Download Prove It 2:1:16:Geometric Concepts: Angles III How do we create

Prove It
How do we create truth?
2:1:16:Geometric Concepts: Angles III
TITLE OF LESSON
Geometry Unit 1 Lesson 16 – Geometric Concepts: Angles III
Prove it! What’s on the outside? What’s on the inside? of Geometry
TIME ESTIMATE FOR THIS LESSON
One class period
ALIGNMENT WITH STANDARDS
California – Geometry
Introductory lesson necessary for:
4.0 Students prove basic theorems involving congruence and similarity.
5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts
of congruent triangles.
6.0 Students know and are able to use the triangle inequality theorem.
7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of
quadrilaterals, and the properties of circles.
MATERIALS
Pre-cut angles
Note: For this you can just cut triangles out of paper, but try to have them the approximate measure that you name
them. You’ll need one per student, no two measurements the same. Keep them between 5° and 180° or 270°. Also, be
sure to cut out your two demo angles—90° and 35°.
LESSON OBJECTIVES
• To apply understanding of angles to doing problems associated with angles
• To add angles and to recognize the different types of angles.
FOCUS AND MOTIVATE STUDENTS
1) Homework Check – Stamp/initial complete homework assignment. Pass back graded work and have students
place in the appropriate sections of their binders.
2) Binders – Pass back graded binders.
3) Agenda – Have students copy the agenda.
4) Present Homework – (5 minutes) Have individual students present one or two observations from their
homework. For instance ask one student to describe an object she has selected in her homework and explain
what type of angle describes that object. Have her draw the angle on the board and write the type of angle (right,
acute, obtuse, etc.) Ask her to estimate the measure of the angle. Have five or so students do this, and collect the
homework from all students, or stamp and sign it and have them leave it in their binders for when you collect
them.
ACTIVITIES – INDIVIDUAL AND GROUP
1.
Lecture: Angle Addition Postulate – (5 minutes): Introduce the angle addition postulate. Write the postulate below
on the board and have the students copy it into the postulates and theorems section of their binders.
→
If AC is interior to the angle BAD then the measure of angle BAD = the measure of angle BAC plus the
measure of angle CAD.
Draw the following diagram associated with this definition and have them copy the drawing:
B
C
D
A
1
© 2001 ESubjects Inc. All rights reserved.
Prove It
How do we create truth?
2:1:16:Geometric Concepts: Angles III
2.
Practice Problems – (1 minutes) Do a few practice problems to understand the angle addition postulate. Ask
questions such as: If the measure of angle BAD is 70° and the measure of angle CAD is 40° what is the measure or
angle BAC? (30°). If the measure of angle BAD is 80° and the measure of angle BAC is 35° what is the measure of
angle CAD (45°). If the measure of angle BAC is 30° and the measure of angle CAD is 35° what is the measure of
angle BAD? (65°). What types of angles are these? (These are adjacent angles.)
3.
Pair Work – (4 minutes) Next, have two demo angles to use (one 90° and one 35°), and explain that they’re going to
have 3 minutes to quickly do this next activity. Showing them with your demo angles, explain that they will be
pairing up with each other student. In each pairing, one angle will be smaller than the other. Both students will write
the other student’s name, then decide whose angle is smaller. The smaller angle will become one of the two adjacent
angles in the larger angle. Demonstrating with the 90° and 35° angles, place the 35° angle inside the 90° angle.
Explain that their jobs will be to determine the measurement of the other, missing adjacent angle in their set. In the
case of the demo, the missing angle is 55°. Then, have students take out a blank piece of paper and number it with
one fewer than the number of students in the class. Give each student one of the pre-made angles and tell them they
have 3 minutes to pair up with another student in the class and complete the assignment.
4.
Review – (5 minutes) Review the types of angles that you discussed during the previous angle lesson (acute, right,
obtuse, straight, reflex, adjacent, vertical, congruent, complimentary, and supplementary). Ask for student volunteers
to give the definition of each of these types of angles. They may read them from their notes.
5.
Acute Angles – (5 minutes) Problems associated with acute angles. Acute angles are angles less than 90°. Ask
questions such as: What is the largest acute angle? (Answer: 89°) Refer to the diagram in number 1 of today’s
activities. If angle BAD is 90° what type of angle is angle BAC? (Answer: Acute). Why must this be true? (Answer:
Since an acute angle is less than 90° and the measure of BAD (90°) is the sum of the measures of BAC and CAD,
then BAC must be less than 90° and is therefore acute.) What about angle CAD? (Answer: It is also acute for the
same reason.)
6.
Right Angles – (5 minutes): Problems associated with right angles. A right angle is an angle that measures 90°.
B
C
A
D
Draw this diagram on the board. Point out that the angle for a right angle is drawn with a rectangle rather than
semicircular angle designator. Any time you see this designation, you can assume that the angle is 90°. Ask: If angle
BAC measures 40° and angle CAD measure 45°is angle BAD a right angle? (No they do not add up to 90° but
instead add up to 85° and therefore BAD is not a right angle.) If BAC is 40°, what would the measurement of CAD
have to be for BAD to be a right angle assuming that? (50°). Have some students form the original set of angles
forming BAC, CAD and BAD adding up to (approximately) 85°. What adjustments could be made to create a right
angle BAD? (You can increase either BAC or CAD by 5°) Is there any other way you could do it? (You could
increase both BAC and CAD by 2.5°) Have the students demonstrate each of these questions, forming angles by
standing in the appropriate configuration. Have one student draw the appropriate diagram on the board and label the
size of each angle.
7.
Obtuse Angles – (5 minutes): Problems associated with obtuse angles. An obtuse angle is an angle that measures
more than 90° but less than 180°. Draw the diagram below on the board and ask which angle is the obtuse angle?
(BAD) How do we know that it is obtuse? (CAD is 90° since it has the rectangular angle designator and BAC
appears to be acute. We need BAD to be greater than 90° but we also need for it to be less than 180° implying that
BAC is acute since CAD is a right angle and is 90°) Do we know this for sure? (Not really since we are deducing this
from the appearance of the diagram and a diagram is not the geometrical object but is only a representation of the
2
© 2001 ESubjects Inc. All rights reserved.
Prove It
How do we create truth?
2:1:16:Geometric Concepts: Angles III
object. If CAD is 20° can BAC be 60° and still imply that BAD is obtuse? (No BAC must be greater than 70° for
BAD to still be obtuse.)
C
B
A
D
8.
Straight Angles – (5 minutes): Problems associated with straight angles. A straight angle is an angle that measures
180°. Ask a student to draw a straight angle on the board. Have three students form a straight angle by standing in the
room. Ask the question: If we divide a straight angle (a line) into two adjacent angles and one of the two adjacent
angles is an acute angle what type of angle is the other angle? (Obtuse). If we divide a straight angle into two
adjacent angles and one of the two adjacent angles is a right angle what type of angle is the other angle? (Right
angle.) Have a student draw this on the board. Be sure to have them designate the right angles with rectangular angle
designators. What types of objects can be drawn using straight angles and right angles?
9.
Reflex Angles – (5 minutes) Problems associated with reflex angles. An angle that measures more than 180° but less
than 360°is a reflex angle. Have a student draw a reflex angle on the board. It should look similar to the angle below.
Note that it is the outside angle that is the reflex angle.
If an angle is 60° and we add an adjacent angle to it that is 130°, what is the measure of the two angles formed by
adding these two together? (190°) Is this a reflex angle? (Yes because it is greater than 180° and less than 360°) If
you add two obtuse angles together what do you get? (A reflex angle)
10. Homework – (5 minutes) Explain today’s homework and have students copy the necessary information from the
board.
HOMEWORK
Add the following angles together and describe the result by recording its measurement (when possible) and the type
of angle. For instance a 45° added to a 45° is a 90° angle, which is a right angle. A 10° angle added to a 10°angle
produces an acute angle of 20°. Have the students give the type of angle, the measure of the angle (if appropriate) and
create an approximate drawing.
10° added to 20°
10° added to 40°
40° added to 50°
60° added to 30°
70° added to 30°
70° added to 70°
90° added to 90°
100° added to 80°
130° added to 50°
130° added to 70°
3° added to 20°
Right added to acute
Right added to obtuse
Obtuse added to obtuse
Straight added to right
Straight added to obtuse
Straight added to acute
Acute added to acute (There’s
leeway here for the answer to be
acute or not.)
GROUP ROLES
None
DOCUMENTATION FOR PORTFOLIO
None
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© 2001 ESubjects Inc. All rights reserved.