north haven high school - pc\|macimages.pcmac.org/sisfiles/schools/ct/northhavenschools/...parallel...
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NORTH HAVEN HIGH SCHOOL
221 Elm Street
North Haven, CT 06473
Applied Geometry (Level 1)
Summer Assignment 2017 June 2017
Dear Parents, Guardians, and Students,
The Geometry curriculum builds on geometry concepts and vocabulary that were introduced in middle school and also
uses a number of techniques learned in Algebra I. Completing this packet will keep foundational geometry concepts and
vocabulary fresh in studentsโ minds and provide a starting point for Geometry at the beginning of the year. Reference
sheets are included in the packet to help complete the work.
Please be sure that the completed packet is brought to school on the first day of class. The teacher will check the packet
and students will receive a grade based on completion. Students must show work in this packet and complete all of the
problems to receive credit for this assignment. Calculators may be used when needed, but this cannot serve as a
replacement for showing work. If students have trouble with an item, they should skip it, and come back to it later, but
persevere in trying to solve the problem.
The mathematics department thanks you for your support and wishes you and your family a happy and restful summer!
Sincerely,
Mr. Hughes
Applied Geometry Teacher
Mrs. Romberg
Mathematics Program Coordinator
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SAT Reference Information:
The number of degrees of arc in a circle is 360.
The number of radians of arc in a circle is
The sum of the measures in degrees of the angles of a triangle is 180.
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Part 1 โ Geometry Review/Preview
You should know the following vocabulary from previous math classes. Please use the following reference sheets, to complete the problems on the subsequent geometry review/preview pages. The three undefined terms in geometry are: point, line, and plane. These are also called the โBuilding
Blocks of Geometryโ because everything is based on these three ideas. We are able to describe them but
not define them.
Point โ is a specific location in space with no size or shape.
Line โ is a set of points that go on indefinitely in both directions. Points on the same line are said to be collinear.
symbolic notation โท
*named with any two points on the line OR an italicized single letter at the end of the line
Plane โ is a flat surface with no edges and no boundaries. It has two dimensions. Objects on the same plane are said to be coplanar.
*named by an upper case script letter OR any three non-collinear (not on the same line) points on the plane.
Line segment โ is part of a line containing two endpoints and all points between the endpoints. symbolic notation โ
*named with the two endpoints only
Congruent segments โ two segments that have the same length. Congruent segments are marked in diagrams by tick marks or hashes. symbolic notation โ ๐จ๐ฉฬ ฬ ฬ ฬ โ ๐ช๐ซฬ ฬ ฬ ฬ (said as โsegment ๐จ๐ฉ is congruent to segment ๐ช๐ซ)
Possible names:
Plane ๐
Plane ๐๐ ๐
Plane ๐๐๐
๐ต ๐ด
Q
๐
๐ ๐
๐
Q
๐ด
๐ถ ๐ท
Possible names:
๐ด๐ต โก
๐ต๐ด โก
Line ๐
๐
Possible names:
๐ถ๐ทฬ ฬ ฬ ฬ
๐ท๐ถฬ ฬ ฬ ฬ
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40ยฐ
40ยฐ
Ray โ is a portion of a line that extends from one point indefinitely in one direction.
symbolic notation ๐ฌ๐ญ symbolic notation ๐ฎ๐ฏ
symbolic notation ๐ธ๐น
*endpoint always named first then any other point on the ray. Opposite Rays โ share the same end point and are collinear.
๐ช๐จ and ๐ช๐ฉ are opposite rays
Angle โ two rays with the same endpoint. The common endpoint is called the vertex and the rays are called sides.
symbolic notation โ ๐ฑ is the vertex
*when naming angles the vertex is always the middle letter, the angle can also be named by just the vertex letter or number inside the angle at the vertex.
*You cannot name any of these angles โ ๐จ because the vertex is shared by three different angles.
The most common unit of measure for angles is the degree. A protractor is used to measure angles. Use the 90o angle as your reference angle when using a protractor.
Congruent angles โ two angles that have the same measure. Congruent angles are marked in diagrams using arcs.
symbolic notation โ โ ๐ฎ๐ฐ๐ฉ โ โ ๐พ๐ถ๐ช (said as โ Angle ๐ฎ๐ฐ๐ฉ is
congruent to angle ๐พ๐ถ๐ชโ)
๐ธ ๐น ๐ป ๐บ
๐
๐
๐ถ ๐ด ๐ต
Possible names:
โ ๐ผ๐ฝ๐พ
โ ๐พ๐ฝ๐ผ
โ ๐ฝ
โ 2
๐ฝ
๐พ
๐ผ
2
๐ต
๐ด
๐ท
๐ถ
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Angles can be classified by their degree measure.
Right Angle:
measures exactly 90ยฐ
Acute Angle:
Measures more than 0ยฐ
and less than
90ยฐ
Obtuse Angle:
Measures more than 90ยฐ
and less than 180ยฐ
โStraight Angleโ:
Measures 180ยฐ
Angle Pairs:
Adjacent angles โ two angles that share a common vertex and side, but have no common interior Points.
*โ ๐จ๐ถ๐ฉ and โ ๐ฉ๐ถ๐ช are adjacent angles
Complementary angles โ two angles whose measures have the sum of 90ยฐ
Supplementary angles โ two angles whose measures have the sum of 180ยฐ
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Linear Pair โ two adjacent angles whose non-common sides are opposite rays (or form a straight angle)
*โ ๐ซ๐จ๐ช and โ ๐ฉ๐จ๐ช are a linear pair
*linear pairs are supplementary Vertical angles โ a pair of non-adjacent angles formed by the intersection of two lines
*โ ๐จ๐ฉ๐ซ and โ ๐ฌ๐ฉ๐ช are vertical angles
*vertical angles are congruent Angle Bisector โ a line, line segment, or ray which divides an angle into two equal parts.
*๐ฒ๐ด is an angle bisector of โ ๐ณ๐ฒ๐ฑ Intersecting Figures Two lines intersect at one point only. Line and plane intersect at one point only. Two planes intersect at a line.
Parallel lines โ coplanar lines that never intersect. Parallel lines are marked in diagrams using arrowheads.
symbolic notation โฅ
๐ฑ๐ฒ โก โฅ ๐ณ๐ด โก (said as โline ๐ฑ๐ฒ is parallel to line ๐ณ๐ดโ)
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Perpendicular lines โ coplanar lines that intersect forming four right angles.
Symbolic notation โฅ *๐ โฅ ๐ (said asโline ๐ is perpendicular to line ๐) Skew lines โ are not parallel, do not intersect and are not on the same plane.
*๐ท๐ธ โก is skew to ๐ผ๐ฝ โก
Polygons- are closed figures, made up of line segments that meet only at their end points and are on one plane.
# of Sides Polygon Name # of Sides Polygon Name
3 Triangle 8 Octagon
4 Quadrilateral 9 Nonagon
5 Pentagon 10 Decagon
6 Hexagon 11 Hendecagon
7 Heptagon 12 Dodecagon
Convex polygon โ a polygon that has all interior angles less than 180ยฐ. All the vertices point โoutwardsโ, away from the center.
Concave polygon โ a polygon that has one or more interior angle greater than 180ยฐ. Some vertices point โinwardsโ, towards the center.
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Triangles โ There are special kinds of triangles. Triangles may be classified by their angle measures.
Triangles may also be classified by their side lengths.
Quadrilaterals โ There are special kinds of quadrilaterals.
Trapezoid: has one pair of parallel
sides (called basesโฆ shown to be
parallel by use of arrows)
Isosceles Trapezoid: has one pair of
parallel sides and the other two
sides are the same length
Parallelogram: has two pairs of
parallel sides
Rectangle: parallelogram with four
right angles
Rhombus: parallelogram with four
sides that are the same length
Square: parallelogram with four
right angles and four sides that are
the same length
All sides
measure
5 feet
Obtuse Triangle: has
one obtuse angle and
two acute angles
Right Triangle: has one
right angle and two
acute angles
Acute Triangle: has
three acute angles
Equilateral Triangle: special
kind of acute triangle, all 3
angles measure 60ยฐ
Scalene Triangle: no sides are
the same length
Isosceles Triangle: at least two
sides are the same length
Equilateral Triangle: all three
sides are the same length
60
60
60
8 in
8 in
5 in5 cm
5 cm
5 cm
5 in
7 in
3 in
8 ft 8 ft
8 ft
8 ft
8 ft
8 ft
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Part 2 โ Geometric Figures
Sketch and label each of the following geometric figures.
1. ๐บ๐ธฬ ฬ ฬ ฬ 2. ๐๐ โก 3. Plane ๐๐ ๐ 4. ๐ผ๐
5. right โ ๐๐ถ๐ฟ 6. ๐๐บ โก 7. Obtuse โ ๐ด๐๐ท 8. Plane FUN
9. Pentagon 10. Acute โ ๐ท๐๐ป 11. Scalene โ๐ฟ๐ด๐ 12. Obtuse isosceles
โ๐๐ป๐
13. Octagon 14. Straight โ ๐ฟ๐๐ 15. Right โ ๐ฝ๐ท๐บ 16. Hexagon
17. Opposite rays ๐๐ฝ and
๐๐ฟ
18. Parallel lines ๐ฟ๐ผ โก and
๐ถ๐ด โก
19. Perpendicular lines
๐ฟ๐ด and ๐ ๐ โก
20. A pair of vertical
angles, โ ๐ฝ๐๐ and โ ๐พ๐๐ถ
21. โ ๐๐ ๐ bisected by
๐ ๐
22. A pair of adjacent
angles, โ ๐ท๐๐บ and
โ ๐ฟ๐๐ท
23. Collinear points ๐ด, ๐ต,
๐ถ, and ๐ท
24. A linear pair, โ ๐ถ๐ด๐ต
and โ ๐ท๐ด๐ต
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You have been given one piece of information for each of the rows below. Complete the chart with the appropriate vocabulary term, definition, diagram or symbol.
Vocabulary Word Description / Definition Diagram Symbol
25. Line JK
a. b. c.
26.
d. e.
f.
27.
g. Angle ABC whose measure
is greater than 90ห
and less than 180ห
h. i.
28. Ray PQ
j. k. l.
29.
m. A flat surface that contains the
non-co-linear points SAG and
extends infinitely in all
directions
n. o.
30. Acute Angle
JKL
p. q. r.
31.
s. t. u.
A
B
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12 cm
5 cm
Part 3 โ Area
Use the SAT formula sheet to assist you in calculating the area of the following figures. Show formula(s) used and
work to support your answer. The first problem is complete and a few problems have been started for you.
39.
Shape(s) :Rectangle
Formula: ๐ด๐๐๐ = ๐๐๐ ๐ โ โ๐๐๐โ๐ก
Work:๐ด๐๐๐ = 12 ๐๐. โ 5 ๐๐.
Answer:๐ด๐๐๐ = 60 ๐๐2
40.
Shape(s) :
Formula:
Work:
Answer:
41.
Shape(s) :Triangle
Formula: ๐ด๐๐๐ = 1
2 ๐ โ
Work:
Answer:
42.
Shape(s) :
Formula:
Work:
Answer:
43.
Shape(s) :
Formula:
Work:
Answer:
44.
Shape(s) :
Formula:
Work:
Answer:
9 ft
8 ft
10 in
9.3 in
5 m22 cm
23.6 in
14 in
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45.
Shape(s) :Trapezoid
Formula:
๐ด๐๐๐ =1
2 (๐๐๐ ๐1 + ๐๐๐ ๐2)โ๐๐๐โ๐ก
Work:
Answer:
46.
Shape(s) :
Formula:
Work:
Answer:
47.
Shape(s) :
Formula:
Work:
Answer:
15 cm
9 cm
height = 13 cm
8 cm
16
cm
height =
10 cm
10 cm
38 cm
8 cm