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8.1 Angle Measures in Polygons
In a polygon, two vertices that are endpoints of the same side are called ___________________________.
A __________________ of a polygon is a segment that joins two nonconsecutive vertices.
Diagonals from one vertex form __________________.
8.1 Angle Measures in Polygons
Polygon Interior Angles Theorem –
The sum of the measures of the interior angles of a convex n-gon is ________________
8.1 Angle Measures in Polygons
Interior Angles of a Quadrilateral –
The sum of the measures of the interior angles of a quadrilateral is __________.
8.1 Angle Measures in Polygons
Find the sum of the measures of the interior angles of a convex octagon.
The sum of the measures of the interior angles of a convex polygon is 2340°. Classify the polygon by the number of sides.
8.1 Angle Measures in Polygons
Polygon Exterior Angles Theorem –
The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex is ______________
8.1 Angle Measures in Polygons
A convex hexagon has exterior angles with measures 34°, 49°, 58°, 67°, and 75°. What is the measure of an exterior angle at the sixth vertex?
8.1 Angle Measures in Polygons
180( 2)n
n
8.2 Properties of Parallelograms
A _____________________ is a quadrilateral with both pairs of opposite sides
____________________.
The term “parallelogram PQRS can be written as _____________.
In _____________, ____________ and ____________ by definition.
8.2 Properties of Parallelograms
Theorem 8.3 – If a quadrilateral is a parallelogram, then its opposite sides are _________________.
Theorem 8.4 – If a quadrilateral is a parallelogram, then its opposite angles are ________________.
8.2 Properties of Parallelograms
Find the values of x and y.
D E
G F53°
y°
4x-1
15
8.2 Properties of Parallelograms
Find the values of x and y.
D E
G F53°
2x
y+2
16
10
8.2 Properties of Parallelograms
Theorem 8.5 – If a quadrilateral is a parallelogram, then its consecutive angles are _________________.
Solve for the variable.
42°
D E
G F2x°
8.2 Properties of Parallelograms
Solve for the variable.
4(p+3)°D E
G F135°
8.2 Properties of Parallelograms
Theorem 8.6 – If a quadrilateral is a parallelogram, then its ______________ bisect each other.
P Q
S R
T
8.2 Properties of Parallelograms
Solve for PR, ST, and the measures of angles SRQ and PQR.
P Q
S R
T
8.3 Showing that a Quadrilateral is a Parallelogram
Theorem 8.7 – If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a _________________.
Theorem 8.8 – If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a ________________________.
D E
FG
8.3 Showing that a Quadrilateral is a Parallelogram
Given:Prove: ABCD is a parallelogram D E
FG
8.3 Showing that a Quadrilateral is a Parallelogram
Theorem 8.9 – If one pair of opposite sides of a quadrilateral are __________________________, then the quadrilateral is a _________________.
Theorem 8.10– If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a ________________________.
D E
FG
8.3 Showing that a Quadrilateral is a Parallelogram
For what value of x is quadrilateral DEFG a parallelogram?
D E
FG
8.4 Properties of Rhombuses, Rectangles, and Squares
Rhombus –
Rectangle –
Square –
8.4 Properties of Rhombuses, Rectangles, and Squares
Rhombus Corollary – A quadrilateral is a rhombus if and only if it has
_____________________________.
Rectangle Corollary – A quadrilateral is a rectangle if and only if it has
_____________________________.
Square Corollary – A quadrilateral is a square if and only if it is a _____________
and a ____________________.
8.4 Properties of Rhombuses, Rectangles, and Squares
For any rectangle ABCD, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning.
a.
b.
AB CD
AB BC
8.4 Properties of Rhombuses, Rectangles, and Squares
For any rhombus ABCD, decide whether the statement is always, sometimes, or never true. Draw a sketch and explain your reasoning.
a.
b.
AB BC
mA mB
8.4 Properties of Rhombuses, Rectangles, and Squares
Diagonal Theorems –
Theorem 8.11 – A parallelogram is a rhombus if and only if its diagonals are _____________________
Theorem 8.12 –A parallelogram is a rhombus if and only if each diagonal ___________________ a pair of opposite angles.
Theorem 8.13 –A parallelogram is a rectangle is and only if its diagonals are ____________________
8.4 Properties of Rhombuses, Rectangles, and Squares
For any rhombus DEFG, decide whether the statement is always, sometimes, or never true. Draw a sketch and explain your reasoning.
1.
2.
3.
DEG FEG
DEG EFD
DG GF
8.4 Properties of Rhombuses, Rectangles, and Squares
Classify the special quadrilateral. Explain your reasoning.
8.4 Properties of Rhombuses, Rectangles, and Squares
Sketch rhombus ABCD. List everything you know about it.
8.4 Properties of Rhombuses, Rectangles, and Squares
Sketch square ABCD. List everything you know about it.
8.5 Using Properties of Trapezoids and Kites
A _________________ is a quadrilateral with exactly one pair of parallel sides.
These parallel sides are called the __________________.
For each of the bases, there is a pair of _____________________________.
The nonparallel sides are called the _________________.
If the legs are congruent, then the trapezoid is an _______________________.
8.5 Using Properties of Trapezoids and Kites
Show that ABCD is a trapezoid.A (1,2), B (4,5), C (7,3), D (7,-2)
8.5 Using Properties of Trapezoids and Kites
Theorem 8.14 –If a trapezoid is isosceles, then each pair of base angles is __________________.
Theorem 8.15 –If a trapezoid has a pair of congruent base angles, then it is an _______________trapezoid.
Theorem 8.16 –A trapezoid is isosceles if and only if its diagonals are __________________.
8.5 Using Properties of Trapezoids and Kites
Theorem 8.17 Midsegment Theorem for Trapezoids –
The midsegment of a trapezoid is ________________ to each base and its
length is ______________ the sum of the lengths of the bases.
A B
D C
P Q
8.5 Using Properties of Trapezoids and Kites
In this diagram, ABCD is an isosceles trapezoid, and PQ is the midsegment.
a. Find b. Find
A B
D C
P Q
m B
PQ
8.5 Using Properties of Trapezoids and Kites
A ____________ is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are __________________.
8.5 Using Properties of Trapezoids and Kites
Theorem 8.18 – If a quadrilateral is a kite, then its diagonals are ______________________.
Theorem 8.19 – If a quadrilateral is a kite, then exactly one pair of opposite angles are ____________.
A
B
C
D
A
B
C
D
8.5 Using Properties of Trapezoids and Kites
In the diagram, PQRS is a kite. Find
S
P
Q
R
m Q
83° 41°
8.5 Using Properties of Trapezoids and Kites
In a kite, the measures of the angles are 6x°, 24°, 84°, and 126°. Find the value of x.What are the measures of the angles that are congruent?
8.6 Identifying Special Quadrilaterals
8.6 Identifying Special Quadrilaterals
Quadrilateral ABCD has at least one pair of opposite angles congruent.What types of quadrilaterals meet this condition?
8.6 Identifying Special Quadrilaterals
Quadrilateral WXYZ has at least one pair of opposite sides that are parallel.What types of quadrilaterals meet this condition?
8.6 Identifying Special Quadrilaterals
What is the most specific name for quadrilateral DEFG?
D
E
F
G
8.6 Identifying Special Quadrilaterals
Is enough information given in the diagram to show that quadrilateral ABCD is a rhombus? Explain.
A
B
C
D
8.6 Identifying Special Quadrilaterals
Give the most specific name for the quadrilateral. Explain your reasoning.
A B
CD
8.6 Identifying Special Quadrilaterals
Give the most specific name for the quadrilateral. Explain your reasoning.
A B
CD
8.6 Identifying Special Quadrilaterals
Give the most specific name for the quadrilateral. Explain your reasoning.
A B
CD
8.6 Identifying Special Quadrilaterals
You are given the following information about quadrilateral ABCD.
Is enough information given to conclude that quadrilateral ABCD is a trapezoid? Explain.
6
12
115
65
AB
CD
m A
m D
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