Download Polygons 7.1 Triangle Application Theorems

Chapter 7 - Polygons
7.1 Triangle Application Theorems:
Theorem 50: The ______ of the measures of the ______ angles of a
______________ is 180°.
m ∠ A + m ∠ B + m ∠ C = 180°
Proof: According to the Parallel Postulate, there exists exactly one line through
point A parallel to line BC, in the figure below.
A
1
2
3
B
C
How can we prove that angles 1 + 2 + 3 = 180° as well as the sum of the angles of
the triangle equal 180° ?
Exterior angles
Definition # 49: An exterior angle of a polygon is an angle that is ________
to and ______________ to an interior angle of the polygon.
2
1
Theorem 51: The measure of an exterior angle of a triangle is ________ to
the sum of the measures of the _________ interior angles.
In the two triangles above label the remote angles.
You should see that Theorem 50 and the Definition of exterior angles are the
keys to the proof of Theorem 51.
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Theorem 52: The measure of an exterior angle of a triangle is equal to the
______ of the measures of the __________ interior angles.
Theorem 52: A segment joining the midpoints of two sides of a triangle is
_______________ to the third side, and its ____________ is _______ the
length of the third side. ( Midline Theorem)
G
Given: H and M are a midpoints
H
M
J
K
Prove: a) Prove HM || JK
b) HM =
1
( JK)
2
To begin, extend line HM to a point P such that MP ≅ HM. Now we have established
P so draw a line between P and K.
Statements
Reasons
1) GM ≅ _____
1)
2) ∠ GMH ≅ ∠ _____
2)
3) Δ GMH ≅ Δ _______
3)
4) ∠ G ≅ ∠ _______
4)
5) PK || _____
5)
6) GH ≅ PK
6)
7) GH ≅ ______
7)
8) PK ≅ _____
8)
9)PK AND HJ are || and ≅ so
PKJH is a ____________.
Therefore, _____ || _____
9)
10) HP =JK
10)
11) MP = HM, HM =
12) HM =
1
( ____)
2
1
(HP)
2
11)
12) Substitution
3
Pg. 301 # 20
Prove that if the midpoints of a quadrilateral are joined in order, the figure
formed is a parallelogram.
D Put in midpoints Q, P, S, R in order starting with AB
A
going in a clockwise direction. Then draw in BD.
B
#3
C
C
D
A
Given: ∠ CAB = 80°, ∠ CBA = 60°, AE and BD are altitudes
Find : m ∠ C _____ and m ∠ AFB_____
F
E
B
#5
The measures of the three angles of a triangle are in the ratio of 4 : 5 : 6.
Find the measure of each.
#6
Given: ∠ ORS = ( 4x + 6)°
∠ P = ( x + 24)°
∠ O = ( 2x + 4)°
Find: m ∠ O
O
S
R
P
4
7.2 Two Proof - Oriented Triangle Theorems:
After studying this section you will be able to apply the No-Choice Theorem and
the AAS Theorem.
Theorem 53: If two angles of one triangle are __________ to two angles of
a second triangle, then the third angles are _____________.
(No-Choice Theorem)
C
F
Given: ∠ A ≅ ∠ D , ∠ B ≅ ∠ E
Conclusion: ∠ C ≅ ∠ F
A
B
D
E
Note: The two triangles do not have to be congruent for us to apply this No-Choice
Theorem
Theorem 54: If there exists a correspondence between the _________of
two triangles such that two ______and a ___________side of one are congruent
to the corresponding parts of the other, then the triangles are congruent. ( AAS)
Given: ∠ G ≅ ∠ K
∠ H ≅ ∠ M
Prove: Δ GHJ ≅ Δ KMO
Statement
1)
2)
3)
4)
5)
J
G
1)
2)
3)
4)
5)
Reason
O
H K
M
5
Look at each pair of triangles and decide if we can prove they are congruent. State
your justification for why they are congruent beside the triangles.
1)
2)
3)
#3
Given: PD and PC line in plane m
BP ⊥ m
∠C ≅ ∠D
Prove: ∠ PBC ≅ ∠ PBD
Statements
B
C
m
P
D
Reasons
C
# 2 Given CB ⊥ AB
DE || AB
∠ CDE = 40°
Find: m ∠ A=_____ ,m ∠ C=_____ , m ∠ CED=_____
D
A
E
B
#7 Given ∠ A ≅ ∠ X
∠ AVZ ≅ ∠ XYB
∠ ZVB ≅ ∠ YBX
Prove: VBYZ is a parallelogram
Statements
A
Z
Y
V
B
X
Reasons
#8
The measures of the angles of a triangle are in the ratio of 3:4:8. Find the
measure of the supplement of the largest angle.___________
#9 Given Triangle as marked
Find: m ∠ 1 =_______
50°
(2x +10)°
1
(4x-10)°
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# 16 a) If the perimeter of Δ DEF is 145,
find the perimeter of Δ GHJ.
b) Can you state a generalization
based on your solution to part a ?
F
H
G
D
7
E
J
# 17 Give the most descriptive name to the figure formed by connecting
consecutive midpoints of each of the following figures.(Be prepared to defend
your answer in each case,)
a) Rhombus
b) Kite
c) Square
d) Rectangle
e) Parallelogram
f) Quadrilateral
g) Isosceles trapezoid
# 18 Given: EF is the median to AC.
∠ CBD ≅ ∠ ADB;
CD is the base of isosceles Δ FDC
C.
Prove: ABCD is a rectangle.
Statement
8
B
A
F
D
E
Reason
7.3 Formulas Involving Polygons:
Can you name all the polygons with 3 sides through 12 sides? Try now
3sided___________ 4sided__________5sided_________6sided__________
7sided___________8sided__________9sided_________10 sided_________
11sided___________12sided_________15sided_________n-sided_________
What is the sum of the measures of the
five angles in the figure?___________
A
E
D
B
How many different ways can you come up with to do it?
C
Theorem 55 : The sum (S1) of the measures of the angles of a polygon
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with n sides is given by the formula S1 = (n − 2)180 .
We will refer, at times, to the angles of a polygon
as interior angles of the polygon. In the diagram
at the right extend each one of the sides at the
vertex to show the exterior angles. Number the
exterior angles, beginning at angle A, 1,2,3....
At vertex A the m ∠ 1 + m ∠ EAB =_______
What would be the total of all 5 supplementary
angles.___________
D
E
A
C
B
Use Theorem 55 to find the total number of degrees in the pentagon._________
Now, subtract the number of degrees in the pentagon from the total number of
degrees in all 5 supplementary angles.______________________
What is the difference?___________
Theorem 56: If one exterior angle is taken at each vertex, the sum Se of
the measure of the exterior angles of a polygon is given by the formula Se = 360
•
Draw all the possible diagonals from every vertex.
•
What is the total your got_________
•
At each vertex put the number of diagonals that
started at that point.
Theorem 57: The number of d diagonals that can be drawn in a polygon of
n sides is given by the formula d =
n(n − 3)
2
10
Homework:
1)Find the sum of the measures of the angles of
a) quadrilateral______
c) octagon________
e)93-gon__________
A
2) Given: m ∠ A = 160, m ∠ B= 50
m ∠ C = 140, m ∠ D = 150
E
Find: m ∠ E =______
B
D
C
3)How many diagonals can be drawn in each figure below? ( use your text)
a)________
b)________
c)________
5)Given: K is a midpoint
P is a midpoint
d)________
J
m ∠ M= 70
K
P
m ∠ JKP= y +15
m ∠ JPK = y – 10
M
O
Find: m ∠ JKP= ________ , m ∠ JPK= ________, m ∠ J= ________
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6)Find the sum of the measures of the exterior angles, one per vertex, of each of
these polygons.
a) triangle_______
b) heptagon__________
c) nonagon________
d) 1984-gon________
7) What is the fewest number of sides a polygon can have?_____
8) On a clock a segment is drawn connecting the mark at the 12 and the mark at
the 1; then another segment connecting the mark at the 1 and the mark at the 2;
and so forth, all the way around the clock.
a) What is the sum of the measures of the angles of the polygon formed?
b) What is the sum of the measures of the exterior angles, one per vertex,
of the polygon?
10) How many sides does a polygon have if the sum of the measures of its angles
is
a) 900
c) 2880
e)436
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7.4 Regular Polygons:
Definition 50: A regular polygon is a polygon that is BOTH equilateral and
equiangular.
Please draw the first 5 regular polygons below. Make each diagram to show the
congruent parts.
Please find the measure of one of the exterior angles in each of the figures above.
Did you use Theorem 56 which says the sum of the exterior angles is always
360°? or Did you divide 360 by the number of angles in the polygon? They are
very compatible theorems.
Theorem 57: The measure E of each exterior angle of an equiangular polygon
( regular polygon) of n sides is given by the formula
E=
360
n
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a) How many degrees are there in each exterior angle of an equiangular heptagon ?
( A heptagon has how many sides?)
b) If each exterior angle of a polygon is 18°, how many sides does the polygon
have?
c) If each angle of a polygon is 108°, how many side does the polygon have?
d) Find the measure of each angle of a regular octagon.
e) Find the measure of each exterior angle of an equilateral quadrilateral.
(Did it say equiangular? yes or no)
Now to homework