Bell Work 9/13/11 1) Find the midpoint of segment PQ if P(1, -3) and Q(3, 7). 2) Find the other...
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Transcript of Bell Work 9/13/11 1) Find the midpoint of segment PQ if P(1, -3) and Q(3, 7). 2) Find the other...
![Page 1: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062718/56649e725503460f94b70918/html5/thumbnails/1.jpg)
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.
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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.
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AgendaBell WorkOutcomesAgenda – Reminder: Honors App due by Fri.Quiz Results ReviewConstruction ActivityFinish 1.5 -1.6IPExit Ticket
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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.
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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.
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School of Athens by Raphael
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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.
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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.
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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.
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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.
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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.
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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
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Examples
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Examples
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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
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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
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Special Angle Relationships
1
2
3
45
6
1 3 5 6
2 4
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Examples
• Try Example 1 on your OWN.
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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.
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Examples
• Try example 3 on your OWN.
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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
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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
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Special Angle Relationships• Complementary Angles: Angles whose sum is
90°
33
75
xCAD
xBAD
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Special Angle Relationships• Supplementary Angles: Angles whose sum is
180°
51
2111
xDEF
xABC
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Examples
Since A is the complement of Z, we know they must add to 90°.
Try examples B and C on your OWN.
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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