honors geometry
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Measuring Segments and Angles
HONORS GEOMETRY
Postulates
RULER POSTULATEThe points of a line can be put into 1 to 1 correspondence with the real numbers so that the distance between any 2 points is the absolute value of the difference of the corresponding numbers.
A B
AB = |2 – 5| = 3
RULER POSTULATE
•Same measure•Notation: •Mark up the picture with the same marks
Congruent
AB CD A B C D
Let’s Practice
P T Q
Given: PT = 5x + 3 and TQ = 7x – 9Find: PT.
x = 6, PT = 33 Answers
Postulates
SEGMENT ADDITION POSTULATEIf 3 points A, B, and C are collinear and B is between A and C, then AB + BC = AC.
Let’s Practice
D S T
Given:DT = 60 and DS = 2x – 8 and ST = 3x – 12Find: x, DS, & ST.
x = 16, DS = 24, ST = 36 Answers
Let’s Practice
E F G
Given:EG = 100 and EF= 4x – 20 and FG = 2x + 30Find: x, EF, & FG.
x = 15, EF = 40, FG = 60 Answers
•A point that divides a segment into two congruent parts.•A line, a ray, or a segment can bisect another segment.
Midpoint
•Cut through at the midpoint•A line, a ray, or a segment can bisect another segment.
Bisect
•Cut into three equal parts•A line, a ray, or a segment can trisect another segment.
Trisect
Let’s PracticeA C B
GivenOC bisects AB, AC = 2x + 1and CB = 3x - 4Find: AC, CB, & AB.
O
x = 5, AC = 11, CB = 11, AB = 22 Answers
Postulates
PROTRACTOR POSTULATELet OA and OB be opposite rays in a plane. OA, OB, and all rays with endpoint O that can be drawn on one side of AB can be paired with the real numbers 0 to 180 so that OA is paired with 0 and OB is paired with 180. If OC is paired with x and OD is paired with y, then mCOD = |x – y|.
PROTRACTOR POSTULATE
O
A B
CD
mCOD = |x – y| = |50 – 120| = 70
Types of Angles
Type Angle Range
Sketch
Acute 0 < x < 90
Right x = 90
Obtuse 90 < x < 180
Straight
x = 180
Let’s use a Ruler and a Protractor!
Complete the Measurement worksheet.
Postulates
ANGLE ADDITION POSTULATEIf point B is in the interior of AOC, then mAOB + mBOC = mAOC.
If AOC is a straight angle, then mAOB + mBOC = 180.
Let’s Practice
R
T
S
Given:mRST = 50 and mRSW = 125.Find:mTSW
W
mTSW = 75Answers
Let’s Practice
E
G
D
Given:DEF is a straight angle and mDEG = 145.Find:mGEF
F
mGEF = 35Answers
Angles and the Clock
Estimate the measure of the angle formed by the hands of a clock at:
6:004:405:20
Given:mMQV = 90 and mVQP = 35.Find:mMQP
Let’s PracticeP
M N
QV
mMQP = 125Answers
Given:mMVQ = 55Find:mQVP
Let’s PracticeP
M N
QV
mQVP = 125Answers
Judging by appearance, name two acute angles.
Let’s PracticeP
M N
QV
Judging by appearance, name two obtuse angles.
Given:mAOC = 7x – 2,mAOB = 2x + 8 and mBOC = 3x + 14.Find:x
Let’s Practice
CO D
A B
x = 12Answers
Given:mAOB = 28 ,mBOC = 3x – 2 and mAOD = 6x.Find:x
Let’s Practice
CO D
A B
x = 18Answers
•The segment joining the midpoint of a side to the opposite vertex.
Median of a Triangle
C is the midpoint of AB. D is the midpoint of AC. E is the midpoint of AD. F is the midpoint of ED. G is the midpoint of EF. H is the midpoint of DB.
If DC = 16, find GH.
Challenge
A BCDE FG H
1616
8 8
32
48
2424
44
2
GH = GF + FD + DHGH = 2 + 4 + 24GH = 30
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