Measuring Segments and Angles Honors Geometry Measuring Segments and Angles

Postulates RULER POSTULATE The 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.

RULER POSTULATE A B AB = |2 – 5| = 3

Congruent Same measure Notation:  Mark up the picture with the same marks AB  CD A B C D    

Let’s Practice    P T Q Given: PT = 5x + 3 and TQ = 7x – 9 Find: PT. x = 6, PT = 33 Answers

SEGMENT ADDITION POSTULATE Postulates SEGMENT ADDITION POSTULATE If 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 – 12 Find: 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 + 30 Find: x, EF, & FG. x = 15, EF = 40, FG = 60 Answers

Midpoint A point that divides a segment into two congruent parts. A line, a ray, or a segment can bisect another segment.

Bisect Cut through at the midpoint A line, a ray, or a segment can bisect another segment.

Trisect Cut into three equal parts A line, a ray, or a segment can trisect another segment.

Let’s Practice A C B    Given OC bisects AB, AC = 2x + 1and CB = 3x - 4 Find: AC, CB, & AB. O  x = 5, AC = 11, CB = 11, AB = 22 Answers

Postulates PROTRACTOR POSTULATE Let 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 mCOD = |x – y|.

PROTRACTOR POSTULATE  O A B C D mCOD = |x – y| = |50 – 120| = 70

Type Angle Range Sketch 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.

ANGLE ADDITION POSTULATE Postulates ANGLE ADDITION POSTULATE If point B is in the interior of AOC, then mAOB + mBOC = mAOC. If AOC is a straight angle, then mAOB + mBOC = 180.

Let’s Practice  W T   S  R Given: mRST = 50 and mRSW = 125. Find: mTSW mTSW = 75 Answers

Let’s Practice  G    D E F Given: DEF is a straight angle and mDEG = 145. Find: mGEF mGEF = 35 Answers

5:20 4:40 6:00 Angles and the Clock Estimate the measure of the angle formed by the hands of a clock at: 5:20 4:40 6:00 Place all three solutions here

Let’s Practice Q P V N M Given: mMQV = 90 and mVQP = 35. Find: mMQP mMQP = 125 Answers

Let’s Practice Q P V N M Given: mMVQ = 55 Find: mQVP mQVP = 125 Answers

Judging by appearance, name two acute angles. Let’s Practice Q P V N M Place both answers here. Judging by appearance, name two acute angles. Judging by appearance, name two obtuse angles.

Let’s Practice A B C O D Given: mAOC = 7x – 2,mAOB = 2x + 8 and mBOC = 3x + 14. Find: x x = 12 Answers

Let’s Practice A B C O D Given: mAOB = 28 ,mBOC = 3x – 2 and mAOD = 6x. Find: x x = 18 Answers

Median of a Triangle The segment joining the midpoint of a side to the opposite vertex.

Challenge 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.

GH = GF + FD + DH GH = 2 + 4 + 24 GH = 30 E G F D C H A B       32 16 16 2 E G F D 24 C H 24 A B       4 4 48 8 8