Download Polygons and Quadrilaterals

Polygons and Quadrilaterals
Honors Geometry Unit 6
Warm Up for 1/31
• Fill in the missing angle measures and add up all the angles in the figure.
Do you notice a pattern between number of sides and the sum?
Polygons
• A Polygon is any closed plane figure made by 3 or more sides
• Plane Figure:
• Closed Figure:
• Examples:
Classifying Polygons
• We can classify all polygons into several categories
• Classifying by shape:
• Convex Polygons:
• Concave Polygons:
Classifying Polygons
• Classifying by Side or Angle Properties:
• Equilateral Polygon:
• Equiangular Polygon:
• Regular Polygon:
Classifying Polygons
• Classifying by Side or Angle Properties:
Classifying Polygons
• Classifying by Number of Sides
•3
—4
•5
—6
•7
—8
•9
—10
• 11+
Polygons
• Polygon Angle Sum-Theorem
• Thinking about the bell work, how did the number of triangles drawn relate to
the number of sides?
• What do the angles of a triangle add up to?
Polygons
• Polygon Angle-Sum Theorem
• The sum of the interior angles of any convex polygon can be found using the
Polygon Angle-Sum Theorem
Polygons
• What is the sum of the interior angles of a heptagon?
• Of a 17-gon?
• Of a 102-gon?
Polygons
• If the sum of the interior angles of a polygon is 1980°, how many
sides does the polygon have?
Warm Up for 2/1
1. What is the sum of the interior angles of a convex nonagon?
2. What is the sum of the interior angles of a convex 23-gon?
3. How many sides does a convex polygon have if the sum of the
interior angles is 3960°? If the sum is 9720°?
Regular Polygons
• Because all the angles of a Regular Polygon are congruent, we can
find the measure of each interior angle:
Regular Polygon
• What is the measure of each interior angle of a regular Pentagon?
• Of a regular 12-gon?
Using the Theorem
Using the Theorem
Using the Theorems
• Find the missing angle measures
Using the Theorems
• Solve for the variables, then find the angle measures
Using the Theorems
• Solve for the variables, then find the angle measures
Exterior Angles
• Find the Sum of the Exterior Angles of each polygon
Exterior Angles
Exterior Angles
• How could we figure out the measure of each exterior angle of a regular n-gon?
Exterior Angles
• What is the measure of an exterior angle of a regular pentagon?
• Of a regular nonagon?
• Of a regular 18-gon?
• Each polygon is a regular polygon, solve for the
variables
Exit Card
Warm Up for 2/2
• Solve for the variables
Quadrilaterals
• A quadrilateral is any polygon with four sides and four angles
• Quadrilaterals can be classified into 4 main groups, each with
different properties
•
•
•
•
Parallelograms
Trapezoids
Kites
Other Quadrilaterals
Quadrilaterals
Parallelograms
• Parallelograms are quadrilaterals with two pairs of parallel sides
• Rectangles, Rhombuses, and Squares are all parallelograms
Rectangles
• Rectangles are parallelograms with 4 right angles
• They have the same properties as parallelograms
Rhombus
• A Rhombus is a parallelogram with 4 congruent sides
• A Rhombus has the same properties as a parallelogram
Square
• A Square is a parallelogram with 4 right angles and 4 congruent sides
• A square is a parallelogram, a rectangle, and a rhombus
• It has the same properties as all three of these
Trapezoid
• A Trapezoid is a quadrilateral with only one pair of parallel sides
Isosceles Trapezoids
• Isosceles Trapezoids are trapezoids with one pair of congruent sides
• The congruent sides are not parallel
Kites
• A Kite is a quadrilateral with no sets of parallel sides and two pairs of
congruent sides
• Any quadrilateral that meets none of these specifications is just a
quadrilateral
Warm Up for 2/6
• Write what properties make each quadrilateral different from the others
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Isosceles Trapezoid
Kite
Properties of Parallelograms
• In addition to having two pairs of parallel sides, parallelograms have a few other
special properties
• First, we need to learn a few more terms
• Opposite Sides—Sides of a quadrilateral that do not share a vertex
• Opposite Angles—Angles of a quadrilateral that do not share a side
Properties of Parallelograms
Properties of Parallelograms
Properties of Parallelograms
Parallelograms
• Find the value of x in each parallelogram
Parallelograms
• Find the value of a
Parallelograms
• Find the value of t
Parallelograms
• Find the value of b
• Then find the lengths of the sides
Exit Card
Warm Up for 2/7
• Find the measures of the missing angles.
Properties of Parallelograms
Properties of Parallelograms
• Find the value of each variable and the length of each diagonal.
Solve for each variable in the parallelograms
Warm Up for 2/8
Solve for each variable in the parallelograms
Properties of Parallelograms
• Fill in the property each part has in a parallelogram
• Opposite Sides
• Opposite Angles
• Consecutive Angles
• Diagonals
Special Parallelograms
Identifying Parallelograms
• State the most specific name for each quadrilateral
• Find the missing angle measures in each rhombus
Warm Up for 2/9
• Give the most precise name to each parallelogram based on the description
1. A parallelogram has congruent diagonals.
2. A parallelogram has perpendicular diagonals and angle measures that are
all 90.
3. A parallelogram has perpendicular diagonals and angle measures of 45,
135, 45, and 135.
• Find the missing angle measures in each rhombus
Exit Card
Warm Up for 2/10
Trapezoids
• Trapezoids are quadrilaterals with only
one pair of parallel sides
• The parallel sides are called the bases
• The non-parallel sides are called the legs
• The angles that touch the same base are
called a pair of base angles
Trapezoids
• Isosceles trapezoids are
trapezoids with congruent legs
• The bases of a trapezoid will never
be congruent
• Isosceles trapezoids have unique
properties
Isosceles Trapezoids
Kites
• A Kite is a quadrilateral with:
• Two pairs of congruent
consecutive sides
• No opposite sides that are
congruent
• No opposite sides that are parallel
Kites
Warm Up for 2/13
Isosceles Trapezoids
• CDEF is a trapezoid with the measures shown. Find
the remaining angle measures.
Isosceles Trapezoids
• PQRS is a trapezoid with the measures shown. Find
the remaining angle measures.
Isosceles Trapezoids
• Find the value of each variable using the given
information
Kites
• Given Kite DEFG, find the values
of the missing labeled angles
Kites
• Given Kite KLMN, find the values
of the missing labeled angles
Kites
Midsegments
Warm Up for 2/14
Exit Card
Warm Up for 2/27
Parallelograms in the Coordinate Plane
• When plotting points, use the distance formula and slopes to to test
for different properties of parallelograms
• Once you find something about the sides, angles, or diagonals, use
the properties to determine what type of parallelogram it is
• If the properties are not met, then it is not a parallelogram
Parallelograms in the Coordinate Plane
• Tell whether the lines are parallel, perpendicular, or neither.
• Find the length and slope of the segment between the given points.
(3,7) & (6,11)
(-3,2) & (5,-4)
Warm Up for 2/28
• Given the following measurements, classify each quadrilateral as precisely as
you can. (Draw and label quadrilaterals if it helps)
ABCD
Slope AB=4
Slope BC=
1
−
4
Slope CD=4
Slope AD=
1
−
4
EFGH
EF=7
WXYZ
Slope WX=-5
1
2
FG=7
Slope XY=
GH=7
Slope YZ=-5
EH=7
EG=11
FH=10
Slope WZ=
1
3
Give the most precise name for each quadrilateral. Explain
Give the most precise name for each quadrilateral. Explain
Give the most precise name for each quadrilateral. Explain
Give the most precise name for each quadrilateral. Explain
• A(-6, 3), B(-2, 0), C(-2, -5), D(-6, -2)
Give the most precise name for each quadrilateral. Explain
• A(3, 4), B(8, 1), C(2, 9), D(3, 6)
Give the most precise name for each quadrilateral. Explain
• A(0, 1), B(1, 4), C(4, 3), D(3, 2)