Download 5.1 Interior angles of a polygon Sides 3 4 5 6 n Number of Triangles

5.1 Interior angles of a polygon
Sides
Number of Triangles
Sum of interiorAngles
3
1
180
4
5
6
n
Sum of the interior angles of a n-sided Polygon = (n-2) 180
What you need to know:
How to use the formula
1) The sum of the measures of the interior angles of a 25-gon is ___________.
How to find one angle in a regular polygon
2) The measure of one angle in a regular octagon is _________.
How to use the formula in reverse
3) How many sides does a polygon have if the sum of its interior angles is 3060?_____________
Review
1
a
b
a+b=_____
_
3
a
5
7
b
a+b+c=___
___
c
6
8
2
4
5.2 Exterior angles of a polygon.
1. Sketch the exterior angles of the octagon
2. m easure each exterior angle of each polygon
3. find the sum of all to the exterior angles of each polygon
The sum of all exterior angles of a polygo n add up to _________________.
What you need to know.
1) The sum of the exterior angles is constant----The sum of the measures of the exterior angles of a 25-gon is
_____________
2) How to use the new knowledge backwards----If the measure of one exterior angle of a regular polygon is 24°,
then the polygon has_____________ sides.
1.
Find all of the m issing angles.
70
110
5.3 Kite and trapezoid properties
Vertex Angle
1. There are tw o sets of congruent Adjacent sides.
2. D iagonals are perpendicular.
N onVertex Angles
3. The line connecting the vertex angles
bisects the vertex angles and
the other diagonal.
4. Tw o isosceles triangles are formed w ith a kite.
---The base angles are congruent.
5. Four right triangles are formed w ith a kite.
6. N onvertex angles are congruent.
Vertex Angle
The following are kites.
1.
x = _____
y = _____
2.
x = _____
y = _____
x
y
151°
73°
Isosceles Trapezoid
BASE
LEG
LEG
BASE
D iagonals are congruent
D iagonals create 4 sets of congruent angles.
1. Bottom base angles are congruent. Top base angles are congruent.
2. Consecutive angles betw een the bases are supplementary. (any trapezoid )
3. Legs are congruent.
1. x = _____ y = _____
2.Perimeter = 105 cm x = _____
23 cm
x
x
121°
y
30 cm
Find the m easures of every letter.
l1
m
l2
l3
l4
k
70°
j
r
n
q
s
30°
i
e
p 40°
t
d
v
f
a
g
c
100°
u
l1
b
h
l2
106°
l3
a=
m=
l4
b=
n=
c=
p=
d=
q=
e=
r=
f=
s=
g=
t=
h=
u=
i=
v=
j= k=
Review
1. x = _____ y = _____
x
121°
y
2. x = _____ y = _____
80
y+3
x-20
3. x = _____ y = _____
100
y
60
x
1. Draw a regular polygon.
1.
2.
3.
4.
5.
2. Draw an equilateral polygon.
If the sum of the measures of two angles is 90°, then the angles
are __________________.
In an isosceles triangle, the base angles are _________________.
The sum of the measures of the angles of an octagon is ________.
Each angle of a regular hexagon measures __________________.
The diagonals of a __________________ are perpendicular bisectors of each other.
5.4 Midsegments. A segment that connects any two midpoints of a triangle and
the nonparallel sides of a trapezoid
properties of midsegments for triangles.
Use your ruler to show the following are true:
1.
2.
3.
4.
Each midsegment bisects the sides with the midpoints.
The midsegment is half the length of the third side.
The midsegment is parallel to the third side.
The three midsegments create four triangles. These triangles are ___________________.
properties of midsegments for triangles. http://www.mathopenref.com/trapezoidmedian.html
Use your ruler and protractor to show the following are true:
1.
2.
3.
4.
The midsegment bisects the legs.
Half the sum of the lengths of the bases is equal to the length of the midsegment.
The midsegment is parallel to the bases.
The angle formed by the leg and base is congruent to the corresponding angle formed by the same leg and
midsegment.
1. Perimeter = 105 cm
x = _____
23 cm
x
30 cm
2–3. Find the missing values in each figure.
2. x = ____ y = ____
z = ____
17
23
x
z
y
60°
40°
26
3. The figure is a trapezoid. q = _____
13
24
q
4. The midsegment of a trapezoid is _________________ to the two bases.
5. The length of a midsegment between two sides of a triangle is _______ the length of the
third side.
6. The length of the midsegment of a trapezoid is ____________________________ of the
lengths of the bases.
7. Draw one median in one triangle and one midsegment in the other triangle and label each as
such.
5.5 Parallelograms. A quadrilateral where opposite sides are parallel.
Use your ruler and protractor to show the following are true:
1.
2.
3.
4.
5.
1.
Opposite angles are congruent.
Consecutive angles are supplementary.
Opposite sides are equal in measure.
The diagonals of a parallelogram bisect each other.
The diagonals form two sets of vertical angles.
a = _____
b = _____
x = _____
y = _____
x
z
y
z
15
30
2. Find the missing coordinate in terms of p and q
,5
p, q
8.
The length of the midsegment of a trapezoid is the ____________ of the lengths of the bases.
9.
The opposite angles of a parallelogram __________________.
10.
11.
The diagonals of a parallelogram __________________.
The consecutive angles of a parallelogram ________________.
5.6 Special parallelograms. Rhombus, Rectangle, and Square.
Properties of a Rhombus (a square is a rhombus)
Use your ruler and protractor to show the following are true:
1. These are all parallelograms, so parallelogram rules above apply.
2. The diagonals are perpendicular to each other.
3. All sides are equal in length.
4. The diagonals bisect the angles.
Properties of a Rectangle (a square is a rectangle)
Use your ruler and protractor to show the following are true:
1. These are all parallelograms, so same rules apply. Apply them to the Rectangle.
2. The diagonals are not perpendicular to each other, but they do bisect each other.
3. The measure of each angle is the same.
4. The diagonals are equal in length.
1. Lengths x=
Angles a=
y=
b=
b
x
y
41
a
105
55
Complete each statement. None of the answers is square.
1. If the sum of the measures of two angles is 90°, then the angles are __________________.
2. In an isosceles triangle, the base angles are _________________.
3. The sum of the measures of the angles of an octagon is _____________________.
4. Each angle of a regular hexagon measures _____________________.
5. The diagonals of a _________________________ are perpendicular bisectors of each other.
6. The midsegment of a trapezoid is _________________________ to the two bases.
7. The length of a __________________between two sides of a triangle is half the length of the third side.
8. The length of the midsegment of a trapezoid is ____________________the sum of the lengths of the bases.
9. The opposite angles of a parallelogram are __________________.
10. The diagonals of a parallelogram _____________________________________.
11. The consecutive angles of a parallelogram are ___________________________________.
12. The diagonals of a ___________________________________ bisect the opposite angles.
13. The diagonals of a _________________________________ are equal in length.
Given diagonal 𝐵𝐾 and ∠𝐵, construct rhombus BAKE.
B
K
B