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Geometry Review
KCAS 7.G.3-6, 8.G.2, 8.G.4, 8.G.5, 8.G.9
W. 11-19
Yellow highlighted need correction before copy.
Red font helps with LRAHD assignments (do #26 and #30 in class).
Blue font helps you find formulas and REMEMBER TO STATE UNITS on answers
KEY for Quiz for Learning is at very end of this document in green font.
TOOLS to measure length:
ruler, yard stick, meter stick, tape measure, etc.
TOOLS to measure distance:
odometer
TOOLS to measure temperature:
thermometer
TOOLS to measure weight:
scale, balance scale, etc.
TOOLS to measure volume:
measuring cup, measuring spoon, graduated cylinder, beaker, etc.
TOOLS to draw circles: COMPASS
TOOLS to measure angle measure:
PROTRACTOR
TOOLS to measure speed:
speedometer, radar gun, etc.
UNITS of length:
feet, inches, yards, miles, meters, decameters, millimeters, etc.
UNITS of temperature: Fahrenheit, Celsius, kelvin, etc.
UNITS of weight:
ounces, pounds #, tons, kilograms, grams, centigrams, etc.
UNITS of volume:
cups, ounces, teaspoons (tsp), pints, quarts, bushels, pecks, liters, etc.
UNITS of angle measure:
degrees °, minutes ‘ (60 ‘ = 1 °), etc.
UNITS of speed:
miles per hour (mph), kilometers per hour, etc.
If asked to find perimeter, etc. , ALWAYS put UNITS on your answers. If no units are given, just say UNITS.
The standard UNIT of LENGTH in the METRIC system is the METER. A meter is about 3” longer than a yard.
The standard UNIT of WEIGHT in the METRIC system is the GRAM. A paper clip weighs about 1 gram.
The standard UNIT of VOLUME in the METRIC system is the LITER. Think of a one liter bottled drink.
Kilo1000
k
Hecto100
h
Deca10
da
-m
-g
-l
deci1/10
d
centi1/100
c
milli1/1000
m
To convert units within the metric system, MOVE DECIMAL point using chart above (direction & # of “jumps”)
Convert 4 dam to mm.
Convert 7.3 l to kl.
UNITS in the ENGLISH system.
Length: inch (in), foot (ft), yard (yd), mile (mi)
12 in = 1 ft
5280 ft = 1 mi
3 ft = 1 yd
1760 yd = 1 mi
Volume: fluid ounce (oz), cup (c), pint (pt), quart (qt), gallon (gal)
2 c = 1 pt
32 oz = 1 qt
2 pt = 1 qt
4 qt = 1 gal
Weight: ounce (oz), pound (lb), ton
16 oz = 1 lb
2000 lb = 1 ton
Time: second (s), minute (min), hour (h), day (d), year (y)
60 s = 1 min
24 h = 1 d
60 min = 1 h
3651/4 d = 1 y
52 weeks = 1 year
7 days = 1 week
Adv. Math Geometry Review W. 11-19
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In math, the word UNIT typically means ONE. To convert units in the English system use UNIT MULTIPLIERS. When you
multiply by 1, due to the multiplicative identity property, you don’t change your value.
Convert 5 miles to inches.
Convert 55 miles per hour to feet per second.
To convert ENGLISH units to METRIC units, use UNIT multipliers.
500 g = 1.1 #
3.785 L = 1 gal
2.54 cm = 1 in
1 km = 0.6 mi
Convert 4 feet to meters.
Convert 5 km to feet.
No-Dimensional (No-D):
An object that has no size.
Example: point
A point is named by a capital letter. It indicates a location in space.
One-Dimensional (1D):
An object that has length only.
LENGTH IS STATED IN LINEAR units like feet, inches, miles, cm, m, km, etc.
A symbol for feet is ‘.
A symbol for inches is “.
How tall are you if you are 5’6”?
Example: Line Segment, ray, or line
Since 2 points determine a line/ray/segment, these are named by 2 points (2 uppercase letters).
Line Segments have finite length, 2 endpoints used to name it, and ∞ points on them.
Rays have infinite length, 1 endpoint, and ∞ points on them. FIRST letter in name is endpt.
Lines have infinite length, no endpoints, and ∞ points on them. They may be named by any 2
points on them, in any order, OR a cursive, lowercase letter.
COLLINEAR points are points that may lie on the same line.
Non-collinear points: points that cannot have a unique line drawn through ALL of them.
Would 2 points always be collinear? Explain.
MIDPOINT: A point in the MIDDLE of a line segment dividing it into 2 CONGRUENT
segments (2 segments of equal length). The symbol for “is congruent to” is
≅
GH ≅ IJ is read in words as segment GH is congruent to segment IJ
If M is the midpoint of segment AB, and AB=10, find AM and make a statement about AM and MB.
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Parallel lines are lines on the SAME PLANE that do NOT intersect. The symbol for “is
parallel to” is ǁ.
Perpendicular lines are lines that intersect to form right, 90°, angles. The symbol for “is perpendicular to” is
AB ǁ b
CD EF
is read in words as Ray AB is parallel to LINE b.
is read in words as line CD is perpendicular to line segment EF.
When 2 lines intersect (cross), their point of intersection is a point.
Skew lines are lines on different planes that do NOT intersect.
AB = 7 in
is read in words: The length of segment AB is equal to 7 Inches.
CD ≅ EF
is read in words as Segment CD is congruent to segment EF.
Can lines or rays be congruent? Explain.
Two-Dimensional (2D): A flat surface that has length and width.
You can find perimeter, circumference, and area of 2-D objects.
Perimeter & Circumference are STATED IN LINEAR units like feet, inches, miles, cm, m, km, etc.
AREA IS STATED IN SQUARE units like square feet, square inches, sq miles, sq cm, sq m, sq km,
etc. We write these as ft2 , m2, but still read them as square feet, square meters, etc.
Example: Planes, Polygons (chart attached and on-line), circles, semicircles, etc.
A plane is a 2-D object (flat surface) with infinite length and width. It is named by a
cursive, uppercase letter or 3 NON-COLLINEAR points on the plane. We represent planes by a
parallelogram.
Planes are determined by 3 non-collinear points. Explain.
When 2 planes intersect, their intersection forms a ______________________.
COPLANAR – points are coplanar if they lie on the same plane.
ANGLES may be named by their vertex if only one angle is at that vertex. Angles may also be named by
3 points with VERTEX IN THE MIDDLE, or angles can be named by a number.
There are 4 types of angles.
Acute Angles: Angles that measure between 0° and 90°.
Right Angles: Angles that measure 90°. They are formed by perpendicular lines and indicated by a small
square on them.
Obtuse Angles: Angles that measure between 90° and 180°.
Straight Angles: Angles that measure 180° and form a STRAIGHT LINE if pictured.
How many degrees are in a complete rotation?
Adjacent Angles: 2 angles that share a common side and common vertex and don’t overlap.
Complementary Angles: 2 angles whose measures sum to 90.
Supplementary Angles: 2 angles whose measures sum to 180.
Vertical Angles: Angles opposite each other when 2 LINES intersect. They are formed by opposite rays
and share a common vertex. VERTICAL ANGLES ARE ALWAYS CONGRUENT!!!
In a plane, lines will either intersect (cross) or be parallel.
If 2 lines are cut by a transversal, certain angles are formed and have relationships.
Interior Angles: Angles on the inside of the lines.
Exterior Angles: Angles on the outside of the lines.
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Corresponding Angles: On same side of transversal and both on top of lines (or below to form another
pair). Corr Angles are congruent if PARALLEL lines are cut by a transversal. Examples of pairs of
corresponding angles are <1 & <5, <3 & <7, <2 and <6, <4 & <8
Alternate Interior Angles: 2 NON-ADJACENT angles on DIFFERENT sides of transversal and inside the
parallel lines (like <3 and <6 pictured below). Alt Int Angles are CONGRUENT if PARALLEL lines are cut by
a transversal.
Alternate Exterior Angles: 2 NON-ADJACENT angles on DIFFERENT sides of transversal and OUTSIDE the
parallel lines. Alt Ext Angles are CONGRUENT if PARALLEL lines are cut by a transversal. <1 and <8 are a
pair of alt ext <s below.
If a ǁ b , please find the measure of all angles below.
t
a
b
A polygon is a closed, 2-D object made of line segments. See polygon flow chart (attached & online).
Polygons can be concave (an internal angle greater than 180°; think “caves” in) or convex (no internal
angles greater than 180°).
EQUILATERAL: All sides are congruent (equal side lengths). We denote this by same # of hash marks on congruent sides.
EQUIANGULAR: All angles are congruent. We denote this by arc with same # of hash marks OR arc with identical #s.
Polygons can be regular. Regular means all angles are congruent AND all sides are congruent.
The sum of the interior angle measures of a polygon is found by the formula (n-2)180 where n is the number of
sides the polygon has.
The sum of the interior angle measures in a triangle is 180.
What is the measure of each interior angle is a REGULAR PENTAGON?
What is the perimeter of a regular octagon with one side of length 20 cm?
Exterior Angles of polygons: The angle between any side of a shape, and a line extended from the next side.
The exterior angle and the interior angle to which it is ADJACENT are SUPPLEMENTARY. All exterior angles of a polygon
(one at each vertex) sum to 360.
Lines of Symmetry: Divide an object into 2 congruent halves.
In a triangle, if two sides are congruent, two angles are congruent and those angles are opposite those sides.
In a triangle, if three sides are congruent, three angles are congruent.
In a triangle, the longest side is opposite the largest angle. The smallest side is opposite the smallest angle.
a. In an equilateral triangle, what would the measure of each angle be?
b. In an isosceles right triangle, what would the measure of each angle be?
c. An equilateral, equiangular triangle could be called a ___________________ triangle.
d. A regular QUADRILATERAL is more commonly referred to as a _______________.
e. What is the perimeter of a regular HEXAGON with one side of length 10 cm?
The following only applies to RIGHT TRIANGLES!!!!
In a right triangle, the sides that form the right angle are called LEGS.
In a right triangle, the side OPPOSITE the right angle is the LONGEST side and it is called the HYPOTENUSE.
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If you know two side lengths in a right triangle, you can find the third side length by using the PYTHAGOREAN
THEOEM. The Pythagorean Theorem states that: If you have a right triangle, then the SUM of the squares of
the lengths of each leg is equal to the length of the hypotenuse squared.
Note the CONVERSE of the Pythagorean Theorem is also true: IF the sum of the squares’ of the leg lengths is
equal to the square of the hypotenuse’s length, THEN the triangle is a right triangle.
Whole Numbers: { 0, 1, 2, 3, … }
Prime numbers: Whole numbers greater than 1 that have only two factors, 1 and the # itself. {2,3,5,7,11,13,17,19,23,…}
Composite Numbers: WN greater than 1 that have more than 2 factors. {4,6,8,9,10,12,14,15,16,18,20,21,22,…}
Relatively Prime: A set of numbers is relatively prime if they collectively share no common factors other than 1.
4 and 15 are relatively prime.
4, 12, and 15 are relatively prime because they ALL share no CFs other than 1.
4, 12, and 18 are NOT relatively prime because they ALL share a common factor of 2.
Primitive Pythagorean Triples are three WHOLE numbers that are relatively prime and satisfy the Pythagorean
Theorem. 6 sets of ppt are listed below.
3, 4, 5
5, 12, 13
7, 24, 25
8, 15, 17
9, 40, 41
20, 21, 29
From each PPT, you can generate other Pythagorean triples. 3,4,5 yields 6,8,10; 9,12,15; 12,16,20; etc.
What other Pythagorean Triples would 5, 12, 13 yield?
Careful as ACT knows you use these short cuts and will try to trick you. The hypotenuse must be longest side!!!
Given the information above, find…
a. The length of the hypotenuse in a right triangle with leg lengths of 9 inches and 12 inches.
b.
The length of a leg in a right triangle with one leg length of 9 meters and hypotenuse length of 12
meters.
c.
Find the perimeter of right triangle ABC where m<B=90, AB=3 and BC=4.
d. Find the perimeter of right triangle ABC where m<B=90, AB=3 and AC=4.
e. Use the PT to find the length of this segment with endpoints at (-2, 1) and (4, 3) on the Cartesian
Plane (Coordinate Plane).
****NOTE: Sides/angles/figures/objects cannot be equal. Only #s can be equal. Since side LENGTHS are #s,
side lengths can be =; angle measures can be =. If side lengths are equal, the sides are said to be congruent.
Adv. Math Geometry Review W. 11-19
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GH = IJ The length of segment GH is equal to the length of segment IJ
GH ≅ IJ
Segment GH is congruent to segment IJ
<J≅<L
Angle J is congruent to angle L
m<J=m<L
the measure of angle J is equal to the measure of angle L
Find all angle measures below if the lines pictured are parallel.
2 figures/objects can be congruent (exact same size and shape – CORRESPONDING sides congruent to each
other and CORRESPONDING angles congruent to each other). 2 Objects can be similar (corresponding side
lengths proportional and corresponding angles congruent – same shape but NOT same size). The symbol for
“is similar to” is ~
∆ABC ≅ ∆DEF
order)
Triangle ABC is congruent to Triangle DEF (NOTE: corresponding parts must be named in
∆GHI ~ ∆MNO Triangle GHI is similar to Triangle MNO.
order!! angles congruent; side lengths proportional)
(NOTE: corresponding parts must be named in
A circle is the locus of all points equidistant from a set point called the center.
Chord: A segment in a circle with its endpoints on the circle.
Diameter: A chord in a circle that contains its center point as its midpoint. The d=2r
Radius: A segment in a circle with one endpoint at the circle’s center and the other on the circle. The
r=(1/2)d.
****Are r2 and 2r equivalent? Explain.
Three-Dimensional (3D): An object with length, width, and height.
You can find lateral surface area, surface area, and volume of 3-D objects.
Example: prism, pyramid, cube, cylinder, cone, sphere, hemisphere, etc.
Any kind of AREA or SURFACE AREA IS STATED IN SQUARE units like square feet, square inches, sq
miles, sq cm, sq m, sq km, etc. We write these as ft2 , m2, but still read them as square feet, square meters, etc.
VOLUME IS STATED IN CUBIC units like cubic feet, cubic cm, etc. We write these as ft3 , cm3, but still
read them as cubic feet, cubic centimeters, etc.
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Face - flat (planar) surface that forms part of the boundary of a solid 3-D object
Lateral Face - The faces that join the bases of a 3-D solid
Base - The faces that join the lateral faces of a 3-D solid
Prism – A 3-D object with TWO congruent POLYGONAL BASES and RECTANGULAR LATERAL FACES.
Pyramid – A 3-D object with ONE POLYGONAL BASE and TRIANGULAR LATERAL FACES.
Cone – A 3-D object with ONE CIRCULAR BASE.
Cylinder – A 3-D object with TWO congruent CIRCULAR BASES
Prisms and pyramids are named by their bases. Be careful with triangular pyramids and rectangular prisms - often
misnamed.
Sphere - a round solid 3-D figure, or its surface, with every point on its surface equidistant from its center. Ex: ball.
Hemisphere – half of a sphere.
Cube – a prism with 6 congruent square faces.
Is a square prism synonymous with cube? Explain.
A triangular PYRAMID has …
____ base(s) that are ______
____ lateral faces that are _____
____ TOTAL FACES
A triangular PRISM has …
____ base(s) that are ______
____ lateral faces that are _____
____ TOTAL FACES
Draw a pentagonal pyramid, pentagonal prism, cylinder, cone, and sphere.
Compare (how are they alike) and contrast (how are they different) pyramids, prisms, cones, and cylinders.
All are 3-D.
Prisms and Cylinders have two bases. Pyramids and cones have one base.
Prisms and Pyramids have polygonal bases. Cones and Cylinders have circular bases.
If we slice a cylinder with a plane, what is formed?
If we slice a pyramid with a plane, what is formed?
Perimeter is the distance AROUND any 2-D object. It is found by adding the side lengths and stated in linear units.
Circumference is the distance AROUND a CIRCLE and stated in linear units. It is found by the formula C = π d where d is
diameter. Pi (π) is an irrational number that is the ratio of the C to the D in a circle. π ≈ 3.14 or π ≈ 22/7.
Area is the number of SQUARES that will COVER a 2-D surface. Make sure to tell how big the squares are!!!
Lateral Surface Area is the sum of the surface areas of all its faces excluding the base of the 3-D solid.
Surface Area is the sum of the surface areas of all the faces in a 3-D solid.
Volume is the number of CUBES that will FILL a 3-D object. Make sure to tell how big the cubes are!!!
7 km2 is read in words as
9 cm3 is read in words as
7 square kilometers.
9 cubic centimeters.
Adv. Math Geometry Review W. 11-19
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What does an area of 300 square inches mean?
An area of 300 in2 means that 300 SQUARES that are 1 inch by 1 inch (1”x1”) would completely COVER the 2-D surface.
What does a volume of 20 cubic meters mean?
A volume of 20 m3 means that 20 CUBES that are 1 meter by 1 meter by 1 meter (1m x 1m x 1m) would completely FILL
the 3-D object.
What is the difference in Surface Area and LATERAL SURFACE AREA?
Surface Area is the number of squares that would cover ALL FACES (bases AND lateral faces) of a 3-D object.
Lateral Surface Area is the number of squares that would cover only all the lateral faces (does NOT include bases).
Does a rectangle, square, polygon, 2-D figure, have volume? Explain.
No, 2-D objects do not have volume because you cannot FILL them with CUBES. They are only surfaces that can be
covered with squares.
Does a rectangle, square, polygon, 2-D figure, have area? Explain.
Yes, 2-D objects do have area because you can COVER them with squares since they are surfaces.
Find the radius and diameter of a circle with circumference of 50 units.
C=πd
50 ≈ 3.14d
15.92 ≈ d
15.92/2 ≈ r ≈ 7.96
Therefore, diameter is approximately 15.92 units and radius is approximately 7.96 units.
C=πd
50 = π d
50/π = d
50/π*(1/2) = r
50/(2π) = r
25/π = r
Therefore, diameter is exactly 50/π units and radius is exactly 25/π units.
Find C and A for circle with radius 15 cm.
d=30 cm
C=πd
C=30π cm or approximately 94.2 cm
A=πr2
A=π (15)2
A=225π cm2 or approximately 706.5 cm2
Find LSA, SA, and V of cylinder with height 4 inches and diameter 6”.
h=4 and d=6 so r=3 INCHES
LSA cylinder = π dh
=24π in2
SA cylinder = 2 π r2 + π dh
18π + 24π = 42π in2
Volume of a cylinder = πr2h
=36π in3
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FORMULAS!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Remember: If 2-D, have area and perimeter; if 3-D, have SA, LSA, and V.
Base and height must be perpendicular to each other.
Formulas that must be memorized are denoted by * (other formulas can be derived from them).
ALL UNITS must be the same before substituting into formulas (if given lengths in cm and m, change to all cm OR all m, etc.)!!!!!!!!!!!
LINEAR UNITS
P
*Perimeter = sum of all side lengths.
C
*C = π d
USING FORMULA ABOVE, Circumference of a semi-circle including the diameter = πr + d
USING FORMULA ABOVE, Circumference of a semi-circle NOT including the diameter = πr
****If you keep π, your answer is exact and you may use =.
****If you SUBSTITUTE in 3.14 for pi, your answer is approximate and you must use ≈ (is approximately equal to).
SQUARE UNITS
A
*Area of a circle = π r2
Area of a semicircle = (π r2 ) / 2
*Area of a square or rectangle = l * w
Area of a triangle = (b*h)/2
Base and height must be perpendicular to each other. Height is perp distance bet a side and opp vertex.
*Area of a trapezoid = (sum of parallel bases)h
2
Height in a trapezoid is perpendicular distance between the bases.
Area of parallelogram = bh
LSA
*Lateral Surface Area = (perimeter of the base)h
LSA cylinder = π dh
LSA rt circ cone = πr √(r2 + h2)
SA
*Surface Area = sum of areas of all exposed faces
SA rect/sq prism or cube = 2lw + 2wh + 2lh
Surface Area of a cube or right rectangular prism = (perimeter of base)(h of prism) + 2(area of base)
SA of pyramid = (perimeter of base)(h of pyramid) + area of the base
SA of cone = π r2 + π r (slant height) OR SA of cone = π r2 + πr √(r2 + h2)
SA cylinder = 2 π r2 + π dh
*SA sphere = 4πr2
CUBIC UNITS
V
*Volume cylinder/prism = (area of the base) h
*Volume cone/pyramid = 1/3 (area of the base) h
* Volume of a sphere = 4/3 π r3
Volume of a cube or right rectangular/square prism = lwh
Volume of a cylinder = πr2h
Volume of a right circular cone = 1/3 πr2h
Adv. Math Geometry Review W. 11-19
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1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Quiz FOR learning
KEY for Quiz for Learning is at very end of this document.
Define and give an example of a No-D figure.
Describe point.
Define area and surface area.
Define volume.
Define perimeter and circumference.
Define radius and diameter.
Define pi.
Define square in three different ways.
Define undecagon.
Define trapezoid.
Draw segment MA and label its midpoint D.
a.
b.
c.
Then, make a congruency statement using the picture you drew.
If MA = 10 cm, then MD = ________
If AD = 6 yds, then MA = __________
12. Draw, label, and name a -e
a.
b.
c.
d.
e.
f.
line
ray
segment
angle
plane
convex polygon
g.
h.
i.
j.
k.
l.
13. Name in all possible ways
a.
concave pentagon
regular hexagon
equiangular triangle
equilateral octagon
4 noncollinear points
5 coplanar points
c. _____________
G
H
b.
d.
14. Fill in the blank with the word linear, square, or cubic.
a.
b.
c.
d.
15.
16.
17.
18.
19.
20.
Units of length are _______ units
Units of surface area are _______ units
Units of circumference are _______ units
Units of volume are _______ units
e.
f.
g.
Units of area are _______ units
Units of lateral surface area are _______ units
Units of perimeter are _______ units
What does an area of 16 square inches mean?
What does a surface area of 17 square kilometers mean?
What does a volume of 18 cubic centimeters mean?
To abbreviate 12 square centimeters, we write ____
What is the sum of the interior angle measures in a dodecagon?
Draw in all lines of symmetry for each object below. If none, write none. If infinitely many, draw some and ∞.
c.
e.
a.
d.
b.
21. How would you read in words?
a.
b.
5‘
10”
c.
d.
75°13’
11 m3
Adv. Math Geometry Review W. 11-19
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e.
m<7=25°
f.
g.
h.
i.
<TUV ≅ <W
TU = 5 mi
t ǁ PQ
RS XY
j.
k.
l.
m.
∆ GHI ≅ ∆ JKL
∆ MNO ~ ∆ PQR
π ≈ 22/7
∞
22. Identify the types of angles below
a.
b. Find the measure of each angle above.
23. Find ALL angle measures in the following triangles; then classify the ∆ by its side lengths & angle measures.
a. ∆WYZ where m<W=2x, m<Y=4x, and m<Z=24
b. ∆ABC where m<A=m<B and m<C=80. Also, make congruency statement about sides and angles.
c. ∆DEF where m<D=x, m<E=2x, and m<F=3x
d. ∆GHI where m<G=m<H=m<I. Make congruency statement about sides and angles.
e.
f.
T
g. T
24. If possible, NAME the object and then FIND P, C, r, d, A, LSA, SA, V:
a.
b.
c.
d.
e.
Adv. Math Geometry Review W. 11-19
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f.
g.
h.
i.
25. Answer the questions below with one of the following words: Perimeter (P), Circumference (C), Area (A), Surface
Area (SA), Lateral Surface Area (LSA), or Volume (V). The word you choose should be the measurement that you
would need to find the requested information below.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
The amount of paint for your bedroom walls
The amount of carpet to carpet your room
the amount of “paste” to texture your ceiling
the amount of border to go around your room
the size of heating/cooling unit for your house
the amount of fencing you’ll need for your rectangular
dog pen
the amount of concrete you’ll need for your
rectangular dog pen
the amount of grass seed to sew your lawn
the amount of water to fill your swimming pool
the amount of wax required to wax your car
the amount of wallpaper required to wallpaper your
den
l.
m.
n.
o.
p.
q.
r.
s.
t.
the amount of fertilizer a farmer needs to cover his
field
the amount of grain a grain bin will hold
the distance you’ll run a circular track
the amount of paint you’ll need to paint the entire
cube that you made in art class
the amount of edging you’ll need to go around your
landscaping.
The amount of mulch you’ll need for your landscaping
The amount of material you’ll need to recover your
couch
The amount of gas your gas tank will hold
The amount of blacktop that will be required to
blacktop your driveway.
26. If the following are polygons, name them by # of sides & state if convex/concave, equilateral, equiangular,
and/or regular. If not a polygon, write “Not p-gon” & tell why it’s not a polygon.
b.
c.
a.
Adv. Math Geometry Review W. 11-19
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d.
g.
e.
f.
27. Fill in the blank.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
n.
o.
p.
q.
r.
s.
t.
u.
v.
w.
x.
A parallelogram with 4 congruent sides is a
___________.
A triangle with no congruent sides is a
_______________ triangle.
A six-sided polygon is a ______________
A quadrilateral with only one pair of parallel sides is a
___________.
A 36-sided polygon is a ______________.
An equilateral, equiangular polygon is a __________
polygon.
Angles whose measures sum to 90 are called
______________.
There are ___ degrees in a right angle, ___ degrees in a
straight angle, and ___ degrees in a complete rotation.
The interior angle measures in a ∆ sum to ____; the int
angle measures in a septagon sum to ______
The exterior angles of ALL polygons sum to ____.
___points determine a line, ___ points are on a line,
and a line has _____ endpoints.
_______ is the symbol for infinity.
If all angles in an object have the same measure, the
object is ______________.
Angles having the same measure are called
__________ angles
The complement of a 60° angle is ___.
The supplement of a 60° angle is ___.
In RS , the endpoint(s) is/are _________.
In TU , the endpoint(s) is/are _________.
In VX , the endpoint(s) is/are _________.
In <XYZ, the vertex is ____
2 lines are _______________ if they are coplanar and
do NOT intersect.
2 lines are __________ if they intersect to form right
angles which measure ______ degrees.
2-D figures made only of line segments that are closed
are called ____________.
A 2-D figure with infinite length and width is called a(n)
_____.
y.
xx.
A 1-D figure with infinite length is called a(n)
_______________
A 3-D figure with 2 circular bases is a
________________
A 3-D figure with all square faces is a
___________________
Half a circle is a ______ and half a sphere is a
____________
In a prism, all lateral faces are ________
In a pyramid, all lateral faces are _____________
A TOOL that measures length is a(n) _____
A TOOL that measures weight is a(n) _____
A TOOL that measures volume is a(n) _____
The basic unit of length in the metric system is the
________.
The basic unit of weight in the metric system is the
________.
The basic unit of volume in the metric system is the
________.
An English unit of length is ____________.
An English unit of weight is ____________.
An English unit of volume is ____________.
A UNIT of length is _________
A point is _____-dimensional.
A triangle is ____-dimensional.
A line is ____-dimensional.
An octagonal pyramid is ____-dimensional.
The metric prefix that means 100 is ____.
The metric prefix that means hundredth is _____.
What shape is formed at the intersection of 2 planes?
___
What are all possible shapes that are formed at the
intersection of a plane and cube? ___
What are all possible shapes that are formed at the
intersection of a plane and cone? ___
Draw in all lines of symmetry for the letter H.
e.
f.
g.
All squares are rhombuses.
All rhombuses are squares.
All squares are pentagons.
z.
aa.
bb.
cc.
dd.
ee.
ff.
gg.
hh.
ii.
jj.
kk.
ll.
mm.
nn.
oo.
pp.
qq.
rr.
ss.
tt.
uu.
vv.
ww.
28. True or False:
a.
b.
c.
d.
29.
30.
31.
32.
33.
34.
All trapezoids are quadrilaterals.
All quadrilaterals are trapezoids.
π = 3.14
All squares are polygons.
Compare and contrast prisms, cylinders, pyramids, and cones.
A trapezoidal prism has ____ base(s), ___ lateral faces, and ___ TOTAL FACES.
A trapezoidal pyramid has ____ base(s), ___ lateral faces, and ___ TOTAL FACES.
A nonagonal prism has ____ base(s), ___ lateral faces, and ___ TOTAL FACES.
A nonagonal pyramid has ____ base(s), ___ lateral faces, and ___ TOTAL FACES.
Name the complement and supplement for each angle.
a.
b.
c.
40°
80°
90°
d.
e.
100°
130°
Adv. Math Geometry Review W. 11-19
Page 13 of 21
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
<A and <3 are supplementary. If m<A=2(x+6) and m<3=3(x+1), find the measure of each angle.
A regular pentagon has a perimeter of 20 ft. What is the length of each side? __
Explain why two points determine a line.
Explain why 3 non-collinear points determine a plane.
Can a triangle have 2 obtuse angles? Explain.
Theorists hypothesize that the 4th dimension is ____
The 3 axes that allow us to graph in 3 dimensions are called the _______________________.
If 2 angle measures are the same, the measures are ____ and the angles are _____.
List 3 PRIMITIVE Pythagorean Triples.
List 3 non-primitive Pythagorean Triples.
Find the perimeter of right triangle DEF where m<E=90, DE=EF, DF=30 cm.
Find the missing side length in the right triangles below and find the length of the segment with endpoints at
(-1, 8) and (5, 2) on the Cartesian Plane (Coordinate Plane).
a.
c.
b.
47. Make a congruence or similarity statement based on the pictures below ensuring that CPCTC.
a.
c.
b.
d. Large ∆: m<S=50, m<T=110, m<U=20
Small ∆: m<V=50, m<W=110, m<X=20
48. Find the missing angle measures
a.
b.
Adv. Math Geometry Review W. 11-19
Page 14 of 21
49. Name, state # of lateral faces, # of base(s), and TOTAL # of faces.
a.
b.
50. Convert
a. 7.23 km = ___ cm
b. 5 dg = ___ hg
c. 2 weeks = ___ secs
d. 10 in = ___ cm
51. Is a triangle with side lengths of 15 cm, 12 cm, and 9 cm a right triangle? Explain.
52. Is a triangle with side lengths of 5 cm, 12 cm, and 10 cm a right triangle? Explain.
*********************KEY******************
Quiz FOR learning
1. A No-D figure has no size. An example is a point.
2. A point is a No-D figure that has no size. It indicates a definite location in space & is named by a capital letter.
3. AREA is the # of squares that will completely COVER a 2-D surface. SURFACE AREA is the # of squares that will
completely COVER all the faces of a 3-D object.
4. VOLUME is the number of cubes that will FILL a 3-D object.
5. Perimeter is the distance AROUND an object. Circumference is the distance AROUND a CIRCLE.
6. Radius is a segment in a circle with endpoints on the circle’s center and circle itself. r=.5d Diameter is a
segment with midpoint at the circle’s center and 2 endpoints on the circle itself. d=2r.
7. Pi is an irrational number that is the ratio of the circumference to the diameter in a circle. π ≈ 3.14 OR π ≈ 22/7
8. A square is a regular quadrilateral OR a rhombus with 4 congruent angles OR a rectangle with 4 congruent sides.
9. An undecagon is a polygon with 11 sides.
10. A trapezoid is a quadrilateral with only one pair of parallel sides.
11. Draw segment MA and label its midpoint D.
a.
MD ≅ AD
b.
c.
If MA = 10 cm, then MD = 5 cm
If AD = 6 yds, then MA = 12 yds
12. Draw, label, and name a -e
a.
b.
c.
d.
e.
f.
line
ray
segment
angle
plane
convex polygon
g.
h.
i.
j.
k.
l.
concave pentagon
regular hexagon
equiangular triangle
equilateral octagon
4 noncollinear points
5 coplanar points
13. Name in all possible ways
a.
Adv. Math Geometry Review W. 11-19
Page 15 of 21
G
H
GH, HG
DE, ED, EF, FE, DF, FD, m
b.
d.
BA
<ABC, <CBA, <B, <1
c. _____________
14. Fill in the blank with the word linear, square, or cubic.
a.
b.
c.
d.
Units of length are LINEAR units
Units of surface area are SQUARE units
Units of circumference are LINEAR units
Units of volume are CUBIC units
e.
f.
g.
Units of area are SQUARE units
Units of lateral surface area are SQUARE units
Units of perimeter are LINEAR units
15. An area of 16 square inches means that 16 squares that are 1” by 1” will completely cover the 2-D surface.
16. A surface area of 17 square kilometers means that 17 squares that are 1 km X 1 km will completely cover the
entire 3-D object (all faces covered including bases).
17. A volume of 18 cubic centimeters means that 18 cubes that are 1 cm x 1 cm x 1 cm will completely fill the 3-D
object.
18. To abbreviate 12 square centimeters, we write 12 cm2
19. The sum of the interior angle measures in a dodecagon is (12-2)180 or 1800.
20. Draw in all lines of symmetry for each object below. If none, write none. If infinitely many, draw some and ∞.
c.
e.
a.
d.
b.
21. How would you read in words?
a.
b.
c.
d.
e.
5 ‘ Five feet
10” Ten inches
75°13’ Seventy-Five degrees 13 minutes
11 m3 Eleven cubic meters
m<7=25°The measure of angle 7 is equal to 25
degrees
f.
g.
h.
<TUV ≅ <W Angle TUV is congruent to angle W
TU = 5 mi The length of segment TU equals 5 miles
t ǁ PQ Line t is parallel to segment PQ
i.
RS XY Ray RS is perpendicular to Line XY
j.
k.
l.
m.
∆ GHI ≅ ∆ JKL Triangle GHI is congruent to Triang JKL
∆MNO ~ ∆PQR Triangle MNO is similar to Triang PQR
π ≈ 22/7 Pi is approximately equal to 22-sevenths
∞
infinity
22. Identify the types of angles below
a.
b. Find the measure of each angle above.
23. Find ALL angle measures in the following triangles; then classify the ∆ by its side lengths & angle measures.
Adv. Math Geometry Review W. 11-19
Page 16 of 21
a.
b.
c.
d.
e.
f.
g.
In ∆WYZ, m<W=52, m<Y=104, and m<Z=24. It is a scalene, obtuse triangle.
m<A=m<B=50 & m<C=80; isosceles, acute. <A≅< 𝑩 𝒂𝒏𝒅 AC ≅ 𝑩𝑪
In ∆DEF, m<D=30, m<E=60, and m<F=90. It is a scalene, right triangle.
m<G=m<H=m<I=60; equilateral, equiangular, & regular. <G≅< 𝑯 ≅< 𝑰 𝒂𝒏𝒅 𝑮𝑯 ≅ 𝑰𝑯 ≅ 𝑰𝑮
m<S=45, m<T=90, m<V=45 isosceles, right
m<A=30, m<B=120, m<C=30 isosceles, obtuse
m<K=20, m<J=80, m<L=80 isosceles, acute
24. If possible, NAME the object and then FIND P, C, r, d, A, LSA, SA, V:
a.
b.
c.
d.
e.
f.
g.
h.
Adv. Math Geometry Review W. 11-19
Page 17 of 21
i.
25. Answer the questions below with one of the following words: Perimeter (P), Circumference (C), Area (A), Surface
Area (SA), Lateral Surface Area (LSA), or Volume (V). The word you choose should be the measurement that you
would need to find the requested information below.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
The amount of paint for your bedroom walls LSA
The amount of carpet to carpet your room A
the amount of “paste” to texture your ceiling A
the amount of border to go around your room P
the size of heating/cooling unit for your house V
the amount of fencing you’ll need for your rectangular
dog pen P
the amount of concrete you’ll need for your
rectangular dog pen V
the amount of grass seed to sew your lawn A
the amount of water to fill your swimming pool V
the amount of wax required to wax your car LSA (SA)
the amount of wallpaper required to wallpaper your
den LSA
l.
m.
n.
o.
p.
q.
r.
s.
t.
the amount of fertilizer a farmer needs to cover his
field A
the amount of grain a grain bin will hold V
the distance you’ll run a circular track C
the amount of paint you’ll need to paint the entire
cube that you made in art class SA
the amount of edging you’ll need to go around your
landscaping. P OR C
The amt of mulch you’ll need for your landscaping V
The amount of material you’ll need to recover your
couch LSA OR SA
The amount of gas your gas tank will hold V
The amt of blacktop that will be required to blacktop
your driveway V.
26. If the following are polygons, name them by # of sides & state if convex/concave, equilateral, equiangular,
and/or regular. If not a polygon, write “Not p-gon” & tell why it’s not a polygon.
f.
a.
d.
g.
b.
e.
c.
27. Fill in the blank.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
A parallelogram with 4 congruent sides is a RHOMBUS.
A triangle with no congruent sides is a SCALENE
triangle.
A six-sided polygon is a HEXAGON
A quadrilateral with only one pair of parallel sides is a
TRAPEZOID
A 36-sided polygon is a 36-GON
An equilateral, equiangular polygon is a REGULAR
polygon.
Angles whose measures sum to 90 are called
COMPLEMENTARY
There are 90 degrees in a right angle, 180 degrees in a
straight angle, and 360 degrees in a complete rotation.
The interior angle measures in a ∆ sum to 180; the int
angle measures in a septagon sum to 900
The exterior angles of ALL polygons sum to 360
2 points determine a line, ∞points are on a line, and a
line has NO endpoints.
∞ is the symbol for infinity.
m.
n.
o.
p.
q.
r.
s.
t.
u.
v.
w.
x.
If all angles in an object have the same measure, the
object is EQUIANGULAR
Angles having the same measure are called
CONGRUENT angles
The complement of a 60° angle is 30°
The supplement of a 60° angle is 120°
In RS , the endpoint(s) is/are POINT R AND POINT S
In TU , the endpoint(s) is/are T
In VX , the endpoint(s) is/are NO ENDPOINTS
In <XYZ, the vertex is y
2 lines are PARALLEL if they are coplanar and do NOT
intersect.
2 lines are PERPENDICULAR if they intersect to form
right angles which measure 90 degrees.
2-D figures made only of line segments that are closed
are called POLYGONS
A 2-D figure with infinite length and width is called a(n)
PLANE
Adv. Math Geometry Review W. 11-19
Page 18 of 21
y.
A 1-D figure with infinite length is called a(n) LINE OR
RAY
z.
A 3-D figure with 2 circular bases is a CYLINDER
aa. A 3-D figure with all square faces is a CUBE
bb. Half a circle is a SEMICIRCLE and half a sphere is a
HEMISPHERE
cc. In a prism, all lateral faces are RECTANGLES
dd. In a pyramid, all lateral faces are TRIANGLES
ee. A TOOL that measures length is a(n) RULER (many ans
ff. A TOOL that measures weight is a(n) SCALE (many ans
gg. A TOOL that measures volume is a(n) MEAS. CUP (ma
hh. The basic unit of length in the metric system is the
METER.
ii.
The basic unit of weight in the metric system is the
GRAM
jj.
The basic unit of volume in the metric system is the
LITER
kk.
ll.
mm.
nn.
oo.
pp.
qq.
rr.
ss.
tt.
uu.
An English unit of length is FOOT (many answers)
An English unit of weight is POUND (many answers)
An English unit of volume is OUNCE (many answers)
A UNIT of length is CM (many answers)
A point is NO-dimensional.
A triangle is 2-dimensional.
A line is 1-dimensional.
An octagonal pyramid is 3-dimensional.
The metric prefix that means 100 is HECTO
The metric prefix that means hundredth is CENTI
What shape is formed at the intersection of 2 planes?
LINE
vv. What are all possible shapes that are formed at the
intersection of a plane and cube? SQUARE, TRIANGLE
ww. What are all possible shapes that are formed at the
intersection of a plane and cone?POINT, CIRCLE, OVAL
xx. Draw in all lines of symmetry for the letter H.
28. True or False:
a.
b.
c.
d.
All trapezoids are quadrilaterals. TRUE
All quadrilaterals are trapezoids. FALSE
π = 3.14 FALSE
All squares are polygons. TRUE
e.
f.
g.
All squares are rhombuses. TRUE
All rhombuses are squares. FALSE
All squares are pent. FALSE
29. Compare and contrast prisms, cylinders, pyramids, and cones.
All are 3-D. Prisms and Cylinders have two bases. Pyramids and cones have one base.
Prisms and Pyramids have polygonal bases. Cones and Cylinders have circular bases.
30.
31.
32.
33.
34.
A trapezoidal prism has 2 base(s), 4 lateral faces, and 6 TOTAL FACES.
A trapezoidal pyramid has 1 base(s), 4 lateral faces, and 5 TOTAL FACES.
A nonagonal prism has 2 base(s), 9 lateral faces, and 11 TOTAL FACES.
A nonagonal pyramid has 1 base(s), 9 lateral faces, and 10 TOTAL FACES.
Name the complement and supplement for each angle.
a.
b.
c.
40° C:50 S:140
80° C:10 S:100
90° C:None S:90
d.
e.
100° C:None S:80
130°
C:None
S:50
35. supplementary. m<A=2(x+6) and m<3=3(x+1). m<A=78 and m<3=102
36. A regular pentagon has a perimeter of 20 ft. What is the length of each side? 4 feet
37. Explain why two points determine a line. Infinitely many lines can be drawn through one point. Only one
unique line can be drawn through any 2 points. 2 points are always collinear.
38. Explain why 3 non-collinear points determine a plane. Infinitely many planes may be drawn through one point,
2 points, or 3 collinear points. Only one unique plane will contain 3 non-collinear points.
39. Can a triangle have 2 obtuse angles? Explain. No because it wouldn’t close if had 2 obtuse <s. Plus, the int <s
measures in a triangle sum to 180. A triangle has 3 <s. If 2 <s are obtuse (over 90), then you’ve already
exceeded the 180 without the third angle.
40. Theorists hypothesize that the 4th dimension is time
41. The 3 axes that allow us to graph in 3 dimensions are called the x-axis, y-axis, and z-axis.
42. If 2 angle measures are the same, the measures are EQUAL and the angles are CONGRUENT
43. List 3 PRIMITIVE Pythagorean Triples. 3,4,5 & 5,12,13 & 8,15,17 (many answers)
44. List 3 non-primitive Pythagorean Triples. 6,8,10 & 9,12,15 & 10,24,26 (many answers)
45. Find perimeter of right ∆DEF m<E=90, DE=EF, DF=30 cm. Perimeter is about 72.42 cm or exactly 30+2√450 cm
46. Find the missing side length in the right triangles below and find the length of the segment with endpoints at
(-1, 8) and (5, 2) on the Cartesian Plane (Coordinate Plane).
a.
approx. 23.43 units or √545 units
b.
Approximately 4.36 units or exactly √19 units
Adv. Math Geometry Review W. 11-19
Page 19 of 21
Exactly √72 units OR approximately 8.49 units.
c.
47. Make a congruence or similarity statement based on the pictures below ensuring that CPCTC.
a.
∆PQR≅∆L JK
c.
b.
∆ ABC ≅ ∆ DEF
∆MNO~∆KLO
d.
∆STU~∆VWX
48. Find the missing angle measures
a.
m<A=35 because it’s SUPPLEMENTARY with the 80 and 65.
m<B=50 because it’s in a triangle with 35 and 95 degree angles; angle meas in TRIANGLES SUM TO 180.
m<C=85 because it’s SUPPLEMENTARY with the 95.
m<D=30 because it’s an interior < in a triangle and all angles sum to 180 (you have 65, 85, so need 30).
b.
m<A=70 because it’s a VERRTICAL ANGLE with a 70 degree angle.
m<B=45 because it’s a STRAIGHT ANGLE with the 135 angle (together they’ll make 180 degrees).
m<C=65 because it forms a TRIANGLE with the angles that measure 70 and 45.
m<D=45 because it’s an ALTERNATE INTERIOR ANGLE with <B, and they’re congruent when lines para.
m<E=70 because it’s supplementary with <s D&C OR because it’s alt int with <A.
m<F=110 because it’s supplementary with <E.
Adv. Math Geometry Review W. 11-19
Page 20 of 21
49. Name, state # of lateral faces, # of base(s), and TOTAL # of faces.
a.
Hexagonal Prism
b. Decagonal Pyramid
6 lateral rectangular faces
10 lateral triangular faces, 1 decagonal base, 11 TOTAL
2 hexagonal bases
FACES
8 TOTAL FACES
50. Convert
a. 7.23 km = 723,000 cm
b. 5 dg = 0.005 hg
c. 2 weeks = 1,209,600 secs
d. 10 in = 25.4 cm
51. Is a triangle with side lengths of 15 cm, 12 cm, and 9 cm a right triangle? Explain. Yes because the Converse of
the Pythagorean Theorem proves it as 92 + 122 = 152
52. Is a triangle with side lengths of 5 cm, 12 cm, and 10 cm a right triangle? Explain. No because the Converse of
the Pythagorean Theorem proves it as 52 + 102 ≠ 122
Adv. Math Geometry Review W. 11-19
Page 21 of 21