11-gon.
125.) Find the measure of each interior angle of a regular 18-gon.
126.) Find the measure of each exterior angle of a regular 15-gon.
In parallelogram PNWL, are diagonals that intersect at point M. NW=12 m, PM=9 m, and
Draw the diagram and find the following measures.
127.) PW 128.)
QRST is a parallelogram. Find each measure. All lengths are in feet.
129.) TQ
130.)
131.) Three vertices of parallelogram are . Find the coordinates of vertex D. Show
your reasoning.
132.) Determine if QWRT must be a parallelogram. Justify your answer.
133.) Show that the quadrilateral with vertices is a parallelogram.
Proof Review
134.) Write a 2-column proof.
Given: RSTU is a parallelogram
Prove:
135.) Write a 2-column proof.
Given: 𝓇 𝓈;
Prove: 𝓂
6
7
5
4
ℓ
𝓂
𝓇
𝓈
136.) Write a 2-column proof.
Given:
Prove:
1
2
Q U
D
A
137.) Write a 2-column proof.
Given:
Prove: is an obtuse triangle
M
L
G S
138.) Write a 2-column proof.
Given:
Prove:
W
L A U R
1 2 3 4
5 6 7
139.) Write a 2-column proof.
Given: K is the midpoint of ;
Prove:
M K R
G
140.) Write a 2-column proof.
Given:
Prove:
L
X T
P
4 3
2 1
R
141.) Write a 2-column proof.
Given:
Prove:
L
X T
P
4 3
2 1
R
142.) Write a 2-column proof.
Given:
Prove:
1 4
3 2
5 6
G
R
V
S
T
143.) Given:
Prove:
144.) Write a 2-column proof.
Given:
Prove:
145.) Write a 2-column proof.
Given:
Prove:
146.) Write a 2-column proof.
Given:
Prove:
147.) Write a 2-column proof.
Given:
Prove:
148.) Write a 2-column proof.
Given: 1 2; p ⊥ q
Prove: p ⊥ r
t
u
t
149.) Write a two-column proof.
Given: C is the midpoint of
;
Prove:
150.) Write a 2-column proof.
Given:
Prove:
151.) Write a 2-column proof.
Given: ;
Prove:
152.) Write a 2-column proof.
Given:
Prove:
153.) Write a 2-column proof.
Given:
Prove: