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Geometry Chapter 1 – Points, Lines, Planes, and Angles

***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. *** ____ Algebraic Equations Review

____ Keystone Vocabulary Writing Assignment

____ (1-1) Points, Lines, and Planes Page 9-10 # 13-20, 21, 22, 25, 26, 27, 30-46

____ (1-2) Linear Measure – Day 1 Page 17 # 12-15, 22 – 27

____ (1-2) Linear Measure – Day 2 Page 17-19 # 28-39, 58, 62

____ (1-3) Distance – Day 1 Page 25 # 13-30

____ (1-3) Midpoints – Day 2 Page 26 # 31-42

____ Take Home Test on Sections 1-1 through 1-3

____ (1-4) Angle Measure – Day 1 Page 33-34 # 4-6, 12-24

____ (1-4) Angle Measure – Day 2 Page 34 # 25-33

____ (1-4) Angle Measure – Day 3 Page 34 # 34 – 39, 52-60, 61, 63, 65

____ (1-5) Angle Relationships – Day 1 Page 42 # 11 – 25 (skip # 17)

____ (1-5) Angle Relationships – Day 2 1-5 Practice WS

____ (1-5) Angle Relationships – Day 3 Page 42 # 17, 27-30, 31-35, 37

____ (1-6) Polygons – Day 1 Page 49 – 50 # 12-25, 29-34

____ (1-6) Polygons – Day 2 Page 49 – 50 #26 – 28

____ Chapter 1 Review WS

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Date: ______________________

Section 1 – 1: Points, Lines, and Planes Notes

A Point: is simply a _______________. Example:

� Drawn as a ________.

� Named by a ______________ letter.

� Words/Symbols:

A Line: is made up of ____________ and has no thickness or __________.

� Drawn with an _________________ at each end.

� Named by the _____________ representing two points on the line or a lowercase

script letter.

� Points on the same _______ are said to be _____________.

� Words/Symbols: Example:

A Plane: is a _______ surface made up of ____________.

� Drawn as a ____________ 4-sided figure.

� Named by a _____________ script letter or by the letters naming three

___________________ points.

� Points that lie on the same plane are said to be _______________.

� Words/Symbols: Example:

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Example #1: Use the figure to name each of the following.

a.) Name a line that contains point P.

b.) Name the plane that contains lines n and m.

c.) Name the intersection of lines n and m.

d.) Name a point not on a line.

e.) What is another name for line n.

f.) Does line l intersect line n or line m? Explain.

Example #2: Draw and label a figure for the following relationship.

a.) Point T lies on WR. b.) AB intersects CD in plane Q at point P.

Example #3:

a.) How many planes appear in this figure?

b.) Name three points that are collinear.

c.) Are points A, B, C, and D coplanar? Explain.

d.) At what point do DBsuur

and CAsuur

intersect?

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CRITICAL THINKING

1.) Why do chairs sometimes wobble?

Include the following in your answer:

• an explanation of how the chair legs relate to points in a plane, and

• how many legs would create a chair that does not wobble.

2.) Complete the figure below to show the following relationship:

Lines a, b, and c are coplanar and lie in plane Q. Lines a and b intersect at

point P. Line c intersects line b at point R, but does not intersect line a.

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Date: ______________________

Section 1 – 2: Linear Measure Notes – Part 1

Measure Line Segments

� A line segment, or ______________, is a measurable part of a line that consists of

two points, called _________________, and all of the points between them.

� A segment with endpoints A and B can be named as _______ or _______.

� The length or _______________ of AB is written as ________.

Example #1: Use a metric ruler to draw each segment.

g.) Draw LM that is 42 millimeters long.

b.) Draw QR that is 5 centimeters long.

Example #2: Use a customary ruler to draw each segment.

a.) Draw DE that is 3 inches long.

b.) Draw FG that is 23

4 inches long.

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Calculate Measures

� Betweenness of Points: Point M is between points P and

Q if and only if P,Q, and M are ______________ and

__________________.

Example #4:

a.) Find LM. b.) Find XZ.

c.) Find DE.

d.) Find x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x – 3.

e.) Find y and PQ if P is between Q and R, PQ = 2y, QR = 3y + 1, and PR = 21. Draw a picture!

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Date: ______________________

Section 1 – 2: Linear Measure Notes – Part 2

Example: Find the value of x and LM if L is between N and M, NL = 6x – 5,

LM = 2x + 3, and NM = 30. Draw a picture!

Measure Line Segments

Key Concept (Congruent Segments):

Two __________________ having the same Ex:

measure are __________________.

Symbol:

Example #1: Name all of the congruent segments found in the kite.

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Example #2: Find the measurement of RS.

Example #3: Use the figures to determine whether each pair of segments is congruent.

a.) ,AB CD b.) ,WZ XY

c.) ,HO HT d.) ,MH TH

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CRITICAL THINKING

1.) Explain the difference between a line and a line segment and why one of these can

be measured, while the other cannot.

2.) Refer to the figure to the right.

a.) Name three collinear points.

b.) Name two planes that contain points B and C.

c.) Name another point in plane DFA.

d.) How many planes are shown?

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Date: ______________________

Section 1 – 3: Distance Notes – Part 1

Distance Between Two Points

Key Concept (Distance Formulas):

� Number Line

� Coordinate Plane

The distance d between two points with coordinates (x1, y1) and (x2, y2) is given by

d =

Example #1: Find the distance between E(-4, 1) and F(3, -1).

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Example #2: Use the number line to find QR.

Example #3: Use the number line to find CD.

Example #4: Use the number line to find AB and CD.

Example #5: Use the Distance Formula to find the distance between the following points.

a.) A(10, -2) and B(13, -7)

b.) X(-5, -7) and Y(-10, 7)

c.) G(-4, 1) and H(3, -1)

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Date: ______________________

Section 1 – 3: Midpoint Notes – Part 2

Midpoint of a Segment

Key Concept (Midpoint):

� The midpoint M of PQ is the point ___________________ P and Q such that

_____________________.

� Number Line: The coordinate of the midpoint

of a __________________ whose endpoints have

coordinates a and b is

Example #1: The coordinates on a number line of J and K are –12 and 16, respectively. Find the

coordinate of the midpoint of JK . Hint: Draw a number line!

Example #2: The coordinates on a number line of T and S are 5 and 8, respectively. Find the

coordinate of the midpoint of TS . Hint: Draw a number line!

� Coordinate Plane: The coordinates of the

_____________________ of a segment

whose endpoints have coordinates (x1, y1) and

(x2, y2) are

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Example #3: Find the coordinates of the midpoint of PQ for P(-1, 2) and Q(6, 1).

Example #4: Find the coordinates of the midpoint of GH for G(8, -6) and H(-14, 12).

Example #5: Find the coordinates of the midpoint of AB for A(4, 2) and B(8, -6).

Example #6: What is the measure of PR if Q is the midpoint of PR ? Segment Bisector: any segment, line, or plane that interests a

segment at its _______________

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CRITICAL THINKING

1.) Which equation represents the following problem?

Fifteen minus three times a number equals negative twenty-two. Find the number.

a.) 15 – 3n = -22

b.) 3n – 15 = -22

c.) 3(15 – n) = -22

d.) 3(n – 15) = -22

2.) Find the distance between points at (6, 11) and ( -2, -4).

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Date: ______________________

Section 1 – 4: Angle Measure Notes – Part 1

Measure Angles

� Degree: a unit of measure used in measuring

______________ and __________. An arc of a

circle with a measure of 1° is ___________ of the

entire circle.

� Ray: is a part of a ___________

� It has one ____________________ and extends

indefinitely in _________ direction.

� Symbols:

� Opposite Rays: two rays _________ and _________ such

that B is between A and C

Key Concept (Angle):

An angle is formed by two ______________________ rays that have a common

__________________.

� The rays are called ____________ of the angle.

� The common endpoint is the ______________.

� Symbols:

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� An angle divides a plane into three distinct parts.

� Points _____, _____, and _____ lie on the angle.

� Points _____ and _____ lie in the interior of the angle.

� Points _____ and _____ lie in the exterior of the angle.

Example #1:

a.) Name all angles that have B as a vertex.

b.) Name the sides of 5∠ .

c.) Write another name for 6∠ .

Example #2:

a.) Name all the angles that have W as a vertex.

b.) Name the sides of 1∠ .

c.) Write another name for WYZ∠ .

d.) Name the vertex of 4∠ .

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Date: ______________________

Section 1 – 4: Angle Measure Notes – Part 2

Measure Angles

Key Concept (Classify Angles):

RIGHT ANGLE: ACUTE ANGLE: OBTUSE ANGLE:

Model: Model: Model:

Measure: Measure: Measure:

Example #1: Measure each angle, then classify as right, acute, or obtuse.

a.) b.)

c.) d.)

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e.) f.)

Example #2: Measure each angle named and classify it as right, acute, or obtuse.

a.) TYV∠

b.) WYT∠

c.) TYU∠

d.) VYX∠

e.) SYV∠

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Date: ______________________

Section 1 – 4: Angle Measure Notes – Part 3

Congruent Angles

Key Concept (Congruent Angles):

Angles that have the same _____________________ are

congruent angles.

Arcs on the figure also indicate which angles are

___________________.

Example #1: State whether each pair of angles is congruent, and if so write a congruence statement.

a.) b.)

Example #2: Find the value of x and the measure of one angle.

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Angle Bisector: a _________ that divides an angle into _________ congruent angles.

Ex:

If PQuuur

is the angle bisector of ___________,

then _____________________.

Example #3: In the figure, QP and QR are opposite rays, and QT bisects RQS∠ .

a.) If 56 +=∠ xRQTm and 27 −=∠ xSQTm , find RQTm∠ .

b.) Find TQSm∠ if 1122 −=∠ aRQSm and 812 −=∠ aRQTm .

Example #4: In the figure, YU bisects ZYW∠ and YT bisects XYW∠ .

a.) If 1051 +=∠ xm and 2382 −=∠ xm , find 2∠m .

b.) If WYZm∠ =82 and 254 +=∠ rZYUm , find r.

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CRITICAL THINKING

1.) Mr. Lopez wants to cover the walls of his unfinished basement with pieces of

plasterboard that are 8 feet high, 4 feet wide, and ¼ inch thick. If the basement

measures 24 feet wide, 16 feet long, and 8 feet tall, how many pieces of plasterboard

will he need to cover all four walls?

2.) Each figure below shows noncollinear rays with a common endpoint.

a.) Count the number of angles in each figure.

b.) Describe the pattern between the number of rays and the number of angles.

c.) Make a conjecture of the number of angles that are formed by 7 noncollinear

rays and by 10 noncollinear rays.

d.) Write a formula for the number of angles formed by n noncollinear rays with a

common endpoint.

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Date: ______________________

Section 1 – 5: Angle Relationships Notes – Part 1

Pairs of Angles

Key Concept (Angle Pairs):

� Adjacent Angles: are two angles that lie in the same ____________, have a common

_____________, and a common ___________, but no common interior ____________

� Examples:

� Vertical Angles : are two non-adjacent angles formed by two __________________ lines

� Examples: Non-example:

� Linear Pair : a pair of ________________ angles whose non-common sides are opposite

__________.

� Example: Non-example:

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Example #1 : Name an angle pair that satisfies each condition.

a.) two angles that form a linear pair

b.) two acute vertical angles

c.) an angle supplementary to VZX∠

d.) two acute adjacent angles

Key Concept (Angle Relationships):

� Complementary Angles: two angles whose measures have a sum of ________

� Examples:

� Supplementary Angles: two angles whose measures have a sum of ________.

� Examples:

Example #2: Find the measures of two supplementary angles if the measure of one angle is 6 less

than 5 times the measure of the other angle.

Example #3: Find the measures of two complementary angles if the difference in the measures of the

two angles is 12.

Example #4: The measure of an angle’s supplement is 33 less than the measure of the angle. Find the

measure of the angle and its supplement.

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Date: ______________________

Section 1 – 5: Angle Relationships Notes – Part 2

Perpendicular Lines

� Lines that form right angles are _____________________.

Key Concept (Perpendicular Lines):

� Perpendicular lines intersect to form _________ right angles.

� Perpendicular lines intersect to form _________________

_______________ angles.

� ________________ and _________ can be perpendicular to lines

or to other line segments and rays.

� The right angle symbol in the figure indicates that the lines are ___________________.

� Symbol: _______ is read is perpendicular to.

Example #1: Find x so that KO HM⊥suur suuur

.

Example #2: Find x and y so that BE and AD are perpendicular.

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Assumptions:

Example #3: Determine whether or not each of the following statements can be assumed or not.

� All points shown are coplanar.

� P is between L and Q.

� PLPN ≅

� QPO∠ and OPL∠ are supplementary.

� PMPN ⊥

� L, P, and Q are collinear.

� LPMQPO ∠≅∠

� POPQ ≅

� PQLP ≅

� LMP∠ and MNP∠ are adjacent angles.

� LPN∠ and NPQ∠ are a linear pair.

� LPMOPN ∠≅∠

� ,,, POPNPM and LQ intersect at P.

Example #4: Determine whether each statement can be assumed from the figure below. Explain.

a.) 90m VYT∠ =

b.) TYW∠ and TYU∠ are supplementary

c.) VYW∠ and TYS∠ are complementary

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Date: ______________________

Section 1 – 5: Angle Relationships Extra Examples

Example #1: Two angles are complementary. One angle measures 24° more than the other. Find the measures of the angles. Example #2: Find the measures of two supplementary angles if the measure of one angle is 4 less than 3 times the measure of the other angle. Example #3: The measure of an angle’s supplement is 22 less than the measure of the angle. Find the measure of the angle and its supplement.

Example #4: Find the value of x so that ACsuur

and BDsuur

are perpendicular.

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CRITICAL THINKING

1.) A counterexample is used to show that a statement is not necessarily true. Find a

counterexample for the statement Supplementary angles for linear pairs.

2.) What kinds of angles are formed when streets intersect?

Include the following in your answer:

• the types of angles that might be formed by two intersecting lines, and

• a sketch of intersecting streets with angle measures and angle pairs identified.

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Date: ______________________

Section 1 – 6: Polygons Notes

Polygons

� A polygon is a ______________ figure whose sides are all segments.

� The sides of each angle in a polygon are called ___________ of the polygon, and the vertex of

each angle is a _____________ of the polygon.

� Examples:

� Polygons can be ________________ or ________________.

� Examples:

_____________________ ________________________

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Regular Polygon: a convex

polygon in which all the ________

are congruent and all the angles are

___________________.

Ex:

Example #1: Name each polygon by the number of sides. Then classify it as convex or concave, regular

or irregular.

a.) b.)

Perimeter

� The perimeter of a polygon is the sum of the _______________ of its sides, which are

_________________.

Example #2: Find the perimeter of each polygon.

a.) b.) c.)

Number of Sides

Polygon

3

quadrilateral

5

6

heptagon

octagon

9

decagon

12

n

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CRITICAL THINKING

1.) Refer to the figure to the right.

• Find the perimeter of pentagon LMNOP.

• Suppose the length of each side of pentagon LMNOP is doubled. What effect

does this have on the perimeter?

2.) Quadrilateral ABCD has a perimeter of 95 centimeters. Find the length of each side

if AB = 3a + 2, BC = 2(a – 1), CD = 6a + 4, and AD = 5a – 5.