Bell Work 9/13/11• 1) Find the midpoint of segment PQ if P(1, -3)

and Q(3, 7).

• 2) Find the other endpoint if one endpoint is at (-2, -4) and the midpoint is at (-3, 2).

• 3) B is the midpoint of AC, find x and the length of each segment.

Outcomes• I will be able to:• 1) Define and Use new vocabulary: midpoint, bisector,

segment bisector, construction, Midpoint Formula and angle bisector.

• 2) Bisect a segment/angle by measuring, by folding, and by algebraic reasoning.

• 3) Use the Midpoint Formula to calculate segment midpoints on a coordinate plane.

• 4) Solve for missing values and angle measures using angle identifications.

AgendaBell WorkOutcomesAgenda – Reminder: Honors App due by Fri.Quiz Results ReviewConstruction ActivityFinish 1.5 -1.6IPExit Ticket

Quiz Review

• Let’s spend a few minutes looking over the quizzes.

• All quizzes must be collected again and are not to go home.

• SO… Open your notebooks and take notes of anything that you might want to study further to get a better grade on next time.

• Collect all quizzes.

Constructions• The compass, like the straight edge, has been a useful

geometry tool for thousands of years. The ancient Egyptians used a compass to mark off distances.

• During the Golden Age of Greece, Greek mathematicians made a game of geometric constructions.

• In his 13 volume work Elements, Euclid (325-265 BC) established the basic rules for constructions using only a compass and straight edge.

• He proposed definitions and constructions about points, lines, angles, surfaces and solids. He also showed why the constructions were correct with deductive reasoning.

• You will learn many of these constructions using the same tools.

School of Athens by Raphael

School of AthensEuclid is represented here teaching while a student is showing a geometric construction to his fellow mathematicians.

Notice the globes being held. This was how they studied the heavens. They thought the earth was the center and the heavens a sphere around them.

Constructions

• Compass and a straight edge only• This game of trying to draw figures with only

these two tools dates back to the classical Greeks.

• Constructions develop deductive reasoning while giving insight into geometry relationships.

Bisecting a Segment• Draw a segment

• Set compass to be more than the midpoint

• Strike an arc above and below the segment from each endpoint.

• DON’T change the compass setting.

• Connect the points where the arcs intersect.

• You have created a perpendicular bisector of the segment.

Bisecting an Angle• Draw an Angle

• Set compass on the vertex and strike an arc that touches both sides of the angle.

• Move the compass to the point made by the first arc touching the side.

• Strike an arc between the sides but beyond the first arc.

• DON’T change the compass setting.

• Repeat with compass on other side of angle.

• Connect the point where the arcs intersect to the vertex of your angle.

• You have created a ray that is an angle bisector.

Write it out Questions• How can we verify that our segment is REALLY

bisected? Describe two things you could do.

• How can we verify that our angle is REALLY bisected? Describe two things you could do.

• Brainstorm: what else do you think you could draw with the rules of construction to challenge a fellow mathematician with?

• Example: Construct a perfect square.

1.5 Angle Bisector

• Angle Bisector – a ray or line that cuts an angle into two congruent pieces

• Example Picture:

• Ray CD is the angle bisector

• Symbols: DCBACD

Examples

Examples

1.6 Special Angle Relationships

• Vertical Angles:• Two angles are ________ __________

if their sides form two pairs of _______ ________.

Vertical Angles share __________________.

Vertical angles are _______________________. • What do we know about vertical angles?• What can we do with the equations?• 4x – 2 = 2x + 14• 2x = 16• X = 8• Find the measure of each angle

142

24

xDCE

xACB

vertical angles

opposite rays

a vertex

always congruent

Special Angle Relationships

• Linear Pair of Angles: Two adjacent angles are a ______ ____

if their non-common sides are _________ _____. The sum of

the measures of angles that form a linear pair is ______.

• What do we know about a linear pair? • How can we find x?• 13x + 3 + 6x + 6 = 180• 19x + 9 = 180• 19x = 171• X = 9• Find each angle measure

66

313

xDBC

xABC

Their sum is 180°

linear pair

opposite rays

180°

add the two angles and set them equal to 180

Special Angle Relationships

1

2

3

45

6

1 3 5 6

2 4

Examples

• Try Example 1 on your OWN.

Examples

Euclid Street

Pythagoras Street

36°x

yz

Label what we know

Find the missing piecesusing what we know about vertical and linear pairs of angles.

Examples

• Try example 3 on your OWN.

Complementary and Supplementary Angles

• Two angles are _________________________ angles if

the sum of their measures is _________.

• Each angle is the _____________________ of the other. They can be adjacent or nonadjacent.

• Two angles are _________________________ angles if the

sum of their measures is _________. • Each angle is the _____________________ of the other. They

can be adjacent or nonadjacent.

complementary

90°

complement

supplementary

180°

supplement

Special Angle Relationships

Each angle is the complementof the other. They can be adjacentor nonadjacent angles

Each angle is the supplementof the other. They can be adjacentor nonadjacent angles

Special Angle Relationships• Complementary Angles: Angles whose sum is

90°

33

75

xCAD

xBAD

Special Angle Relationships• Supplementary Angles: Angles whose sum is

180°

51

2111

xDEF

xABC

Examples

Since A is the complement of Z, we know they must add to 90°.

Try examples B and C on your OWN.

Exit Quiz• Solve for x:• 1)

• 2) Name the angle relationship and solve for x• a) b)

55

34

xRQS

xPQS

75

173

xDAC

xBAD

84

26

xDCE

xBCA