ch. 1 highlights geometry a

Ch. 1 HighlightsGeometry A

Ms. Urquhart

Mrs. Vander Bee

Lesson 1-1 Point, Line, Plane 2

Coplanar Objects

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

H

E

G

DC

BA

F

Are the following points coplanar?

A, B, C ?A, B, C, F ?H, G, F, E ?E, H, C, B ?A, G, F ?C, B, F, H ?

YesNo

YesYesYesNo

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


Front Side – True or False


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 – 8RT = 60

3x-16

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

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


EXAMPLE 2 Use algebra with segment lengths

Point M is the midpoint of VW . Find the length of VM .ALGEBRA


GUIDED PRACTICE

Identify the segment bisector of .

PQ

Then find PQ.

line l


MIDPOINT FORMULA

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

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

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

2 2


EXAMPLE 3 Use the Midpoint Formula

a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.



252

1 + 4 2

– 3 + 2 2 =, M , – 1M

The coordinates of the midpoint M are 1,–5

2 2

ANSWER

SOLUTION

a. FIND MIDPOINT Use the Midpoint Formula.



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

STEP 1 Find x.

1+ x 22

=

1 + x = 4

x = 3

STEP 2 Find y.

4+ y 12

=

4 + y = 2

y = – 2

The coordinates of endpoint K are (3, – 2).ANSWER

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


Distance Formula

The distance between two points A and B

is


SOLUTION

EXAMPLE 4 Standardized Test Practice

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


Naming Angles

Name the three angles in diagram.

Name this one angle in 3 different ways.

WXY, WXZ, and YXZ

What always goes in the middle? The vertex of the angle


EXAMPLE 2 Find angle measures

oALGEBRA Given that m LKN =145 , find m LKM and m MKN.

SOLUTION

STEP 1

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

m LKN = m LKM + m MKN Angle Addition Postulate

Substitute angle measures.

145 = 6x + 7 Combine like terms.

Subtract 7 from each side.138 = 6x

Divide each side by 6.23 = x

145 = (2x + 10) + (4x – 3)o oo



STEP 2

Evaluate the given expressions when x = 23.

m LKM = (2x + 10)° = (2 23 + 10)° = 56°

m MKN = (4x – 3)° = (4 23 – 3)° = 89°

So, m LKM = 56° and m MKN = 89°.ANSWER


GUIDED PRACTICE

Find the indicated angle measures.

3. Given that KLM is straight angle, find m KLN and m NLM.

STEP 1

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

Straight angle

Substitute angle measures.

Combine like terms.

Subtract 2 from each side.

Divide each side by 14.

m KLM + m NLM = 180°

(10x – 5)° + (4x +3)°= 180°14x – 2 = 180

14x = 182

x = 13

SOLUTION


GUIDED PRACTICE

STEP 2

Evaluate the given expressions when x = 13.

m KLM = (10x – 5)° = (10 13 – 5)° = 125°

m NLM = (4x + 3)° = (4 13 + 3)° = 55°

ANSWER m KLM = 125° m NLM = 55°


SOLUTION

EXAMPLE 3 Double an angle measure

In the diagram at the right, YW bisects XYZ, and m XYW = 18. Find m XYZ.

o

By the Angle Addition Postulate, m XYZ = m XYW + m WYZ. Because YW bisects XYZ you know that XYW WYZ. ~

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

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


Example 4


EXAMPLE 2 Find measures of a complement and a supplement

SOLUTION

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

m 2 = 90° – m 1 = 90° – 68° = 22

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



Sports

When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m BCE and m ECD.


SOLUTION


STEP 1 Use the fact that the sum of the measures of supplementary angles is 180°.

Write equation.

(4x+ 8)° + (x + 2)° = 180° Substitute.

5x + 10 = 180 Combine like terms.

5x = 170

x = 34

Subtract 10 from each side.

Divide each side by 5.

mBCE + m ECD = 180°



STEP 2

Evaluate: the original expressions when x = 34.

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

m ECD = (x + 2)° = ( 34 + 2)° = 36°

The angle measures are 144° and 36°.ANSWER


Angles Formed by the Intersection of 2 Lines

Click Me!


SOLUTION

EXAMPLE 4 Identify angle pairs

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

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

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

1 and 5 are vertical angles.ANSWER

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

ANSWER


Example 5

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.


Example 6 Given that m5 = 60 and m3 = 62, use your knowledge of linear pairs and vertical angles to find the missing angles.


EXAMPLE 1 Identify polygons

SOLUTION

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

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

a.

b. The figure is a convex polygon.

d. The figure is a concave polygon.

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

c.

b. c.a. d.


# of sides Type of Polygon

triangle

quadrilateral

pentagon

hexagon

heptagon

octagon

nonagon

decagon

dodecagon

n-gon

5

10

4

n

12

9

3

6

8

7

What is a polygon with 199 sides called? 199-gon

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EXAMPLE 2 Classify polygons

SOLUTION

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

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

a.

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

b.


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.

3x + 6 4x – 2=6 = x – 28 = x

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


EXAMPLE 3 Find side lengths

Then find a side length. Evaluate one of the expressions when x = 8.

303(8) + 6 ==3x + 6

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

ANSWER


Perimeter/Area

Rectangle

Square

Triangle

Circle


Area

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


Perimeter

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.

ch. 1 highlights geometry a

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