4.1 Properties of a Parallelogram 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 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.