Download Lesson 1: 1-1 Using Patterns and Inductive Reasoning

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Lesson 4: 1-4 Measuring Segments and Angles
OVERVIEW/ Expectations
In this lesson, students will find the lengths of segments and the measures of angles.
OBJECTIVES:
Cadets will be able to
1) Find the length and midpoint of a segment
2) Name and classify angles.
3) Identify the vertex and sides of an angle
4) Evaluate the measure of an angle and angle bisectors
5) Solve problems involving segments and/or angles
6) find the lengths of segments and measures of angles in order to solve real-world problems.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
How are properties of geometric figures related to their measurable attributes?
How can you use the Distance Formula to measure distances on a line or coordinate plane?
How can you measure segments and angles?
INDICATORS
2.1.4 construct and/or draw and/or validate properties of geometric figures using appropriate tools and
technology.
Properties and relationships include:
2.1.4.a line/segment relationships, including parallel, perpendicular, intersecting, bisecting,
midpoint, median, and altitude.
INSTRUCTIONAL SEQUENCE
Measuring Length
• formal definitions of length on a number line and congruence of line segments
• investigation of segment addition postulate
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STATE STANDARD
To which part of your curriculum does this lesson relate?
State standard 1: Make sense of problems and persevere in solving them.
State standard 2: Reason abstractly and quantitatively
State standard 4: Model with mathematics
State standard 5: Use appropriate tools strategically
State standard 6:Attend to precision
State standard 7:Look for and make use of structure
State standard 8:Look for and express regularity in repeated reasoning
VOCABULARY:
coordinate
midpoint
distance
length
segment bisector
bisect
construction
between
congruent segments
congruent segments
MATERIALS:

Geometry textbook

Calculators TI 84

Proctrator, Ruler, Patty or Waxed Paper, Compass, Graphic organizer, Aver
CRITERIA OF SUCCESS
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Identify key words in a word problem.
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Use drawings and/or symbols to represent an unknown value in a word problem.
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Use drawings/symbols to write an equation
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Recall some definitions or postulates involve.
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Determine the number of steps needed to solve the problem
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Solve the equation for the unknown value in the word problem.
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Check your answer
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WARM UP:
Check the Skills You’ll Need page 25 or see worksheet
INTRODUCTORY AND DEVELOPMENTAL ACTIVITIES:
Introduce the main concepts of the lesson using the new vocabulary found on page 25. Students should
begin taking notes.
GUIDED PRACTICE:
EXPLORE
Definition 1:
A ruler can be used to measure the distance between two points. A point corresponds to one and only one
number on a ruler. The number is called a coordinate. The following postulate summarizes this concept.
Definition 2:
The distance between any two points is the absolute value of the difference of the coordinates. If the
coordinates of points A and B are a and b, then the distance between A and B is |a – b| or |b – a|. The
distance between A and B is also called the length of AB , or AB.
A
a
B
b
AB = |a – b| or |b - a|
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Example 1: Finding the Length of a Segment
Find each length.
(QUESTIONING TECHNIQUE/WAIT TIME: What are the coordinates of A, B, C?
A. BC
BC = |1 – 3|=2
B. AC
AC = |–2 – 3|=5
Check: you may count along the number line to find lengths
Check It Out! Example 1
Find each length
a. XY
b. XZ
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Example 1 B: Complete example 1 page 26 in your textbook, then exercises 1-7 page29 in your textbook
Definition 3:
Congruent segments are segments that have the same length. In the diagram, PQ = RS, so you can write
PQ  RS . This is read as “segment PQ is congruent to segment RS.” Tick marks are used in a figure to show
congruent segments.
Definition 4:
In order for you to say that a point B is between two points A and C, all three points must lie on the
same line, and AB + BC = AC.
Example 3A: Using the Segment Addition Postulate
G is between F and H, FG = 6, and FH = 11. Find GH.
(QUESTIONING TECHNIQUE/WAIT TIME: Draw a picture. What is the relation between
segments)
Example 3 B: complete example 2 page 26 in your textbook then exercises 8-11 page 29 in your
textbook
(QUESTIONING TECHNIQUE/WAIT TIME:
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Example 3 C: M is between N and O. Find NO.
(QUESTIONING TECHNIQUE/WAIT TIME:
M is between N and O. Find NO.
(QUESTIONING TECHNIQUE/WAIT TIME:
Example 3 D: complete example 3 page 27 then 10-15 pages 29-30 in your textbook
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Definition 5:
The midpoint M of AB is the point that bisects, or divides, the segment into two congruent segments.
If M is the midpoint of AB, then AM = MB.
So if AB = 6, then AM = 3 and MB = 3.
Example 4: Recreation Application
The map shows the route for a race. You are at X, 6000 ft from the first checkpoint C. The second
checkpoint D is located at the midpoint between C and the end of the race Y. The total race is 3 miles.
How far apart are the 2 checkpoints?
XY  35280 ft  Convert race distance to feet.
 15,840 ft
The checkpoints are 4920 ft apart.
Example 5: Using Midpoints to Find Lengths
D is the midpoint of EF , ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF.
D is the midpoint of EF , ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF.
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D is the midpoint of EF, ED = 4x + 6, and DF = 7x – 9. Find ED, DF, and EF.
Check it out !Example 6: S is the midpoint of RT, RS = –2x, and ST = –3x – 2. Find RS, ST, and RT.
INDEPENDENT PRACTICE:
Workbook Practice 1-4
ASSESSMENT/EXIT CARD :
1) A line segment on a number line has endpoints with coordinates –12 and 20.
What is the length of the segment?
F8
G 12
H 20
J 32
2) segment is 32. Which of the following are the possible coordinates of the other
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endpoint?
I. – 44
II. – 20
III. 20
IV. 44
F I and II
G I and III
H II and IV
J III and IV
Look at the figure below:
If the length of PR is 300, what is the length of PQ ?
3) Points P, Q, and R are collinear, with Q between P and R.
PQ  5 x  30; QR  3x  80; PR  10 x  20 What is the length of PR ?
Lesson Quiz 1-4
CLOSURE: 3-2-1
Ask Students: Explain how two of the postulates in this lesson can help you measure segments and angles.
HOMEWORK:
#82-84 page 33 textbook
Geometry pp. 29-33
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Lesson 4: 1-4 Measuring Angles
OVERVIEW/ Expectations
In this lesson, students will find the lengths of segments and the measures of angles.
OBJECTIVES:
Cadets will be able to find Angle Measure
Name and classify angles.
Identify the vertex and sides of an angle
Compute the measure of an angle and angle bisectors
Solve problems involving angles
find the measures of angles in order to solve real-world problems.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
How are properties of geometric figures related to their measurable attributes?
How can you use the Distance Formula to measure distances on a line or coordinate plane?
How can you measure segments and angles?
INDICATORS
2.1.4 construct and/or draw and/or validate properties of geometric figures using appropriate tools and
Technology
2.1.4.c angles and angle relationships, including bisector, obtuse, acute, and right.
INSTRUCTIONAL SEQUENCE
Measuring Angles
• formal definitions of angle measure and angle congruence
• investigation of angle addition postulate
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STATE STANDARD
To which part of your curriculum does this lesson relate?
State standard 1: Make sense of problems and persevere in solving them.
State standard 2: Reason abstractly and quantitatively
State standard 4: Model with mathematics
State standard 5: Use appropriate tools strategically
State standard 6:Attend to precision
State standard 7:Look for and make use of structure
State standard 8:Look for and express regularity in repeated reasoning
VOCABULARY:
Angle , vertex, interior of an angle , exterior of an angle , measure, degree, acute angle , right angle, obtuse
angle, straight angle , congruent angles, angle bisector .
MATERIALS:

Geometry textbook

Calculators TI 84
WARM UP:
Worksheet
INTRODUCTORY AND DEVELOPMENTAL ACTIVITIES:
With the use of the elbow, students will be able to demonstrate an understanding of the concept of angles,
right angles, obtuse angles, acute angles, with 100% accuracy.
Introduce the main concepts of the lesson using the new vocabulary found on page 25. Students should
begin taking notes.
ENGAGEMENT
A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically.
Using a transit, a survey or can measure the angle formed by his or her location and two distant points.
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GUIDED PRACTICE:
EXPLORE
Definition 8: An angle is a figure formed by two rays, or sides, with a common endpoint called the
vertex (plural: vertices). You can name an angle several ways: by its vertex, by a point on each ray and
the vertex, or by a number.
Definition 9:
The set of all points between the sides of the angle is the interior of an angle. The exterior of an angle is
the set of all points outside the angle.
Note: You cannot name an angle just by its vertex if the point is the vertex of more than one angle. In
this case, you must use all three points to name the angle, and the middle point is always the vertex.
Example 1: Naming Angles
A surveyor recorded the angles formed by a transit (point A) and three distant points, B, C, and D.
Name three of the angles.
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Check It Out! Example 1
Write the different ways you can name the angles in the diagram.
Answer: RTQ, T, STR, 1, 2
Note: The measure of an angle is usually given
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360
in degrees. Since there are 360° in a circle, one degree is
of a circle. When you use a protractor to measure angles, you are applying the following postulate.
You can use the Protractor Postulate to help you classify angles by their measure. The measure of an angle is
the absolute value of the difference of the real numbers that the rays correspond with on a protractor.
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Example 2: Measuring and Classifying Angles
Find the measure of each angle. Then classify each as acute, right, or obtuse.
A. WXV
mWXV = 30° , WXV is acute.
B. ZXW
mZXW = |130° - 30°| = 100° , ZXW = is obtuse.
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Check It Out! Example 2
Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse.
a. BOA
b. DOB
c. EOC
mBOA = 40°, BOA is acute. mDOB = 125° , DOB is obtuse. mEOC = 105°, EOC is obtuse.
Definition 10: Congruent angles are angles that have the same measure. In the diagram, mABC =
mDEF, so you can write ABC  DEF. This is read as “angle ABC is congruent to angle DEF.” Arc
marks are used to show that the two angles are congruent.
The Angle Addition Postulate is very similar to the Segment Addition Postulate that you learned in the
previous lesson
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Example 3: Using the Angle Addition Postulate
mDEG = 115°, and mDEF = 48°. Find mFEG
Check It Out! Example 3
mXWZ = 121° and mXWY = 59°. Find mYWZ.
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EXERCISE 1: Complet example 6 page 29 in your textbook then #16-28, 74-78 pages 31-32 Textbook
Definition: An angle bisector is a ray that divides an angle into two congruent angles.
Example 4: Finding the Measure of an Angle
KM bisects JKL, mJKM = (4x + 6)°, and mMKL = (7x – 12)°. Find mJKM.
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Check It Out! Example 4a
Find the measure of each angle.
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Check It Out! Example 4b
INDEPENDENT PRACTICE:
mWYZ = (2x – 5)° and mXYW = (3x + 10)°. Find the value of x.
Workbook Practice 1-4 ( see classwork worksheet for more )
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ASSESSMENT/EXIT CARD :
Lesson Quiz 1-4
1)Which of the following is NOT one of the undefined terms of geometry?
A angle
B line
C point
D plane
2) Look at the figure below:
Angles ABC and ABD form a linear pair. What is the value of x?
A 12.5
B 13.75
C 25
D 50
3) Look at the figure below:
Angles ABD and ABC form a linear pair. What is the measure of angle ABD?
A 2.5°
B 8.75°
C 40°
D 65°
4) Look at the drawing below:
If
is the angle bisector of ∠ PQS , what is the measure, in degrees, of ∠ PQS ?
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5) Look at the drawing below:
If
is the angle bisector of ∠ PQS , what is the measure, in degrees, of∠ PQS ?
CLOSURE:3-2-1
Ask Students: Explain how two of the postulates in this lesson can help you measure segments and angles.
HOMEWORK:# 85-86 page 33 textbook
Geometry pp. 29-33
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