Math 2 Geometry
Based on Elementary Geometry, 3rd ed, by Alexander & Koeberlein
4.1
Properties of a Parallelogram
Informal Definition
A quadrilateral is a polygon that has four
sides.
Informal Definition
A quadrilateral is a polygon that has four
sides.
Implied in this definition is that the four
segments are co-planar.
Informal Definition
A quadrilateral is a polygon that has four
sides.
Implied in this definition is that the four
segments are co-planar.
A closed figure with four sides that does
not have all segments in the same plane is
a skew quadrilateral.
Definition
A parallelogram is a quadrilateral in which
both pairs of opposite sides are parallel.
Theorem 4.1.1
A diagonal of a parallelogram separates it
into two congruent triangles.
Theorem 4.1.1
A diagonal of a parallelogram separates it
into two congruent triangles.
Proof:
Theorem 4.1.1
A diagonal of a parallelogram separates it
into two congruent triangles.
Proof:
Need drawing
Given statement
Prove statement
Theorem 4.1.1
A diagonal of a parallelogram separates it
into two congruent triangles.
Proof:
Need drawing
Given statement
Prove statement
Use ASA
Corollaries:
4.1.2
Opposite angles of a
parallelogram are congruent.
Corollaries:
4.1.2
Opposite angles of a
parallelogram are congruent.
Why?
Corollaries:
4.1.2
Opposite angles of a
parallelogram are congruent.
Why?
CPCTC
Corollaries:
4.1.2
4.1.3
Opposite angles of a
parallelogram are congruent.
Opposite sides of parallelogram
are congruent.
Corollaries:
4.1.2
Opposite angles of a
parallelogram are congruent.
4.1.3
Opposite sides of parallelogram
are congruent.
Why?
Corollaries:
4.1.2
4.1.3
Opposite angles of a
parallelogram are congruent.
Opposite sides of parallelogram
are congruent.
Why?
CPCTC
Corollaries:
4.1.4
Diagonals of a parallelogram
bisect each other.
Corollaries:
4.1.4
Diagonals of a parallelogram
bisect each other.
Why?
Corollaries:
4.1.4
Diagonals of a parallelogram
bisect each other.
Why?
Corollaries:
4.1.4
Diagonals of a parallelogram
bisect each other.
Why?
Corollaries:
4.1.4
Diagonals of a parallelogram
bisect each other.
Why?
Corollaries
4.1.5
Consecutive angles of a
parallelogram are supplementary.
Corollaries
4.1.5
Consecutive angles of a
parallelogram are supplementary.
Why?
Corollaries
4.1.5
Consecutive angles of a
parallelogram are supplementary.
Why?
Corollaries
4.1.5
Consecutive angles of a
parallelogram are supplementary.
Why?
Definition
An altitude of a parallelogram is a line
segment from one vertex that is
perpendicular to the opposite side (or to
an extension of that side).
Definition
An altitude of a parallelogram is a line
segment from one vertex that is
perpendicular to the opposite side (or to
an extension of that side).
Lemma 4.1.6
If two sides of one triangle are congruent to two sides of
a second triangle and the included angle of the
first triangle is greater than the included angle of
the second, …
Lemma 4.1.6
If two sides of one triangle are congruent to two sides of
a second triangle and the included angle of the first
triangle is greater than the included angle of the second,
Lemma 4.1.6
If two sides of one triangle are congruent to two sides of
a second triangle and the included angle of the first
triangle is greater than the included angle of the second,
then the length opposite the included angle of
the first is greater than the length of the side
opposite the included angle of the second.
Observation
If two non-right angles are supplementary
Observation
If two non-right angles are supplementary
• Can they both be acute?
Observation
If two non-right angles are supplementary
• Can they both be acute?
• Can they both be obtuse?
Observation
If two non-right angles are supplementary
• Can they both be acute?
• Can they both be obtuse?
One must be acute and one must be obtuse.
Theorem 4.1.7
In a parallelogram with unequal pairs of
consecutive angles, the longer diagonal
lies opposite the obtuse angle.
Theorem 4.1.7
In a parallelogram with unequal pairs of
consecutive angles, the longer diagonal
lies opposite the obtuse angle.
Theorem 4.1.7
In a parallelogram with unequal pairs of
consecutive angles, the longer diagonal
lies opposite the obtuse angle.