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: