geometry unit 1.3

UNIT 1.3 ANGLE BASICS AND MEASURING

SEGMENTS

Coordinate : The numerical location of a point on a number line.

Length :Length : On a number line length AB = AB = |B - A|

Midpoint :Midpoint : On a number line, midpoint of AB = 1/2 (B+A)

BA C D E

2 4 6 8-2-4-6-8 -1 0

Find the length of each segment.

XY = | –5 – (–1)| = | –4| = 4

ZY = | 2 – (–1)| = |3| = 3

ZW = | 2 – 6| = |–4| = 4

Find which two of the segments XY, ZY, and ZW are

congruent.

Because XY = ZW, XY ZW.

Measuring Segments and Angles

The Segment Addition Postulate

If three points A, B, and C are collinear and B is between A and C,

then AB + BC = AC.

A B C

Use the Segment Addition Postulate to write an equation.

AN + NB = AB Segment Addition Postulate(2x – 6) + (x + 7) = 25 Substitute.

3x + 1 = 25 Simplify the left side. 3x = 24 Subtract 1 from each side. x = 8 Divide each side by 3.

AN = 10 and NB = 15, which checks because the sum of the segment lengths equals 25.

If AB = 25, find the value of x. Then find AN and NB.

AN = 2x – 6 = 2(8) – 6 = 10 NB = x + 7 = (8) + 7 = 15 Substitute 8 for x.

Use the definition of midpoint to write an equation.

5x + 45 = 8x Add 36 to each side.

RM and MT are each 84, which is half of 168, the length of RT.

M is the midpoint of RT. Find RM, MT, and RT.

RM = 5x + 9 = 5(15) + 9 = 84MT = 8x – 36 = 8(15) – 36 = 84

Substitute 15 for x.

RT = RM + MT = 168

RM = MT Definition of midpoint5x + 9 = 8x – 36 Substitute.

45 = 3x Subtract 5x from each side. 15 = x Divide each side by 3.

1. T is in between of XZ. If XT = 12 and XZ = 21,

then TZ = ?

2. T is the midpoint of XZ. If XT = 2x +11 and XZ = 5x + 8,

find the value of x.

Quiz

1. T is in between of XZ. If XT = 12 and XZ = 21, then TZ = ? 21 – 12 = 9, TZ = 9

2. T is the midpoint of XZ. If XT = 2x +11 and XZ = 5x + 8, find the value of x.Since T is a midpoint of XZ, 2*XT = XZ2(2x + 11) = 5x + 84x+22=5x+8X=14

Answers:

Midpoint

On a number lineformula:

2

ba

On a coordinate plane

formula:

2

,2

, 2121 yyxxyx mm

QS has endpoints Q(3, 5) and S(7, -9).

Find the coordinates of its midpoint M.

The midpoint of AB is M(3, 4). One endpoint is A(-3, -2).

Find the coordinates of the other endpoint B.

1) QS has endpoints Q(3, 5) and S(7, -9). Find the coordinates of its midpoint M.((3+7)/2, (5+-9)/2)The midpoint is (5,-2).

2) The midpoint of AB is M(3, 4). One endpoint is A(-3, -2). Find the coordinates of the other endpoint B(-3+x)/2 = 3, -3 + x = 6, x = 9(-2+y)/2 = 4, -2 + y = 8, y = 10The other endpoint is (9,10).

Answers

FAD , FBC, 1 • Right Angle• Obtuse Angle• Acute Angle• Straight Angle• Congruent Angles

• Formed by two rays with the same endpoint. • The rays: sides• Common endpoint: the vertex• Name:

• Measures exactly 90º• Measure is GREATER than 90º• Measure is LESS than 90º• Measure is exactly 180º ---this is a line• Angles with the same measure.

1

2

FAD

ADE

FAB

• Angles

Name the angle below in four ways.

The name can be the vertex of the angle: G.

Finally, the name can be a point on one side, the vertex, and a point on the other side of the angle:

AGC, CGA.

The name can be the number between the sides of the angle: 3

Use the Angle Addition Postulate to solve.

m 1 + m 2 = m ABC Angle Addition Postulate.

42 + m 2 = 88 Substitute 42 for m 1 and 88 for m ABC.

m 2 = 46 Subtract 42 from each side.

Suppose that m 1 = 42 and m ABC = 88. Find m 2.

Use the figure below for Exercises 4–6.

4. Name 2 two different ways.

5. Measure and classify 1, 2, and BAC.

6. Which postulate relates the measures of 1, 2, and BAC?

14

Angle Addition Postulate

Use the figure below for Exercises 1-3.

1. If XT = 12 and XZ = 21, then TZ = 7.

2. If XZ = 3x, XT = x + 3, and TZ = 13, find XZ.

3. Suppose that T is the midpoint of XZ. If XT = 2x + 11 and XZ = 5x + 8, find the value of x.

9

24

90°, right; 30°, acute; 120°, obtuse

DAB and BAD

All rights belong to their respective owners.Copyright Disclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, TEACHING, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, EDUCATIONAL or personal use tips the balance in favor of fair use.