Download journal 6 12314

Polygons and
Quadrilaterals
By: Ou Suk Kwon
• A Polygon is any figure that has more than 3 sides closed, and they
are not curved.
• The exterior angles of every polygon always add up to 360 degrees.
• To get the interior angles of a regular polygon, equiangular and
equilateral, is (n-2) times 180. measure of each angle and n being
the amount of sides that a polygon has.
• A convex polygon is a polygon with all vertices pointing out and can
be regular or irregular.
• A concave polygon is one that has one or more vertices pointing in
so it cannot be a regular polygon.
• An equiangular polygon has all angles equal but the sides can have
different size compared to each other.
• The equilateral is the opposite. It always has congruent sides but
the angles can be anything they want as long as the sides are
congruent and the figure is closed.
• The interior angle theorem for polygons is that
(n-2) times 180 is equal to the total amount of
angles in the polygon. N represent the # of the
sides that a polygon contains, as the # of
sides of the polygons changes, the
measurement of the interior angles are
different in every different types of polygons.
• Triangle, quadrilateral, pentagon, hexagon,
heptagon, nonagon, decagon, dodecagon and
so on.
(3-2)180=180
(6-2)180=720
(4-2)180=360
4 theorems of parallelograms
•Theorem 6-2-1 says If a quadrilateral is a parallelogram then its
opposite sides are congruent.
Converse: If quadrilateral’s opposite sides are congruent then it is a
parallelogram.
•Theorem 6-2-3 says if a quadrilateral is a parallelogram then its
consecutive angles are add up to 180 (supplementary).
Converse: If a quadrilateral’s consecutive angle are supplementary
then it is a parallelogram.
•Theorem 6-2-2 says If a quadrilateral is a parallelogram then its
opposite angles are congruent.
Converse: If a quadrilaterals opposite angles are congruent then it is a
parallelogram.
Theorem 6-2-4 says If a quadrilateral is a parallelogram then its
diagonals bisect each other.
Converse: If a quadrilaterals diagonals bisect each other then it is a
parallelogram.
how to prove that a
quadrilateral is a parallelogram
There are 6 ways to prove that a
quadrilateral is parallelogram.
1) Both opposite sides are congruent
2) Both opposite angles are congruent
3) Both opposite sides are parallel
4) Consecutive angles are supplementary
5) One pair of congruent and parallel sides
6) When the diagonals bisect
A rhombus has its own characteristics, same as
rectangle, but a square is both a rhombus and a
rectangle because of its unique and special
characteristics. A rhombus is a parallelogram where
all sides are congruent and the diagonals
are perpendicular. A rectangle has all angles that are
right angles, and the diagonals are congruent. A
square is equiangular and equilateral, the diagonals
are congruent and are perpendicular. Square can be
parallelogram, rhombus, and rectangle. But by the
converse we can’t say.
example
Trapezoids
• a trapezoid has one pair of parallel sides. They have
two bases and two legs but the legs are never parallel
to each other.
• There is a special trapezoids, called isosceles
trapezoids, and this is where opposite angles are
supplementary, the legs are congruent, the base angles
are congruent and the diagonals bisect.
• Theorem 6-6-3: If a quadrilateral is an isosceles
trapezoid, then each pair of base angles are congruent.
• Theorem 6-6-4: If a trapezoid has one pair of congruent
base angles, then the trapezoid is isosceles.
• To find midsegment you have to use (B1+B2)/2
Examples
Kites!
A kite is a quadrilateral that has two pair of congruent adjacent sides,
the diagonals are perpendicular, the have on pair of congruent angle
which is where the two non congruent sides meet and one diagonal is
bisected which is the one going from non congruent to non congruent
adjacent sides.
The theorems are:
6-6-1 theorem: if a quadrilateral is a kite, then is diagonals are
perpendicular.
Converse: if diagonals are perpendicular, then the quadrilateral is a kite.
6-6-2 theorem: if a quadrilateral is a kite, then exactly one pair of
opposite angles are congruent.
Converse: if exactly of pair of opposite angles are congruent, then the
quadrilateral is a kite.
Examples