Download File - Andrew Busch

Proof Geometry
 All
quadrilaterals have four sides.
 They also have four angles.
 The sum of the four angles totals 360°.
 These properties are what make quadrilaterals
alike, but what makes them different?
 Two
sets of parallel sides . (Definition)
 Two sets of congruent sides. (Theorem)
 The angles that are opposite each other are
congruent . (Theorem)
 The diagonals bisect each other. (Theorem)
 Consecutive angles are supplementary. (Theorem)
A
rectangle is a parallelogram all of
whose angles are right angles. (Def.)

A
Has all properties of quadrilateral and
parallelogram
rectangle also has four right angles.
(Theorem)
 The
diagonals are congruent. (Theorem)
 Prove
it is a parallelogram with 1 right angle
Theorem: If a parallelogram has one right
angle then it is a rectangle.
 Prove
that the diagonals are congruent to
each other and bisect each other.
Theorem: If the diagonals of a quadrilateral
are congruent and bisect each other, then it is
a rectangle.
A
rhombus is a parallelogram all of whose
sides are congruent. (Definition)

All properties of parallelogram
 All
sides are congruent (Theorem)
 The diagonals bisect the angles. (Theorem)
 The diagonals are perpendicular bisectors of
each other (Theorem)
 Prove
it is a parallelogram with all 4
sides congruent.
 Prove it is a parallelogram with each
diagonal bisecting a pair of opposite
angles.
 Prove it is a quadrilateral with
diagonals that bisect each other and
are perpendicular to each other.
A
square is a rectangle all of whose sides are
congruent.(Definition)
All the properties of a parallelogram apply
 All the properties of a rectangle apply
 All the properties of a rhombus apply

 Prove
it is a rectangle and a rhombus
 Trapezoid
has one and only one set of
parallel sides.
 Prove

a set of sides is parallel
Technically, we would need to show the
other pair of sides are not parallel.
We won’t do that.
 An
isosceles trapezoid has two equal
sides. These equal sides are called the
legs of the trapezoid, which are the nonparallel sides of the trapezoid.
 Never
assume a trapezoid is isosceles unless it
is given
 Both
pair of base angles in an isosceles
trapezoid are also congruent.
 The diagonals are congruent
 Prove
that is a trapezoid with the
non parallel set of sides congruent
 Prove that it is a trapezoid with
congruent diagonals
 Prove that it is a trapezoid with base
angles congruent.
 Two
adjacent sets of consecutive congruent
sides (Definition)
 Diagonals are perpendicular to each other.
(Theorem)
 One of the diagonals bisects the other.
 The angles included between the sets congruent
sides are congruent—i.e. one pair of congruent
opp. Angles. (Theorem)
 Prove
that it is a quadrilateral with
two sets of consecutive congruent
sides.
 Prove that one diagonal is the
perpendicular bisector of the other.
p. 289: #4-6, 8-10