Geometry 2014 – 2015 Midterm
Review
Vocabulary available online.
Problems will be set up, not solved.
Cartoons at the end of each chapter.
Chapter 1
Angle pairs, Midpoint and Distance
Formula, Area of Polygons
Angle Pairs
• Angle vocabulary Complementary: Two angles whose sum is 90o.
Supplementary: Two angles whose sum is 180o.
Vertical: Two opposite congruent angles.
Linear Pair: Two adjacent angles whose sum is 180o.
• Distance formula: √((x2 – x1)2 + (y2 – y1)2)
• Midpoint formula: (x1 + x2, y1 + y2)
2
2
Practice
• What is the measure of each angle?
3x + 15
7x + 5
3x + 10
x
2x + 10
5x – 20
Practice
• A = (5, 2)
B = (-3, 6)
What are the length and midpoint of AB?
Cartoon #1
Chapter 2
Key Topics: Statements, Laws, and
Proofs
Conditional Statements
• Symbols: p = hypothesis, /\ = and, ~ = not, → = if, then
q = conclusion, \/ = or, = therefore
•
•
•
•
Conditional: p → q
Converse: q → p
Inverse: ~p → ~q
Contrapositive: ~q → ~p
Practice
• What is the symbolic representation of each statement?
p: A and B are both 45o angles.
q: A and B are complementary.
• If A and B are both 45o angles, then A and B are complementary.
• If A and B are not complementary, then A and B are not both 45o
angles.
• A and B are both 45o angles. Therefore, A and B are both 45o
angles.
Laws of Logic
• Law of Syllogism:
• Law of Detachment:
Practice
• What conclusion can be drawn from each set of statements?
• If x = 5, then 3x + 7 = 22.
x = 5.
• If a number is prime, then it is not a perfect number.
If a number is not perfect, then it is not the sum of its factors.
Therefore, what can be concluded about 5?
Equation Proofs
• Two column proofs: Two columns are constructed
with statements on the left and reasons on the right.
For every step, a postulate or theorem must be given
to justify the step.
• Statements and reasons: Statements are the “work”
side. Reasons are the postulate or theorem
permitting such a step.
Practice
1) Given: 3(x + 2) = 15
Prove: x = 3
Statements Reasons
1) 3(x + 2) = 15
2) ____________
3) 3x = 9
4) ____________
___________
Distributive
___________
Division
2) Given: ½ x + 6y = 12, y = 3 Statements Reasons
Prove: x = -12
1) ½ x + 6y = 12,
y=3
2) ____________
3) ½ x = -6
4) ____________
___________
Substitution
___________
Multiplication
Cartoon #2!
Chapter 3
Lines and Parallel Line Properties
Transversal Angles
• Corresponding: Congruent.
Ex: 1 & 5
• Same-Side Exterior: Supplementary.
Ex: 2 & 8
• Same-Side Interior: Supplementary.
Ex: 4 & 6
• Alternate Interior: Congruent.
Ex: 3 & 6
• Alternate Exterior: Congruent.
Ex: 1 & 8
1 2
3 4
5 6
7 8
a
b
Practice
• For what value of x is a||b?
(7x + 2)
(3x + 20)
a
b
Slope
• Slope formula: m =
y2 – y1
x2 – x1
• y-intercept form: y = mx + b
• point-slope form: (y – y1) = m(x – x1)
• Parallel line: A line with the same slope.
• Perpendicular line: A line with negative reciprocal
slope.
Practice
• What is the slope of 3x + 2y = 12?
• What is the equation of a line through (-2, 3)
perpendicular to y = -3x + 4
Interlude: Compass
Constructions
Combination of chapters 1 – 3
Perpendicular Bisector

Construct a Perpendicular Bisector
A
B

Construct an Angle Bisector
Construct a Congruent Angle
Cartoon #3!
Chapter 4
Properties of a Triangle, Triangle
Congruency, Introduction to Proofs
Basic Triangle Properties
• Triangle angle-sum theorem: The sum of all angles of a
triangle is 180o.
• Triangles by sides: Classification by side length.
Scalene: No congruent sides.
Isosceles: Two congruent sides.
Equilateral: Three congruent sides.
• Triangles by Angle: Classification by angle measure:
Acute: Three acute angles.
Right: One right angle, two acute angles.
Obtuse: One obtuse angle, two acute angles.
Practice
• How would a 45o-45o-90o be classified?
• How would a 30o-60o-90o be classified?
Triangle Segments
• Segments in a triangle: Starting from the vertex of a triangle.
Angle Bisector: A ray that splits an angle into two congruent
smaller angles.
Altitude: A segment from a vertex to the opposite side
forming a right angle.
Median: A segment from a vertex to midpoint.
Perpendicular Bisector: An altitude connecting to the opposite
midpoint.
Practice
• Identify each segment:
Yellow:
Purple:
Green:
Triangle Congruency
• Triangle Congruency: Must have three characteristics
SSS: Three congruent sides.
SAS: A congruent angle between two congruent sides.
AAS: One congruent side not between two congruent
angles.
ASA: One congruent side between two congruent angles.
HL: A right angle, congruent leg, and congruent
hypotenuse.
Practice
• Why is each pair of triangles congruent?
Proofs
• Proof: A logical explanation of the steps used to
reach a conclusion.
• Components commonly used in triangle proofs:
Reflexive sides: When two triangles share a side.
Vertical angles: When two triangles make an X shape.
Alternate interior angles: When two parallel lines are
present.
Perpendicular bisector: Makes two sides reflexive
and two angles congruent.
Practice
• Given: ∠CAB ≅ ∠ACD. AB || CD.
Prove: △ABC ≅ △CDA.
A
D
B
C
Statements
1) ____________
2) ∠ABC ≅ ∠CDA
3) AC ≅ AC
4) ____________
Reasons
Given
____________
____________
____________
Advanced Triangle Properties
• Overlapping triangles: Redraw separately and mark
any previous congruencies.
• Corresponding Parts of Congruent Triangles are
Congruent: After a triangle has been proven
congruent, shows two sides or angles are congruent.
• Exterior angles: An exterior angle is the sum of its
two remote interior angles.
Practice
• What is the value of
the exterior angle?
Given: △ABC ≅ △CDA
Prove: BC ≅ DA.
Statements
65o
55o
12x + 20
1) △ABC ≅ △CDA
2) ____________
Reasons
Given
Cartoon #4
Chapter 5
Midsegment, Centroid, and Triangle
Side Lengyhs
Midsegment and Centroid
• Midsegment: A segment formed by joining two
midpoints. Half the length of the base.
• Centroid: The point where three medians intersect.
Midpoint to centroid: 1/3 of median length.
Vertex to centroid: 2/3 of median length.
Practice
• What is the value of each variable?
A
CD = 21
12
D
E
2x + 20
z
G
y
200
B
F
C
Triangle Inequalities
• Longest-side largest-angle Theorem: The longest side of a
triangle is opposite the largest angle.
• Triangle Inequality Theorem: The sum of the two smaller
sides of a triangle must be greater than the third side.
• Hinge Theorem: If two triangles have two congruent
sides, and the angle between one pair of congruent sides
is larger than the other, the third side is longer on the
triangle with a greater included angle.
Practice
• Which set of side lengths
will make a triangle?
A)
B)
C)
D)
E)
5m, 5m, 8m
3m, 3m, 3m
17m, 28m, 20m
35m, 21m, 14m
11m, 19m, 20m
Which side is greater,
BC or EF?
A
105o
B
C
D
100o
E
F
Cartoon #5
Chapter 6
Polygons, Parallelograms, and
Coordinate Geometry
Properties of Polygons
• Polygon Angle-Sum Theorem: The sum of all angles
of a polygon is (n – 2)*180. The measure of each
angle of an equiangular polygon is (n – 2) * 180
n
• Exterior Angle-Sum Theorem: The sum of all exterior
angles is 360o. The measure of each exterior angle
of an equiangular polygon is 360.
n
Practice
• What is the value of each variable?
• How many sides does a
polygon with a 10o exterior
angle have?
y
x
6 Properties of a Parallelogram
• Parallelogram Properties:
1) Two pairs of congruent, opposite sides.
2) Two pairs of parallel sides.
3) One pair of opposite, parallel, congruent sides.
4) Congruent opposite angles.
5) Supplementary consecutive angles.
6) Bisecting Diagonals.
Practice
• For what value of x is ABCD a parallelogram?
A
B
25o
3x + 2
x
26
30o
D
C
Special Parallelograms
• Rhombus:Equilateral.
Perpendicular diagonals.
Diagonals form angle bisectors.
• Rectangle: Equiangular.
Congruent diagonals.
Diagonals form complementary angles.
• Square:
Regular.
Congruent perpendicular diagonals.
Diagonals form 45o angles at vertices.
Practice
For what values of x is ABCD a rhombus? Rectangle?
Why would a square
B
be impossible?
A
3x + 5
C
6x – 20
D
Coordinate Geometry
• Draw any image first. Then to…
• Prove a quadrilateral is a rhombus: Show the
diagonals are perpendicular (slope formula).
• Prove a quadrilateral is a rectangle: Show the
diagonals are congruent (distance formula).
• Prove a quadrilateral is a square: Show the diagonals
are perpendicular and congruent.
Practice
•  ABCD has vertices A (-4, 2), B (-1, 6), C (4, 6), D (1, 2).
Show  ABCD is a rhombus.
Cartoon #6!
Chapter 7
Ratios and Proportions, Similar
Polygons, Similar Triangles, and
Similar Right Triangles
Ratios and Proportions
• Ratio: A comparison between two objects.
• Extended ratio: A comparison between three or
more objects.
• Proportion: A comparison between two ratios.
Practice
• The ratio of two complimentary angles is 2 : 5. What
is the measure of each angle?
• The perimeter of a triangle is 72m, and the ratio of
its sides is 5 : 6 : 7.
What is the length of each side?
Similar Polygons
• Similar Polygons: Two polygons with congruent
angles and proportional side lengths.
• Scale Factor: The ratio of corresponding sides/A
number each side may be multiplied by to find the
length of the corresponding side.
Practice
• ABCD ~ EFGH. What side does AB correspond to?
What is the scale factor of AB to EF?
What is the value of each variable?
30
F
E
A 20
B
3y
x
15
D
3z
C
H
4z + 25
30
G
Triangle Similarity
• SSS~: All three sides are proportional.
• SAS~: Two sides proportional, and congruent
included angle.
• AA~: Two congruent angles.
• Tip: Look for parallel lines to show AA~.
Practice
• Given: RS||TU.
Why is △QRS ~ △QTU?
Why is △A ~ △B?
Q
9
12
A
R
T
S
U
3
B
4
Similar Right Triangles
A
B
• Similar Right Triangles: An altitude constructed from
the vertex of a right angle forms two smaller similar
right triangles.
Ex: ∆ABD ~ ∆ACB ~ ∆ BCD
A
D
B
C C
B
D
Practice
• ∆ABC is split by an altitude into two smaller right
triangles.
– Draw ∆ABD and ∆DBC
– Write a similarity statement for the three right triangles
A
12
B
5
13
C
Side-Splitter and Angle-Bisector
Theorems
Q
• Side-Splitter Theorem:
If RS||TU, then QR = QS
RT
SU
R
• Angle Bisector Theorem:
If AC bisects ∠DAB, then
AB = AD
BC DC
S
T
U
A
B
C
D
Practice
• What is the value of each variable?
A
Q
5
T
x
7
R
S
6
B
4
U
12
y
C
8
D
Cartoon #7!
Chapter 8
Pythagorean Theorem, Special Right
Triangles, and Trigonometry
Pythagorean Theorem
• Pythagorean Theorem: If ABC is a right triangle, then a2 +
b2 = c2, where c is the hypotenuse.
• Reducing Radicals: Factor the radical. Then, circle any
pairs of factors. Remove one member from any pairs and
leave remaining factors inside the radical.
• Converse of the Pythagorean Theorem:
If a2 + b2 = c2, then ABC is a right triangle.
If a2 + b2 > c2, then ABC is an acute triangle.
If a2 + b2 < c2, then ABC is an obtuse triangle.
Practice
• What is the missing side length of each triangle? Put
solutions in simplest radical form.
4m
6m
10m
5m
Special Right Triangles
45o – 45o – 90o: The hypotenuse is √2 times longer than
another side.
30o – 60o – 90o: The hypotenuse is twice as long as the
side opposite 30o. The side opposite 60o is √3 times
longer than the side opposite 30o.
SOL Question
• What is the length of the
missing side?
Trigonometric Ratios
Sin(θ) = opp
hyp
B
hypotenuse
Cos(θ)= adj
hyp
θ
A
Tan(θ)= opp
adj
adjacent
opposite
C
Practice
sin A =
B
cos B =
tan A =
5
3
4
C
θ
A
Inverse Trigonometric Ratios
• Inverse Trig Ratios: If both side lengths are known,
then use the inverse operation to find the value of an
angle.
B
• Ex: tan(θ) = ¾
tan-1(tan(θ)) = tan-1(¾)
θ = tan-1(¾) = 36.87o
5
3
4
C
θ
A
Practice
• What are the values of ∠B and ∠C?
A
12
5
13
C
B
Cartoon #8