GEOMETRY

Math 7 Unit 4

StandardsStrand 4:

Concept 1PO 1. Draw a model that demonstrates basic geometric relationships such

as parallelism, perpendicularity, similarity/proportionality, and congruence.

PO 6. Identify the properties of angles created by a transversal intersecting two parallel lines

PO 7. Recognize the relationship between inscribed angles and intercepted arcs.

PO 8. Identify tangents and secants of a circle.

PO 9. Determine whether three given lengths can form a triangle.

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.

PO 7. Calculate the length of a side given two similar triangles.

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The hardest part about Geometry

Point: a location in space: think about the tip of your pencil

A

●A

Line

all the points on a never-ending straight path that extends in all directions

AB

A B

Segment

all the points on a straight path between 2 points, including those endpoints

CD

C D

Ray

a part of a line that starts at a point (endpoint) and extends forever in one direction

EF

E F

Angleformed 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.

A

CB

1

Vertex

Side

Side

70

Angle

B

A

CB

15°ABC

1

Planea flat surface without thickness extending in all directionsThink: a wall, a floor, a sheet of paper

A

Parallel Lines

lines that never intersect (meet) and are the same distance apart

AB

A B

C D

║ CD

Perpendicular Lines

lines that meet to form right angles

AB

CD

BA

C

D

Intersecting Lines

lines that meet at a point

BA

C

D

Right AngleAn angle that measures90 degrees.

Straight Angle

An angle that measures180 degrees or 0. (straight line)

Acute Angle

An angle that measuresbetween 1 and 89 degrees

Obtuse Angle

An angle that measures between 91 and 179 degrees

Complementary Angles

Two or more angles whose measures total 90 degrees. 1

2

Supplementary Angles

Two or more angles that add up to 180 degrees.

1 2

*****Reminders******

SupplementaryStraight angleComplimentaryCorner

Adjacent Angles

Two angles who share a common side

B

A

C

D

Example 1

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

Example 1

• Angles 1 and 2 are complementary. If

• m 1 = 60, find m 2.

1 2

60

1 + 2 = 90 2 = 90 - 1 2 = 90 - 60 2 = 30

Example 3

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

1 2

< 1 + < 2 = 180 < 2 = 180 - < 1 < 2 = 180 - 114 < 2 = 66

114

7.2 Angle Relationships

1 2

3 4

5 6

7 8

t

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

1 32 4

Vertical Angles• Example 1: Find the measures of

the missing angles

55

?

?

125

t

125

PARALLEL LINES• Def: line that do not intersect.

• Illustration: l

m

A

B

C

D

AB || CD l || m

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.

Examples of Parallel Lines• Streets: Belmont & School

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

• Illustration:

Transversal

t

Supplementary Angles/Linear Pair

• Two angles that form a line (sum=180)

1 2

3 4

5 6

7 8

t5+6=1806+8=1808+7=1807+5=180

1+2=1802+4=1804+3=1803+1=180

Supplementary Angles/Linear Pair

• Find the measures of the missing angles

? 72

?

t

108 108

Alternate Exterior Angles• Two angles that lie outside parallel lines

on opposite sides of the transversalt

2 71 8

1 2

3 4

5 6

7 8

Alternate Interior Angles• Two angles that lie between parallel

lines on opposite sides of the transversalt

3 64 5

1 2

3 4

5 6

7 8

Corresponding Angles• Two angles that occupy

corresponding positions.Top Left

t

Top Left

Top Right

Top Right

Bottom Right

Bottom Right

Bottom Left

Bottom Left

1 52 63 74 8

1 2

3 4

5 6

7 8

Same Side Interior Angles

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

t

3 +5 = 1804 +6 = 180

1 2

3 4

5 6

7 8

List all pairs of angles that fit the description.

a. Correspondingb. Alternate Interiorc. Alternate Exteriord. Consecutive Interior

1234

5 6 7 8 t

Find all angle measures

1 67

3

t

113 180 - 67

2

5

6 7

8

67

67

67

113

113

113

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

Algebraic Angles• Name the angle relationship

– Are they congruent, complementary or supplementary?

– Complementary• Find the value of x x + 36 = 90

-36 -36

x = 54

= 90

x

36

Example 2

• Name the angle relationship– Vertical– Are they congruent, complementary or

supplementary?• Find the value of x

115

x

x = 115

Example 3• Name the angle relationship

– Alternate Exterior– Are they congruent, complementary or

supplementary?• Find the value of x

125

t

5x

5x = 1255 5x = 25

Example 4

• Name the angle relationship– Corresponding– Are they congruent, complementary or

supplementary?• Find the value of x

2x + 1

t

151

2x = 1502 2x = 75

2x + 1 = 151- 1 - 1

Example 5

• Name the angle relationship– Consecutive Interior Angles– Are they congruent, complementary or

supplementary?• Find the value of x

81

t

7x + 15

supp

7x = 847 7x = 12

7x + 96 = 180 - 96 - 96

7x + 15 + 81 = 180

Example 6

• Name the angle relationship– Alternate Interior Angles– Are they congruent, complementary or

supplementary?• Find the value of x

3x

t

2x + 20

20 = x

2x + 20 = 3x- 2x - 2x

The World Of Triangles

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.

Colors Does it make a triangle? Y/N

a. orange, blue dark greenb. light green, yellow, dark

greenc. red, white, blackd. yellow, brown, light greene. dark green, yellow, redf. purple, dark green, whiteg. orange, blue, whiteh. black, dark green, red

YesYesNoNoYesNoNoYes

• Can you use two of the same color rods and make a triangle? Explain and give an example.

Makes a triangle Does not make a triangle

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.

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

Triangles

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

Triangle Inequality:

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

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

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

TRIANGLESTriangles can be classified according to the size oftheir angles.

Right Triangles

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

Obtuse Triangles

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

Acute Triangles

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

TrianglesTriangles can be classified according to the length of theirsides.

Scalene Triangles

• A triangle with three unequal sides.

Isosceles Triangles

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

Equilateral Triangles

• An equilateral triangle is a triangle with all three sides of equal length.

• Equilateral triangles are also equilangular. (all angles the same)

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

• Examples: Find the missing angle:

7050

x

x

42

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

• Examples: Find the missing angle:

x

8080

60

x

7.5 NOTES Congruent and Similar

• Def’n - 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.

Symbol:

EX. 1 A

B

C

_______AB _______ABC

_______BC EFD________

_______CAB

All corresponding parts are congruent so

F

D

E

ABC EDF

EF_________

Similar

• Def’n – similar – Figures that have the same shape but differ in size are similar.

• Corresponding angles are equal.

Symbol: ~

Example 2

A B

C

D

E F

________________ ~ _________________

Example 3: Find the value of x in each pair of figures.

• Corresponding sides are equal so

STLROB MIKE JOSH

16 ft

B

R O

T S

L

2x ft

H S

62 in

E

I

M

K

J O3x + 32 in

2x = 16 2 2x = 8 ft

-32 -323x + 32 = 62

3x = 30 3 3x = 10 in

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.

NOTES on Similar Figures/Indirect Measurement

• Recall that similar figures have corresponding angles that are CONGRUENT but their sides are PROPORTIONAL.

• Def’n – 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.

EX. 1 The triangles below are similar.

c) Find the measure of <VWU.

VW

S

6 in.

TR105

V

5 in. WU

m SRT

65

RT STUW

~RST VUWm

a) Find the ratio of the corresponding side lengths.

b) Complete each statement.i.) ii.) iii.)

UVW

105

EX. 2 Write a mathematical statement saying the figures are similar.Show which angles and sides correspond.

B H A I

JD K C

ABCD~

IHKJ

You can use similar triangles to find the measure of objects we can’t measure.

• Use a proportion to solve for x.• Example If find the value of x.

DEFABC ~

30x = 240

30 30x = 8 ft

Example 2:

5x = 70

5 5

x = 14 mm

1057 x

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

x12

510

Example 4: Use similar triangles to find the distance across the pond.

10x = 360

10 10

x = 36 m

45108

x

CIRCLES

Radius (or Radii for plural)• The segment

joining the center of a circle to a point on the circle.

O

A

Example: OA

ChordA segment joining

two points on a circleB

CA

Example: AB

Diameter• A chord that

passes through the center of a circle.

O

A

BExample: AB

SecantA line that intersects

the circle at exactly two points.

D

C

B

A

O

Example: AB

Tangent

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

B

A

Example: AB

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. A

B

Example: AB

(

Central Angle• An angle whose

vertex is at the center of a circle.

G

Q

HExample: <GQH

Inscribed Angle• An angle whose

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

M

N

TExample: <MTN

Intercepted Arc• An arc that lies in

the interior of an inscribed angle.

M

N

T

Example: MN

(

Important Information

An inscribed angle is equal in measure to half of the measure of its intercepted arc.

M

N

T

So the measure < MTN of is equal to ½ of the measure of MN

(

EX. 1 Refer to the picture at the right.

e) Give the measure of arc AB.

a) Name a tangent:

b) Name a secant:

c) Name a chord:

d) Name an inscribed angle:

EF

BD

AD

<ADB

54

B

A

C

D

E

F27