Download lesson 1.5

Susan Chaffee
Geometry Chapter 1
Spring 2009
Lesson 3 – Sections 1.5-1.6
Unit: Chapter 1 Tools of Geometry
Lesson: Section 1.5
Section 1.6
Measuring Segments
Measuring Angles
Lesson Goals: Students will understand how to find segment and angle measures.
Lesson Objectives:
Materials and/or Special Notes:
 Angle Names worksheet
 (optional) Video
Introduction: Congruent – Two figures are both equal in size and similar in shape
Motivation: Recall the meaning of the word geometry, comes from the Greek geo: earth and metrein:
to measure.
Geometry originated as the science of measuring land. One of the wonders of the world, the
pyramids, depended on geometry to assure the sides of the pyramid all meet at the same point.
Babylonian tablets dating back to 2500 B.C. demonstrate their understanding of what later
became known as Pythagorean theorem (ask if anyone remembers what it’s used for).
Section 1-5 Measuring Segments
Today we look at measuring segments and angles. What seems like simple and intuitive
properties, measuring, are included in Euclid’s Postulates. Ask if students remember what a
postulate is. How is it different than a theorem?
Recall a segment. Do you think a line or ray has length?
TEACHER and STUDENT ACTIVITIES :
The measurement of a segment is defined by the following postulate (fancy words for what we already
know to be true).
SB-1: Show line with arrows.
SB-2: Ruler Postulate: The points on a line can be assigned the real numbers, called coordinates, such
that the distance is defined as follows: If A has coordinate a and B has coordinate b on a line. Then the
distance between A and B is the number given by: |𝑎 − 𝑏|
|9-4|=5
SB-3: Def. Length of a segment – Defined by the ruler postulate as the absolute value of the distance
between coordinates.
Notation: 𝐴𝐵 is the notation to represent of the length of segment ̅̅̅̅
𝐴𝐵
̅̅̅̅ with coordinates a and b has length |𝑎 − 𝑏|
Example: A segment 𝐴𝐵
SB-4:
GD: Congruent – Two figures are both equal in size and similar in shape.
̅̅̅̅ ≅ 𝐶𝐷
̅̅̅̅ implies that the segments have the same length, i.e. AB=CD
Notation: 𝐴𝐵
Susan Chaffee
Geometry Chapter 1
Spring 2009
Example: Geometric shapes can be congruent, such as triangles, segments, angles, etc…
NonExample: The equal sign “=” not used to compare geometric figures.
Important: AB represents a real number, ̅̅̅̅
𝐴𝐵 is a geometric shape, a segment
4 equivalent statements of congruence, show on board.
GD: Postulate 1-6: Segment Addition Postulate. If 3 points A, B, and C are collinear and B is between A
and C then AB+BC=AC
Example 1: Find XY and YX.
Example 2: XY=4 and YZ=5, what is XZ?
Example 3: AB=2 and AC=10, what is BC?
Example 4: Line AB=25, Find x
2x-6
x+7
GD: Midpoint – A midpoint divides a segment into two congruent segments.
Example: If M is midpoint of ̅̅̅̅
𝐴𝐵, then ̅̅̅̅̅
𝐴𝑀 ≅ ̅̅̅̅̅
𝑀𝐵
NonExample: A ray or line can not have a midpoint.
Example 5:
In section 1-7 we will look at distance in 2 dimensions.
Brain Break or video http://studio4learning.tv/sub_subject.php?pl=182&id=630
Section 1-6 Measuring Angles
Angle – formed by two rays with same endpoint
⃗⃗⃗⃗⃗ 𝑎𝑛𝑑 𝐵𝐶
⃗⃗⃗⃗⃗ .
Notation: ∠𝐴𝐵𝐶 is an angle with vertex B and rays, 𝐵𝐴
Example: 4 ways to name the angle as shown.
Postulate 1-7: Protractor Postulate. Let OA and OB be opposite rays in a plane. All rays with endpoint
⃡⃗⃗⃗⃗ are assigned unique real numbers between 0 and 180. Assign 𝑂𝐴
⃗⃗⃗⃗⃗ to
O, lying on one side of 𝐴𝐵
coordinate 0 and ⃗⃗⃗⃗⃗
𝑂𝐵 coordinate to 180.
⃗⃗⃗⃗⃗
If a ray 𝑂𝐶 has coordinate c and a second ray ⃗⃗⃗⃗⃗⃗
𝑂𝐷 has coordinate d, then 𝑚∠𝐶𝑂𝐷 = |𝑐 − 𝑑|
Example : 𝑚∠𝐶𝑂𝐷 = 80. Must put “m” in front of angle.
Demonstrate how to measure with protractor – line up origin of protractor at end of the ray.
When comparing angles, we call them congruent if they have the same measure (just as with segments)
For example : 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐷𝐸𝐹 same measure, implies ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐸𝐹. Tik marks are also used.
Worksheet.
Susan Chaffee
Classify angles
Angle Measure
Geometry Chapter 1
Angle name
Between 0 and 90
Acute
90
Right
Between 90 and 180
Obtuse
180
Straight
Spring 2009
Example
Worksheet: Use protractor to measure and classify angles.
Given a right angle, find the missing:
Only makes sense if ray is between the other 2 rays.
⃗⃗⃗⃗⃗ is in between 𝑂𝐴
⃗⃗⃗⃗⃗ and 𝑂𝐶
⃗⃗⃗⃗⃗ , then 𝑚∠𝐴𝑂𝐶 = 𝑚∠𝐴𝑂𝐵 +
Postulate 1-8: Angle Addition Postulate . If 𝑂𝐵
𝑚∠𝐵𝑂𝐶
Example: Suppose m∠ABC=88 and 𝑚∠1=42, what is measure of angle 2?
Angle Pairs – have special names.
2 angles with special relationships to each other.
NAME
Description
Congruent
2 angles with same measure
Vertical angles
2 angles whose sides are
opposite rays
Adjacent
2 coplanar angles with common
side, vertex and no common
interior points
Complementary
2 angles whose measurements
sum to 90
Supplementary
2 angles whose measures sum to
180
We will revisit these names often, so know them well!
Example
In the following diagram identify
1. vertical pairs, 2. Supplementary pairs 3. complementary pairs 4. Adjacent pairs 5. Congruent
Learning Log: Measuring Angles – Write a step by step description of how you use the protractor to
measure an angle. Draw a diagram and label with points, refer to this in your description.
Susan Chaffee
Geometry Chapter 1
Spring 2009
Lab exercise: If time allows, we will use the computer lab for experimenting with angle constructions,
bisectors, and perpendicular bisectors. At end of lesson, students will write learning log describing what
they have observed.
Lesson Closure: We learned about congruence, a word used in geometry to describe equivalence
between angles and segments. Two angles are not equal, they are congruent. In our next lesson we will
learn how to construct segments and angles using compass and straightedge!
Assignments: p.33,#1-15(odd),16-19,29-32, p. 40,#1,9-13,16-21,24,25,27,28,31,42
Assessment:
 Formative:
 Summative:
Lesson Accommodations: