GEOMETRY

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GEOMETRY

• Math 7 Unit 4

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Standards
Strand 4 Concept 1 PO 1. Draw a model that demonstrates basic geometric relationships such as parallelism, perpendicularity, similarity/proportionality, and congruence.
Strand 4 Concept 1 PO 6. Identify the properties of angles created by a transversal intersecting two parallel lines
Strand 4 Concept 1 PO 7. Recognize the relationship between inscribed angles and intercepted arcs.
Strand 4 Concept 1 PO 8. Identify tangents and secants of a circle.
Strand 4 Concept 1 PO 9. Determine whether three given lengths can form a triangle.
Strand 4 Concept 1 PO 10. Identify corresponding angles of similar polygons as congruent and sides as proportional.
Strand 4 Concept 4 PO 6. Solve problems using ratios and proportions, given the scale factor.
Strand 4 Concept 4 PO 7. Calculate the length of a side given two similar triangles.
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GEOMETRY IS EVERYWHERE
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IN FLAGS
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IN NATURE
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IN SPORTS
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IN MUSIC
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IN SCIENCE
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IN Games
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IN BUILDINGS
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The hardest part about Geometry
Vocabulary
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Point
a location in space
think about the tip of your pencil
Notation
?A
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Line
all the points on a never-ending straight path
that extends in all directions
Notation
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Segment
all the points on a straight path between 2
points, including those endpoints
Notation
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Ray
a part of a line that starts at a point
(endpoint) and extends forever in one direction
Notation
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Angle
formed by 2 rays that share the same endpoint.
The point is called the VERTEX and the rays are
called the sides. Angles are measure in degrees.
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Angle
Notation
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Plane
a flat surface without thickness extending in all
directions
Think a wall, a floor, a sheet of paper
Notation
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Parallel Lines
lines that never intersect (meet) and are the
same distance apart
Notation

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Perpendicular Lines
lines that meet to form right angles
Notation
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Intersecting Lines
lines that meet at a point
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Right Angle
An angle that measures 90 degrees.
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Straight Angle
An angle that measures 180 degrees or 0.
(straight line)
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Acute Angle
An angle that measures between 1 and 89 degrees
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Obtuse Angle
An angle that measures between 91 and 179 degrees
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Complementary Angles
Two or more angles whose measures total 90
degrees.
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Supplementary Angles
Two or more angles that add up to 180 degrees.
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Reminders
Supplementary Straight angle
Complimentary Corner
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Adjacent Angles
Two angles who share a common side
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Example 1

• Estimate the measure of the angle, then use a
  protractor to find the measure of the angle.

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Example 1

• Angles 1 and 2 are complementary. If
• m 1 60?,
• find m 2.

1 2 90?
2 90? - 1
2 90? - 60?
2 30?
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Example 3

• Angles 1 and 2 are supplementary. If m 1 is
  114?, find m 2.

lt 1 lt 2 180?
lt 2 180? - lt 1
lt 2 180? - 114?
lt 2 66?
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7.2 Angle Relationships
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Vertical Angles

• Two angles that are opposite angles.
• Vertical angels are always congruent!

• 1 ? ? 3
• 2 ? ? 4

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Vertical Angles

• Example 1 Find the measures of the missing
  angles

125 ?
55 ?
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PARALLEL LINES

• Def line that do not intersect.
• Illustration

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Examples of Parallel Lines

• Hardwood Floor
• Opposite sides of windows, desks, etc.
• Parking slots in parking lot
• Parallel Parking
• Streets Arizona Avenue and Alma School Rd.

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Examples of Parallel Lines

• Streets Belmont School

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Transversal

• Def a line that intersects two lines at
  different points
• Illustration

t
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Supplementary Angles/Linear Pair

• Two angles that form a line (sum180?)

• 5?6180
• 6?8180
• 8?7180
• 7?5180

• 1?2180
• 2?4180
• 4?3180
• 3?1180

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Supplementary Angles/Linear Pair

• Find the measures of the missing angles

t
?
72 ?
108 ?
?
108 ?
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Alternate Exterior Angles

• Two angles that lie outside parallel lines on
  opposite sides of the transversal

t

• 2 ? ? 7
• 1 ? ? 8

1
2
3
4
5
6
7
8
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Alternate Interior Angles

• Two angles that lie between parallel lines on
  opposite sides of the transversal

t

• 3 ? ? 6
• 4 ? ? 5

1
2
3
4
5
6
7
8
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Corresponding Angles

• Two angles that occupy corresponding positions.

t

• 1 ? ? 5
• 2 ? ? 6
• 3 ? ? 7
• 4 ? ? 8

Top Left
Top Right
Bottom Left
Bottom Right
Top Left
Top Right
Bottom Left
Bottom Right
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Same Side Interior Angles

• ?3 ?5 180
• 4 ?6 180

t

• Two angles that lie between parallel lines on the
  same sides of the transversal

1
2
3
4
5
6
7
8
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• List all pairs of angles that fit the
  description.
• Corresponding
• Alternate Interior
• Alternate Exterior
• Consecutive Interior

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Find all angle measures
t
180 - 67
67 ?
113 ?
1
3
2
67 ?
113 ?
5
8
67 ?
113 ?
6
7
67 ?
113 ?
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Example 5

• find the m? 1, if m? 3 57?
• find m? 4, if m? 5 136?
• find the m? 2, if m? 7 84?

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Algebraic Angles
90?

• Name the angle relationship
• Are they congruent, complementary or
  supplementary?
• Complementary
• Find the value of x

x 36 90
-36 -36
x 54?
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Example 2

• Name the angle relationship
• Vertical
• Are they congruent, complementary or
  supplementary?
• Find the value of x

?
x 115 ?
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Example 3

• Name the angle relationship
• Alternate Exterior
• Are they congruent, complementary or
  supplementary?
• Find the value of x

?
5x 125
5 5
x 25
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Example 4

• Name the angle relationship
• Corresponding
• Are they congruent, complementary or
  supplementary?
• Find the value of x

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2x 1 151
- 1 - 1
2x 150
2 2
x 75
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Example 5

• Name the angle relationship
• Consecutive Interior Angles
• Are they congruent, complementary or
  supplementary?
• Find the value of x

supp
7x 15 81 180
7x 96 180
- 96 - 96
7x 84
7 7
x 12
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Example 6

• Name the angle relationship
• Alternate Interior Angles
• Are they congruent, complementary or
  supplementary?
• Find the value of x

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2x 20 3x
- 2x - 2x
20 x
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The World Of Triangles
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Pick Up Sticks

• For each given set of rods, determine if the rods
  can be placed together to form a triangle. In
  order to count as a triangle, every rod must be
  touching corner to corner. See example below.

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Colors Does it make a triangle? Y/N
a. orange, blue dark green
b. light green, yellow, dark green
c. red, white, black
d. yellow, brown, light green
e. dark green, yellow, red
f. purple, dark green, white
g. orange, blue, white
h. black, dark green, red
Yes
Yes
No
No
Yes
No
No
Yes
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• Can you use two of the same color rods and make a
  triangle? Explain and give an example.

• Now find five new sets of three rods that can
  form a triangle. Find five new sets of rods that
  will not make a triangle.

Makes a triangle Does not make a triangle





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• Without actually putting them together, how can
  you tell whether or not three rods will form a
  triangle?

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Triangles

• A triangle is a 3-sided polygon. Every triangle
  has three sides and three angles, which when
  added together equal 180.

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Triangle Inequality

• In order for three sides to form a triangle, the
  sum of the two smaller sides must be greater than
  the largest.

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Triangle Inequality

• Examples
• Can the following sides form a triangle? Why or
  Why not?
• A. 1,2,2 B. 5,6,15

Yes!
No!
5 6 gt 15
1 2 gt 2
Given the lengths of two sides of a triangle,
state the greatest whole-number measurement that
is possible for the third. A. 3,5 B. 2,8
7
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TRIANGLES
Triangles can be classified according to the
size of their angles.
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Right Triangles

• A right triangle is triangle with an angle of 90.

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Obtuse Triangles

• An obtuse triangle is a triangle in which one of
  the angles is greater than 90.

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Acute Triangles

• A triangle in which all three angles are less
  than 90.

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Triangles
Triangles can be classified according to the
length of their sides.
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Scalene Triangles

• A triangle with three unequal sides.

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Isosceles Triangles

• An isosceles triangle is a triangle with two
  equal sides.

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Equilateral Triangles

• An equilateral triangle is a triangle with all
  three sides of equal length.
• Equilateral triangles are also equilangular.
  (all angles the same)

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The sum of the interior angles of a triangle is
180 degrees.

• Examples Find the missing angle

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The sum of the interior angles of a quadrilateral
is 360 degrees.

• Examples Find the missing angle

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7.5 NOTES Congruent and Similar

• Defn - congruent In geometry, figures are
  congruent when they are exactly the same size and
  shape.
• Congruent figures have corresponding sides and
  angles that are equal.

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EX. 1
All corresponding parts are congruent so
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Similar

• Defn similar Figures that have the same
  shape but differ in size are similar.
• Corresponding angles are equal.

Symbol
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Example 2
________________ _________________
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Example 3 Find the value of x in each pair of
figures.
3x 32 62

• Corresponding sides are equal so

-32 -32
3x 30
2x 16
2 2
3 3
x 8 ft
x 10 in
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Example 4

• Sketch both triangles and properly label each
  vertex. Then list the three pairs of sides and
  three pairs of angles that are congruent.

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NOTES on Similar Figures/Indirect Measurement

• Recall that similar figures have corresponding
  angles that are CONGRUENT but their sides are
  PROPORTIONAL.
• Defn ratio of the corresponding side lengths
  of similar figures (a.k.a. SCALE FACTOR)
  corresponding sides of congruent triangles are
  proportional. One side of the first triangle
  over the matching side on the second triangle.

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EX. 1 The triangles below are similar.
a) Find the ratio of the corresponding side
lengths.
b) Complete each statement. i.) ii.) iii.)

• c) Find the measure of ltVWU.

105?
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EX. 2 Write a mathematical statement saying the
figures are similar.Show which angles and sides
correspond.
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You can use similar triangles to find the measure
of objects we cant measure.

• Use a proportion to solve for x.

• Example If
• find the value of x.

30x 240
30 30
x 8 ft
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Example 2
5x 70
5 5
x 14 mm
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Example 3 A basketball pole is 10 feet high and
casts a shadow of 12 feet. A girl standing
nearby is 5 feet tall. How long is the shadow
that she casts?
10x 60
10 10
x 6 ft
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Example 4 Use similar triangles to find the
distance across the pond.
10x 360
10 10
x 36 m
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CIRCLES
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Radius (or Radii for plural)

• The segment joining the center of a circle to a
  point on the circle.

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Chord

• A segment joining two points on a circle

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Diameter

• A chord that passes through the center of a
  circle.

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Secant

• A line that intersects the circle at exactly two
  points.

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Tangent

• A line that intersects a circle at exactly one
  point.

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Arc

• A figure consisting of two points on a circle and
  all the points on the circle needed to connect
  them by a single path.

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Central Angle

• An angle whose vertex is at the center of a
  circle.

Example ltGQH
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Inscribed Angle

• An angle whose vertex is on a circle and whose
  sides are determined by two chords.

Example ltMTN
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Intercepted Arc

• An arc that lies in the interior of an inscribed
  angle.

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Important Information
An inscribed angle is equal in measure to half of
the measure of its intercepted arc.
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EX. 1 Refer to the picture at the right.
a) Name a tangent
b) Name a secant
c) Name a chord
d) Name an inscribed angle
ltADB
e) Give the measure of arc AB.
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