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Math 070 – Chapter 5: Discovering and Proving Polygon Properties
5.1: Polygon Sum Conjecture
Ex. Draw a quadrilateral with one diagonal. Label all six angles in the diagram.
Can you determine the sum of the measures of the angles of the quadrilateral?
Conjecture C-29: Quadrilateral Sum Conjecture
The sum of the measures of the four interior angles of any quadrilateral is ______.
Ex. Draw a pentagon. Choose one vertex and draw the two diagonals from that
vertex. Label all nine angles in the diagram. Can you determine the sum of the
measures of the angles of the pentagon?
Conjecture C-30: Pentagon Sum Conjecture
The sum of the measures of the five interior angles of any pentagon is _________.
Definition: If a polygon has n sides, it is called an n-gon.
Ex. Continuing the same pattern, complete the table:
# of sides
Sum of Angles
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5
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Conjecture C-31: Polygon Sum Conjecture
The sum of the measures of the n interior angles of an n-gon is given by the
formula _________________________________.
Ex. Do example from Condensed Lesson 5.1.
5.2: Exterior Angles of a Polygon
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Math 070 – Chapter 5: Discovering and Proving Polygon Properties
Definition: A set of exterior angles of an n-gon is a collection of n exterior angles,
with each having a different vertex of the polygon as its vertex.
Ex. Draw a pentagon, with a set of exterior angles illustrated, then use algebra to
determine the sum of the measures of the set of exterior angles.
Conjecture C-32: Exterior Angle Sum Conjecture
For any polygon, the sum of the measures of a set of exterior angles is _________.
Ex. Recall that an equiangular quadrilateral is called a rectangle. Use the polygon
sum conjecture to find the measure of each angle of a rectangle.
Ex. Using the formula from the Polygon Sum Conjecture, how can we calculate the
measure of each angle in an equiangular n-gon?
Conjecture C-33: Equiangular Polygon Conjecture
You can find the measure of each interior angle of an equiangular n-gon by using
the formula ________________________________________________________.
Ex. Do example from Condensed Lesson 5.2.
5.3: Kite and Trapezoid Properties
Definitions: In a kite, the angles that are between each pair of congruent sides are
called vertex angles, and the other two angles are called nonvertex angles.
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Math 070 – Chapter 5: Discovering and Proving Polygon Properties
Ex. Construct a kite and label the vertex angles and nonvertex angles, then study it
in order to complete the following conjectures
Conjecture C-34: Kite Angles Conjecture
The nonvertex angles of a kite are ______________________________________.
Ex. Write a flowchart (or two-column) proof of the Kite Angles Conjecture.
Conjecture C-35: Kite Diagonals Conjecture
The diagonals of a kite are ____________________________________________.
Conjecture C-36: Kite Diagonal Bisector Conjecture
The diagonal connecting the vertex angles of a kite is the ____________________
of the other diagonal.
Conjecture C-37: Kite Angle Bisector Conjecture
The vertex angles of a kite are _________________________ by the diagonal
connecting the vertices of the vertex angles.
Ex. Construct an isosceles trapezoid, then study it in order to complete the
following conjectures.
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Math 070 – Chapter 5: Discovering and Proving Polygon Properties
Conjecture C-38: Trapezoid Consecutive Angles Conjecture
The consecutive angles between the bases of a trapezoid are _________________.
Ex. Write a flowchart (or two-column) proof for the Trapezoid Consecutive Angles
Conjecture.
Conjecture C-39: Isosceles Trapezoid Conjecture
The base angles of an isosceles trapezoid are ______________________________.
Conjecture C-40: Isosceles Trapezoid Diagonals Conjecture
The diagonals of an isosceles trapezoid are _______________________________.
Ex. Write a flowchart (or two-column) proof for the Isosceles Trapezoid Diagonals
Conjecture (use the Isosceles Trapezoid Conjecture).
5.4: Properties of Midsegments
Ex. Draw a triangle, construct the three midsegments, then study it in order to
complete the following conjectures.
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Math 070 – Chapter 5: Discovering and Proving Polygon Properties
Conjecture C-41: Three Midsegments Conjecture
The three midsegments of a triangle divide it into __________________________.
Conjecture C-42: Triangle Midsegment Conjecture
A midsegment of a triangle is ___________________________ to the third side
and ________________________ the length of the third side.
The midsegment of a trapezoid is the line segment connecting the midpoints of the
two nonparallel sides.
Ex. Draw a trapezoid, construct the midsegment, then study it in order to complete
the following conjectures.
Conjecture C-43: Trapezoid Midsegment Conjecture
The midsegment of a trapezoid is _____________________________ to the bases
and the length is equal to _____________________________________________.
Ex. Do example from Condensed Lesson 5.4.
5.5: Properties of Parallelograms
Ex. Construct a parallelogram, then study it in order to complete the following
conjectures.
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Math 070 – Chapter 5: Discovering and Proving Polygon Properties
Conjecture C-44: Parallelogram Opposite Angles Conjecture
The opposite angles of a parallelogram are _______________________________.
Conjecture C-45: Parallelogram Consecutive Angles Conjecture
The consecutive angles of a parallelogram are _____________________________.
Conjecture C-46: Parallelogram Opposite Sides Conjecture
The opposite sides of a parallelogram are ________________________________.
Ex. Write a flowchart (or two-column) proof for the Parallelogram Opposite Sides
Conjecture.
Conjecture C-47 Parallelogram Diagonals Conjecture
The diagonals of a parallelogram _______________________________________.
Ex. Do example from Condensed Lesson 5.5.
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