Download Angles in Polygons

8-4
Angles in Polygons
Connection: Geometric Drawings
Essential question: How can you draw shapes that satisfy given conditions?
TE ACH
Standards for
Mathematical Content
1
CC.7.G.2 Draw (freehand, with ruler and
protractor, and with technology) geometric shapes
with given conditions. Focus on constructing
triangles from three measures of angles or sides,
noticing when the conditions determine a unique
triangle, more than one triangle, or no triangle.
Materials
• Ruler
• Protractor
Questioning Strategies
• What is a “unique” triangle? It is a triangle of
a specific shape and size. A triangle that has a
different position, but the same shape and size,
is not a different triangle. Two triangles are
considered the same unique triangle if they have
the same shape and size, regardless of their
positions or orientations.
Prerequisites
Measuring line segments and angles
Properties of triangles
Math Background
Students will construct triangles based on different
conditions and using different tools. Some sets of
conditions will determine a unique triangle. In this
lesson, students will find that the length of two sides
and the measure of the angle between the two sides
describes a unique triangle. Also, knowing the three
sides of a triangle will result in a unique triangle, as
long as the three sides connect to make a triangle.
Students will find that there are no unique triangles
for knowing the lengths of the two sides and the
measure of an angle not included between the two
sides. This lesson will have students explore these
sets of conditions and other sets (SAS and AAA) in
the practice set.
• How do you know that the triangle in this Explore
is always unique? No matter the position or
orientation that you begin constructing it in, you
will always end up with a triangle with the same
shape and size.
2
EXPLORE
Materials
• Ruler
Connect to previous learning by first reviewing
the definition and properties of a triangle. Then
review with students how to draw and measure
line segments and angles. Explain to students that
they will learn to construct triangles and learn what
information they need in order to do so.
• Protractor
• Compass
Questioning Strategies
• How do you know that the given information
does not describe one unique triangle? When you
use the compass to find where the second and
third segments will meet in step 3, you find there
are two possibilities. Both possibilities satisfy the
given information.
325
Lesson 4
© Houghton Mifflin Harcourt Publishing Company
Differentiated Instruction
To help students understand that the uniqueness
of a triangle does not depend on its position or
orientation, have students cut out a triangle shape
and move it around on their desktop. Point out that
no matter where the triangle cut-out is located or
arranged, it is still the same shape and size.
IN T RO DUC E
Chapter 8 EXPLORE
Name
Class
8-4
Date
Notes
Angles in Polygons
Connection: Geometric Drawings
Essential question: How can you draw shapes that satisfy given conditions?
CC.7.G.2
1
EXPLORE
video tutor
Two Angles and Their Included Side
Draw each triangle with the given conditions.
Triangle 1
Angles: 30° and 80°
Included side: 2 inches
Triangle 2
Angles: 55° and 50°
Included side: 1 inch
Use a ruler and a protractor to draw each triangle with the given angles
and included side length.
A
Draw Triangle 1.
Step 1: Use a ruler to draw a line that is 2 inches long. This will be the
included side.
Step 2: Place the center of the protractor on
the left end of the 2-in. line. Then
make a 30°-angle mark.
Step 3: Draw a line connecting the left side of
the 2-in. line and the 30°-angle mark.
This will be the 30° angle.
© Houghton Mifflin Harcourt Publishing Company
30˚
2 in.
Step 4: Repeat Step 2 on the right side of the
triangle to construct the 80° angle.
B
Step 5: The side of the 80° angle and the side
of the 30° angle will intersect. This is
Triangle 1 with angles of 30° and 80°
and an included side of 2 inches.
Draw Triangle 2.
75˚
50˚
1 in.
55˚
325
Chapter 8
Lesson 4
7_MNLESE876535_C08L04.indd 325
3/14/12 7:09:10 PM
REFLECT
1a.
Conjecture When you are given two angle measures and the length of the included side,
do you get a unique triangle?
Sample answer: Yes, the two angles and the length of the included side
CC.7.G.2
2
EXPLORE
Two Sides and a Non-Included Angle
Use a ruler, protractor, and compass to construct a triangle with given
1 inches and a non-included angle of 45°.
lengths of 2 inches and 1__
2
A non-included angle is the angle not between the two given sides.
Step 1: Use a ruler to draw a straight line. This will be part of the triangle, but
does not have to measure a specific length.
Step 2: As in
1
, place the center of the protractor on the left end of the
2 in.
45˚
line. Then make a mark at the correct 45-degree point. Use your ruler
to make this side of the triangle 2 inches long.
Step 3: Make your compass the width of 1__12 inches. Place the sharp point
on the end of the 2-inch side that you just drew in Step 2. Rotate
the compass until it intersects, or meets, the bottom line twice
(see figure).
2 in.
45˚
Step 4: The point where the compass crosses the bottom line shows
where a line can be drawn that is exactly 1__12 inches long. Use your ruler to
verify the length and draw the line.
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
determine at what point the sides meet. The triangle is unique.
REFLECT
2a.
Is there another triangle that can be drawn with the given conditions?
Yes, you could connect the line to the other point where the compass
crossed the line.
2b. When you are given two side lengths and the measure of a non-included
angle, do you get a unique triangle? Explain.
Sample answer: No, the length of the second side could meet the bottom
of the triangle in two places, so you could get two different possible triangles.
Chapter 8
7_MNLESE876535_C08L04.indd 326
Chapter 8 326
Lesson 4
3/13/12 10:59:57 AM
326
Lesson 4
CLOS E
Highlighting the
Standards
Essential Question
How can you draw shapes that satisfy
given conditions?
Explores 2 and 3 provide an opportunity to
address Standard 5 (Use appropriate tools
strategically.). Students use geometric tools
and software in order to make conjectures
about whether or not they create a unique
triangle. Students must work precisely because
the activities involve drawing angles with
specific measures.
3
Possible answer: You can use a ruler, protractor,
compass, and software to help you draw shapes.
Some conditions result in unique triangles, while
other conditions create more than one triangle, or
no triangle at all.
Summarize
Have students complete a chart like the one shown
here to summarize what they have learned about
triangles in this lesson. The first row is started. Have
students add as many rows as needed.
EXPLORE
Questioning Strategies
• Do three given lengths always create a triangle?
Conditions
No, the three sides may not always connect.
Angle – Side - Angle
• If three given sides make a triangle, is the triangle
always unique? Yes
Teaching Strategies
If geometry software is not available, have students
use a ruler to draw each side separately. Have
students cut out and label each segment, then
manipulate the segments to form a triangle.
PR ACTICE
Where skills are
taught
327
Where skills are
practiced
1 EXPLORE
EX. 1
2 EXPLORE
EXS. 2–4
3 EXPLORE
EXS. 2–4
© Houghton Mifflin Harcourt Publishing Company
Chapter 8 Unique Triangle
(Y or N)
Lesson 4
Notes
CC.7.G.2
3
EXPLORE
Drawing Three Sides
Use geometry software to draw a triangle whose sides
have the following lengths: 2 units, 3 units, and 4 units.
Step 1: Draw three line segments of 2, 3, and 4 units
of length.
E
F
c=4
C
D
b=3
A
B
a=2
___
Step 2: Let AB be the base of the triangle. Place
endpoint C on top of endpoint B and
endpoint E on top of endpoint A. These will
become two of the vertices of the triangle.
F
D
c=4
E
B
A a=2 C
Step 3: Using the endpoints C and E as fixed
vertices, rotate endpoints F and D to see
if they will meet in a single point.
D
c=4
The line segments of 2, 3, and 4 units do /do not
form a triangle.
b=3
F
b=3
E
A a=2 C
B
© Houghton Mifflin Harcourt Publishing Company
TRY THIS!
3a.
Repeat Steps 2 and 3, but start with a different base length. Do the line segments make
the exact same triangle as the original?
Yes, the line segments make the same size and shape triangle as the original.
3b. Use geometry software to draw a triangle with given sides of 2, 3, and 6 units. Do these
line segments form a triangle?
No, these line segments do not connect to form a triangle.
REFLECT
3c.
Conjecture When you are given three side lengths that form a triangle,
do you get a unique triangle or more than one triangle?
You get a unique triangle. The triangle can only be formed one way with that
particular size and shape or no triangle can be formed at all.
327
Chapter 8
Lesson 4
7_MNLESE876535_C08L04.indd 327
3/13/12 11:00:11 AM
practice
The given conditions will make a unique triangle ; check students’ work.
2. Use geometry software to determine if the given side lengths can be used to
form one unique triangle, more than one triangle, or no triangle.
Construction
1
Construction
2
Construction
3
Construction
4
Side 1 (units)
5
8
20
1
Side 2 (units)
5
9
20
1
Side 3 (units)
Triangle
Formation?
10
10
20
7
No triangle
One
One
No triangle
3. On a separate piece of paper, draw a triangle that has degrees of 30°, 60°, and
90°. Measure the side lengths. Check student’s work.
a. Can you draw another triangle with the same angles but different side lengths?
Sample answer: Yes, I can draw several triangles of different sizes
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
1. On a separate piece of paper, draw a triangle that has side lengths of 3 cm
and 6 cm with an included angle of 120°. Determine if the given information
makes a unique triangle, more than one triangle, or no triangle.
that have 30°, 60°, and 90° angles.
b. If you are given 3 angles in one triangle, will the triangle be unique?
No, several differently sized triangles can be drawn with the
same angles.
4. Draw a freehand sketch of a triangle with three angles
that have the same measure. Explain how you made
your drawing.
Sample answer: Since a triangle has 180°,
I drew angles that were about 60° then
connected them with lines that were
about the same length.
Chapter 8
7_MNLESE876535_C08L04.indd 328
Chapter 8 328
Lesson 4
3/13/12 11:00:22 AM
328
Lesson 4
Add i t i o na l P r ac ti c e
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
Answers
Additional Practice
1. Check students’ drawings; one triangle
2.Check students’ drawings; more than
one triangle
Problem Solving
1. Check students’ drawings; no triangle
2. Check students’ drawings; one triangle
© Houghton Mifflin Harcourt Publishing Company
Chapter 8 329
Practice and Problem Solving
Name
Class
Notes
8-4
Date
Name ________________________________________ Date __________________ Class __________________
GeometricPractice
Figures
Additional
Chapter
Practice B: Angles in Polygons
Use a ruler, protractor, and compass to construct each figure.
1. Draw a triangle that has side lengths of 3 cm and 4 cm with
an included angle of 90°. Determine if the given information
makes a unique triangle, more than one triangle, or no triangle.
_________________________________________________________________________________________
© Houghton Mifflin Harcourt Publishing Company
2. Draw a triangle that has angles that measure 45°, 45°,
and 90°. Determine if the given information makes a unique
triangle, more than one triangle, or no triangle.
_________________________________________________________________________________________
Chapter 8
329
Practice and Problem Solving
43
Holt McDougal Mathematics
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7_MNLESE876535_C08L04PP.indd 329
Grade_7-B3.indd 329Name
6/25/12 5:50:19 PM
________________________________________ Date __________________ Class __________________6/25/12
Geometric
Figures
Problem
Solving
5:37:11 PM
Name ________________________________________ Date __________________ Class _________________
Geometric Figures
Chapter
Problem Solving: Angles in Polygons
Practice B: Congruent Figures
Describe each translation.
1. Draw a triangle that has sides that measure 5 cm, 5 cm,
and 11 cm. Determine if the given information makes a
unique triangle, more than one triangle, or no triangle.
1.
2.
FPO
FPO
_________________________________________________________________________________________
2. Draw a triangle that has two angles that measure 30° and 40°
with an included side length of 5 cm. Determine if the given
information makes a unique triangle, more than one triangle,
or no triangle.
________________________________________
_______________________________________
________________________________________
_______________________________________
________________________________________
_______________________________________
Graph each transformation.
3. Translate ΔABC 7 units to the right and 1 unit down.
4. Reflect ΔABC 7 across the x-axis.
_________________________________________________________________________________________
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Use a ruler, protractor, and compass to construct each figure.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Chapter 8
Grade_7-B3.indd 330
7_MNLESE876535_C08L04PP.indd 330
Chapter 8 43
330
FPO
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Practice
Problem
Solving
Holtand
McDougal
Mathematics
Holt McDougal Mathemati
6/25/12 5:38:09 PM
6/25/12 5:50:20 PM
330
Practice and Problem Solving