1.4: Measuring Segments and Angles

Prentice Hall Geometry

Coordinate :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.

GEOMETRY LESSON 1-4GEOMETRY LESSON 1-4

Measuring Segments and AnglesMeasuring Segments and Angles

The Segment Addition PostulateThe 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

Coordinate Plane

Parts of Coordinate Plane

x-axis

y-axis

origin

Quadrant IQuadrant II

Quadrant IVQuadrant III

( +, + )( - , + )

( - , - )( + , - )

DistanceDistanceOn a number line

formula: d = | x2 – x1 |

On a coordinate plane

formula:

212

21 )()(

2yyxxd

Find the distance between T(5, 2) and R( -4. -1) to the nearest tenth.

Find the distance between T(5, 2) and R( -4. -1) to the nearest tenth.

AB has endpoints

A (1, -3) and B (-4, 4).

Find AB to the nearest tenth.

MidpointMidpoint

On a number line

formula: 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.

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

Homework

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