Bell Work 1) Sketch a ray with an initial point at B going through A 2) Sketch collinear points A, B, C that are not collinear to D 3) Find the next 3.

Bell Work

• 1) Sketch a ray with an initial point at B going through A

• 2) Sketch collinear points A, B, C that are not collinear to D

• 3) Find the next 3 in the sequence and describe the pattern: -3, 3, 9, 15,…

• 4)

Outcomes

I will be able to:• 1)Measure and Calculate segment lengths• 2) Calculate Distances using the Distance

Formula• 3) Use the Angle Postulates• 4)Classify Angles as acute, right, obtuse or

straight

Agenda

• 1) Bell Work• 2) Outcomes• 3) Building Blocks continued• 4) Finding Segment Length• 5) Using the Distance Formula

Geometric Proof

• Most things in Geometry must be proven.• Theorems – Rules that must be proven

• However, a few things exist that do not need to be proven.

• Postulates or Axioms – Rules that are accepted without proof.

Segment Measure

• Ruler Postulate – Points on a line can be matched one to one with real numbers. The real number that corresponds to a point is the Coordinate.

• The Distance between two points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B.

• AB is also called the length of AB.• *Note: Absolute values are always positive. This is because

absolute values represent the distance from 0. See Page 17

x₂x₁ AB =l x₂ - x₁l

Segment Measure• What is the measure of Segment MN?• What is the measure of Segment NP?• What is the measure of Segment MP?

• When three points are on one line, you can say that one point is between the other two.

• Segment Addition Postulate- If a point is between another two, we can find the distance of the larger segment by adding the two smaller segments.

Segment MeasureOR…• Segment Addition Postulate- • If B is between A and C, then AB + BC = AC.• And…• If AB + BC = AC , then B is between A and C.

Label segments AB, BC and AC.

A B C

Examples

• 1. Two friends leave their homes and walk in a straight line towards the other’s house. When they meet, one has walked 578 feet and the other has walked 498 feet. How far apart are the two homes?

Examples

• 2. A U-haul with a trailer has a total length of 35 feet. If the trailer is 29 feet, how long is the cab?

Examples

• 3. Suppose J is between H and K. Use the Segment Addition Postulate to solve for x. Then find the length of each segment.

• HJ = 2x + 4• JK = 3x + 3• KH = 22

Distance Formula

• Distance Formula –

• How can we relate the distance formula to measurement of segments and Pythagorean Theorem?

212

212 )()( yyxxd

If A (x1, y1) and B (x2, y2) are points in a coordinate plane, then the distance between A and B is…

Length

• ***Note: If we are finding the length of the segment between points, we denote it by:

• mAB not AB

• ***This is so we know we are talking about the measurement and not the segment itself.

Congruent Segments

• Congruent Segments – Segments that have the same length

Lengths are equal Segments are congruent• AB = CD AB CD“The length of AB is equal to the length of CD” “AB is congruent to CD”

Example

• Plot A(-1, 1) B(-4, 3), C(3, 2) and D(2, -1). Draw line segments AB, AC, and AD. Find the length of each segment. Are any of them congruent?

• Yes, AB is congruent to AD

Distance Formula• With a partner find distance between:

1)A and B 2)D and E 3)A and C 4)E and C 5)B and D

• ***Be ready to share your answers

Simplifying Radicals• When simplifying radicals, sometimes it helps to

use a Prime Factoring Tree.• Look for factors that are written twice and circle

them. When we take the square root, we write only one of the circled numbers. The numbers uncircled, without a pair, remain under the radical, multiplied back together.

• Square root of 72• 2 x 2 x 2 x 3 x 3 = 6√2 • Take 10 minutes to work

1.4 Angles and their Measures

• An Angle consists of two different rays that have the same initial point. The rays are the sides of the angle. The initial point is the vertex.

Measure of an Angle

• The measure of an ∠A is denoted as m∠A• The measure of an angle can be measure with

a protractor in degrees.• Write it as m∠BAC = 50°• Congruent Angles – Angles that have the same

measure• Note: Measures are equal, and angles are congruent• m∠BAC = m∠DEF ∠BAC ∠DEF• Say “is equal to” “is congruent to”

Classify Angles

• Acute 0° < m∠A <90° DRAW

• Right m∠A = 90°

• Obtuse 90° < m∠A < 180°

• Straight m∠A = 180°

Protractor Postulate

• Consider a point A on one side of line OB. The rays that form OA can be matched with the real numbers from 0 to 180.

• The measure of ∠AOB is equal to the absolute value of the difference between the real numbers for OA and OB.

Protractor Postulate

• Interior point – A point that lies between points that lie on each side of the angle.

• Exterior points – A point that does not lie on the angle or in its interior.

Interior

Exterior

Angle Addition Postulate

• Angle Addition Postulate – If P is in the interior of ∠RST, then m∠RSP + m∠PST = m∠RST

T

S

R

P

Examples

• 1) In the figure on your paper, m∠CDE = 62° and m∠EDF = 18° .

• Find the measure of ∠CDF.• 2) Plot the points and classify the angles with

your table partner.

• 3) Discuss and solve the third example on your paper with your table partner.

Adjacent Angles

• Adjacent Angles – Are two angles that share a common vertex and side but have no common interior points

• Complete the drawings described in examples 1 and 2 on your paper.

Exit Quiz• Plot the following points on the graph on the

exit quiz.• S at (-3, -1); T at (-4, 1); M at (1, 1); G at (1, -2);• E at (-1, 4); N at (3, -4); and E at (4, 3)• Using the distance formula and starting at S, use

the following distances to find the word hidden in the letters. The correct order of distances is as follows:

• You must show all distance formula work in the exit quiz.

74,50,13,3 ,40,29