Transcript Lesson 6-1 - Erica Whalen

Five-Minute Check (over Chapter 5)
CCSS
Then/Now
New Vocabulary
Theorem 6.1: Polygon Interior Angles Sum
Example 1: Find the Interior Angles Sum of a Polygon
Example 2: Real-World Example: Interior Angle Measure of
Regular Polygon
Example 3: Find Number of Sides Given Interior Angle
Measure
Theorem 6.2: Polygon Exterior Angles Sum
Example 4: Find Exterior Angle Measures of a Polygon
Over Chapter 5
State whether this sentence is always, sometimes,
or never true. The three altitudes of a triangle
intersect at a point inside the triangle.
A. always
B. sometimes
C. never
Over Chapter 5
Find n and list the sides of ΔPQR in order from
shortest to longest if mP = 12n – 15,
mQ = 7n + 26, and mR = 8n – 47.
A. n = 8;
B. n = 8;
C. n = 6;
D. n = 6;
Over Chapter 5
State the assumption you would make to start an
indirect proof of the statement.
If –2x ≥ 18, then x ≤ –9.
A. x is positive.
B. –2x < 18
C. x > –9
D. x < 9
Over Chapter 5
Find the range for the measure of the third side of
a triangle given that the measures of two sides
are 43 and 29.
A. n > 14
B. 14 < n < 72
C. 29 < n < 43
D. n > 0
Over Chapter 5
Write an inequality relating mABD and mCBD.
A. mABD < mCBD
B. mABD ≤ mCBD
C. mABD > mCBD
D. mABD = mCBD
Over Chapter 5
Write an equation that you can use to find the
measures of the angles of the triangle.
A. x – 5 + 3x = 180
B. x – 5 + 3x + 111 = 180
C. x – 5 + 3x = 69
D. x – 5 + 3x = 111
Content Standards
G.MG.1 Use geometric shapes, their
measures, and their properties to describe
objects (e.g., modeling a tree trunk or a
human torso as a cylinder).
Mathematical Practices
4 Model with mathematics.
3 Construct viable arguments and critique
the reasoning of others.
You named and classified polygons.
• Find and use the sum of the measures of the
interior angles of a polygon.
• Find and use the sum of the measures of the
exterior angles of a polygon.
• diagonal
Find the Interior Angles Sum of a Polygon
A. Find the sum of the measures of the interior
angles of a convex nonagon.
A nonagon has nine sides. Use the Polygon Interior
Angles Sum Theorem to find the sum of its interior
angle measures.
(n – 2) ● 180 = (9 – 2) ● 180
= 7 ● 180 or 1260
n=9
Simplify.
Answer: The sum of the measures is 1260.
Find the Interior Angles Sum of a Polygon
B. Find the measure of each interior angle of
parallelogram RSTU.
Step 1 Find x.
Since
the sum of the measures of the interior
angles is
Write an equation to
express the sum of the measures of the interior angles
of the polygon.
Find the Interior Angles Sum of a Polygon
Sum of measures
of interior angles
Substitution
Combine like
terms.
Subtract 8 from
each side.
Divide each side
by 32.
Find the Interior Angles Sum of a Polygon
Step 2 Use the value of x to find the measure of
each angle.
mR = 5x
= 5(11) or 55
mS = 11x + 4
= 11(11) + 4 or 125
mT = 5x
= 5(11) or 55
mU = 11x + 4
= 11(11) + 4 or 125
Answer: mR = 55, mS = 125, mT = 55,
mU = 125
A. Find the sum of the measures of the interior
angles of a convex octagon.
A. 900
B. 1080
C. 1260
D. 1440
B. Find the value of x.
A. x = 7.8
B. x = 22.2
C. x = 15
D. x = 10
Interior Angle Measure of Regular
Polygon
ARCHITECTURE A mall is designed so that five
walkways meet at a food court that is in the shape
of a regular pentagon. Find the measure of one of
the interior angles of the pentagon.
Interior Angle Measure of Regular
Polygon
Understand
Look at the diagram of the situation.
The measure of the angle of a corner in
between two walkways is the interior
angle of a regular pentagon.
Plan
Use the Polygon Interior Angles Sum
Theorem to find the sum of the
measures of the angles. Since the
angles of a regular polygon are
congruent, divide this sum by the
number of angles to find the measure
of each interior angle.
Interior Angle Measure of Regular
Polygon
Solve
Find the sum of the interior
angle measures.
(n – 2) ● 180 = (5 – 2) ● 180
n=5
= 3 ● 180 or 540 Simplify.
Find the measure of one interior
angle.
Substitution
Divide.
Interior Angle Measure of Regular
Polygon
Answer: The measure of one of the interior angles of
the food court is 108.
Check
To verify that this measure is correct,
use a ruler and a protractor to draw a
regular pentagon using 108 as the
measure of each interior angle. The
last side drawn should connect with the
beginning point of the first segment
drawn.
A pottery mold makes bowls that are in the shape
of a regular heptagon. Find the measure of one of
the interior angles of the bowl.
A. 130°
B. 128.57°
C. 140°
D. 125.5°
Find Number of Sides Given Interior Angle
Measure
The measure of an interior angle of a regular
polygon is 150. Find the number of sides in the
polygon.
Use the Interior Angle Sum Theorem to write an
equation to solve for n, the number of sides.
S = 180(n – 2)
(150)n = 180(n – 2)
150n = 180n – 360
0 = 30n – 360
Interior Angle Sum
Theorem
S = 150n
Distributive Property
Subtract 150n from each
side.
Find Number of Sides Given Interior Angle
Measure
360 = 30n
12 = n
Add 360 to each side.
Divide each side by 30.
Answer: The polygon has 12 sides.
The measure of an interior angle of a regular
polygon is 144. Find the number of sides in the
polygon.
A. 12
B. 9
C. 11
D. 10
Find Exterior Angle Measures of a Polygon
A. Find the value of x in the diagram.
Find Exterior Angle Measures of a Polygon
Use the Polygon Exterior Angles Sum Theorem to
write an equation. Then solve for x.
5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) +
(5x + 5) = 360
(5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) + 3 +
(–12) + 3 + 5] = 360
31x – 12 = 360
31x = 372
x = 12
Answer:
x = 12
Find Exterior Angle Measures of a Polygon
B. Find the measure of each exterior angle of a
regular decagon.
A regular decagon has 10 congruent sides and
10 congruent angles. The exterior angles are also
congruent, since angles supplementary to congruent
angles are congruent. Let n = the measure of each
exterior angle and write and solve an equation.
10n = 360
Polygon Exterior Angle
Sum Theorem
n = 36
Divide each side by 10.
Answer:
The measure of each exterior angle of a
regular decagon is 36.
A. Find the value of x in
the diagram.
A. 10
B. 12
C. 14
D. 15
B. Find the measure of each exterior angle of a
regular pentagon.
A. 72
B. 60
C. 45
D. 90