math 2 geometry based on elementary geometry, 3 rd ed, by alexander & koeberlein 4.1 properties...
TRANSCRIPT
Math 2 GeometryBased 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 drawingGiven statementProve statement
Theorem 4.1.1
A diagonal of a parallelogram separates it into two congruent triangles.
Proof:Need drawingGiven statementProve 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 Opposite angles of a parallelogram are congruent.
4.1.3 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 Opposite angles of a parallelogram are congruent.
4.1.3 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.