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Ch. 1 Highlights Geometry A. Ms. Urquhart Mrs. Vander Bee. Coplanar Objects. **Remember: Any 3 non-collinear points determine a plane!. Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible.

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Ch 1 highlights geometry a

Ch. 1 HighlightsGeometry A

Ms. Urquhart

Mrs. Vander Bee


Coplanar objects
Coplanar Objects

**Remember: Any 3 non-collinear points determine a plane!

Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible.

Are the following points coplanar?

A, B, C ?

Yes

A, B, C, F ?

No

H, G, F, E ?

Yes

E, H, C, B ?

Yes

A, G, F ?

Yes

C, B, F, H ?

No

Lesson 1-1 Point, Line, Plane


Front side true or false
Front Side – True or False

Lesson 1-1 Point, Line, Plane


Example 3
Example 3

Point S is between point R and point T. Use the given information to write an equation in terms of x. Solve the equation. Then find both RS and ST.

RS = 3x – 16

ST = 4x – 8

RT = 60

I---------------60 -------------------I

3x-16

I-----------4x-8-----------I

Lesson 1-1 Point, Line, Plane


ALGEBRA

Point Mis the midpoint of VW. Find the length of VM .

EXAMPLE 2

Use algebra with segment lengths

Lesson 1-1 Point, Line, Plane


Guided practice
GUIDED PRACTICE

linel

Identify the segment bisector of .

Then find PQ.

Lesson 1-1 Point, Line, Plane


Midpoint formula
MIDPOINT FORMULA

The midpoint of two points P(x1, y1) and Q(x2, y2) is

M(X,Y) = M(x1 + x2, x2 +y2)

2 2

Think of it as taking the average of the x’s and the average of the y’s to make a new point.

Lesson 1-1 Point, Line, Plane


a. FIND MIDPOINTThe endpoints ofRSare R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.

EXAMPLE 3

Use the Midpoint Formula

Lesson 1-1 Point, Line, Plane


1

,

,

M

M

=

2

5

2

The coordinates of the midpoint Mare

1

5

,

2

2

ANSWER

1 + 4

– 3 + 2

2

2

EXAMPLE 3

Use the Midpoint Formula

SOLUTION

a. FIND MIDPOINTUse the Midpoint Formula.

Lesson 1-1 Point, Line, Plane


FIND ENDPOINTLet (x, y) be the coordinates of endpoint K. Use the Midpoint Formula.

b. FIND ENDPOINTThe midpoint of JKis M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.

STEP 1

Find x.

STEP 2

Find y.

4+ y

1+ x

1

2

=

=

2

2

ANSWER

The coordinates of endpoint Kare (3, – 2).

EXAMPLE 3

Use the Midpoint Formula

4 + y = 2

1 + x = 4

y =–2

x =3

Lesson 1-1 Point, Line, Plane


Distance formula
Distance Formula

The distance between two points A and B

is

Lesson 1-1 Point, Line, Plane


EXAMPLE 4

Standardized Test Practice

SOLUTION

Use the Distance Formula. You may find it helpful to draw a diagram.

Lesson 1-1 Point, Line, Plane


Naming angles

Name the three angles in diagram.

Name this one angle in 3 different ways.

Naming Angles

WXY, WXZ, and YXZ

The vertex of the angle

What always goes in the middle?

Lesson 1-1 Point, Line, Plane


o

ALGEBRAGiven that m LKN =145 , find m LKM andm MKN.

STEP 1

Write and solve an equation to find the value of x.

mLKN = m LKM + mMKN

o

o

o

145 = (2x + 10)+ (4x – 3)

EXAMPLE 2

Find angle measures

SOLUTION

Angle Addition Postulate

Substitute angle measures.

145 = 6x + 7

Combine like terms.

138 = 6x

Subtract 7 from each side.

23 = x

Divide each side by 6.

Lesson 1-1 Point, Line, Plane


STEP 2

Evaluate the given expressions when x = 23.

mLKM = (2x+ 10)° = (2 23+ 10)° = 56°

mMKN = (4x– 3)° = (4 23– 3)° = 89°

So, m LKM = 56°and m MKN = 89°.

ANSWER

EXAMPLE 2

Find angle measures

Lesson 1-1 Point, Line, Plane


3. Given that KLMis straight angle, find mKLN andm NLM.

STEP 1

Write and solve an equation to find the value of x.

m KLM + m NLM

= 180°

(10x – 5)° + (4x +3)°

= 180°

= 180

14x – 2

= 182

14x

x

= 13

GUIDED PRACTICE

Find the indicated angle measures.

SOLUTION

Straight angle

Substitute angle measures.

Combine like terms.

Subtract 2 from each side.

Divide each side by 14.

Lesson 1-1 Point, Line, Plane


STEP 2

Evaluate the given expressions when x = 13.

mKLM = (10x– 5)° = (10 13– 5)° = 125°

mNLM = (4x+ 3)° = (4 13+ 3)° = 55°

mKLM

= 125°

mNLM

= 55°

ANSWER

GUIDED PRACTICE

Lesson 1-1 Point, Line, Plane


In the diagram at the right, YWbisects XYZ, and mXYW = 18. Find m XYZ.

o

By the Angle Addition Postulate, m XYZ =mXYW + m WYZ. BecauseYW bisects XYZyou know thatXYW WYZ.

So, m XYW = m WYZ, and you can write

M XYZ = m XYW + m WYZ = 18° + 18° = 36°.

~

EXAMPLE 3

Double an angle measure

SOLUTION

Lesson 1-1 Point, Line, Plane


Example 4
Example 4

Lesson 1-1 Point, Line, Plane


a. You can draw a diagram with complementary adjacent angles to illustrate the relationship.

m 2 = 90° – m 1 = 90° – 68° = 22

EXAMPLE 2

Find measures of a complement and a supplement

a. Given that 1 is a complement of 2 and m1 = 68°,

find m2.

SOLUTION

Lesson 1-1 Point, Line, Plane


Sports

When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find mBCEand mECD.

EXAMPLE 3

Find angle measures

Lesson 1-1 Point, Line, Plane


Use the fact that the sum of the measures of supplementary angles is 180°.

STEP1

mBCE+m∠ ECD=180°

EXAMPLE 3

Find angle measures

SOLUTION

Write equation.

(4x+ 8)°+ (x +2)°= 180°

Substitute.

5x + 10 = 180

Combine like terms.

5x = 170

Subtract10 from each side.

x = 34

Divide each side by 5.

Lesson 1-1 Point, Line, Plane


STEP angles is 2

Evaluate: the original expressions when x = 34.

m BCE = (4x + 8)° = (4 34 + 8)° = 144°

m ECD = (x + 2)° = ( 34 + 2)° = 36°

ANSWER

The angle measures are144°and36°.

EXAMPLE 3

Find angle measures

Lesson 1-1 Point, Line, Plane


Angles formed by the intersection of 2 lines
Angles Formed by the Intersection of 2 Lines angles is

 Click Me!

Lesson 1-1 Point, Line, Plane


1 angles is and 4 are a linear pair. 4 and 5 are also a linear pair.

Identify all of the linear pairs and all of the vertical angles in the figure at the right.

ANSWER

1 and 5 are vertical angles.

ANSWER

EXAMPLE 4

Identify angle pairs

SOLUTION

To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays.

To find vertical angles, look or angles formed by intersecting lines.

Lesson 1-1 Point, Line, Plane


Example 5
Example 5 angles is

Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle.

Lesson 1-1 Point, Line, Plane


Example 6
Example 6 angles is

Given that m5 = 60 and m3 = 62, use your knowledge of linear pairs and vertical angles to find the missing angles.

Lesson 1-1 Point, Line, Plane


a. angles is

b.

c.

d.

a.

Some segments intersect more than two segments, so it is not a polygon.

The figure is a convex polygon.

b.

Part of the figure is not a segment, so it is not a polygon.

c.

d.

The figure is a concave polygon.

EXAMPLE 1

Identify polygons

Tell whether the figure is a polygon and whether it is convex or concave.

SOLUTION

Lesson 1-1 Point, Line, Plane


3 angles is

4

5

6

7

8

9

10

12

n

What is a polygon with 199 sides called?

199-gon

Lesson 1-1 Point, Line, Plane


a. angles is

b.

a.

The polygon has 6 sides. It is equilateral and equiangular, so it is a regular hexagon.

b.

The polygon has 4 sides, so it is a quadrilateral. It is not equilateral or equiangular, so it is not regular.

EXAMPLE 2

Classify polygons

Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning.

SOLUTION

Lesson 1-1 Point, Line, Plane


A table is shaped like a regular hexagon.The expressions shown represent side lengths of the hexagonal table. Find the length of a side.

ALGEBRA

3x + 6

4x – 2

=

6

x – 2

=

8

x

=

EXAMPLE 3

Find side lengths

SOLUTION

First, write and solve an equation to find the value of x. Use the fact that the sides of a regular hexagon are congruent.

Write equation.

Subtract 3x from each side.

Add 2 to each side.

Lesson 1-1 Point, Line, Plane


3 hexagon.The expressions shown represent side lengths of the hexagonal table. Find the length of a side.x + 6

3(8) + 6

=

=

30

The length of a side of the table is 30inches.

ANSWER

EXAMPLE 3

Find side lengths

Then find a side length. Evaluate

one of the expressions when x = 8.

Lesson 1-1 Point, Line, Plane


Perimeter area
Perimeter/Area hexagon.The expressions shown represent side lengths of the hexagonal table. Find the length of a side.

Rectangle

Square

Triangle

Circle

Lesson 1-1 Point, Line, Plane


Area hexagon.The expressions shown represent side lengths of the hexagonal table. Find the length of a side.

The area of the triangle is 14 square inches and its height is 7 inches. Find the base of the triangle.

Lesson 1-1 Point, Line, Plane


Perimeter
Perimeter hexagon.The expressions shown represent side lengths of the hexagonal table. Find the length of a side.

The perimeter of a rectangle 84.6 centimeters. The length of the rectangle is twice as long as its width. Find the length and width of the rectangle.

Lesson 1-1 Point, Line, Plane


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