Math 2 Geometry Based on Elementary Geometry, 3rd ed, by Alexander

1
Math 2 GeometryBased on Elementary Geometry,
3rd ed, by Alexander Koeberlein

• 4.1
• Properties of a Parallelogram

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Informal Definition

• A quadrilateral is a polygon that has four
  sides.

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Informal Definition

• A quadrilateral is a polygon that has four
  sides.
• Implied in this definition is that the four
  segments are co-planar.

4
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.

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Definition

• A parallelogram is a quadrilateral in which both
  pairs of opposite sides are parallel.

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Theorem 4.1.1

• A diagonal of a parallelogram separates it into
  two congruent triangles.

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Theorem 4.1.1

• A diagonal of a parallelogram separates it into
  two congruent triangles.
• Proof

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Theorem 4.1.1

• A diagonal of a parallelogram separates it into
  two congruent triangles.
• Proof
• Need drawing
• Given statement
• Prove statement

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Theorem 4.1.1

• A diagonal of a parallelogram separates it into
  two congruent triangles.
• Proof
• Need drawing
• Given statement
• Prove statement Use ASA

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Corollaries

• 4.1.2 Opposite angles of a parallelogram are
  congruent.

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Corollaries

• 4.1.2 Opposite angles of a parallelogram are
  congruent.
• Why?

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Corollaries

• 4.1.2 Opposite angles of a parallelogram are
  congruent.
• Why? CPCTC

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Corollaries

• 4.1.2 Opposite angles of a parallelogram are
  congruent.
• 4.1.3 Opposite sides of parallelogram are
  congruent.

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Corollaries

• 4.1.2 Opposite angles of a parallelogram are
  congruent.
• 4.1.3 Opposite sides of parallelogram are
  congruent.
• Why?

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Corollaries

• 4.1.2 Opposite angles of a parallelogram are
  congruent.
• 4.1.3 Opposite sides of parallelogram are
  congruent.
• Why? CPCTC

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Corollaries

• 4.1.4 Diagonals of a parallelogram bisect
  each other.

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Corollaries

• 4.1.4 Diagonals of a parallelogram bisect
  each other.
• Why?

18
Corollaries

• 4.1.4 Diagonals of a parallelogram bisect
  each other.
• Why?

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Corollaries

• 4.1.4 Diagonals of a parallelogram bisect
  each other.
• Why?

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Corollaries

• 4.1.4 Diagonals of a parallelogram bisect
  each other.
• Why?

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Corollaries

• 4.1.5 Consecutive angles of a parallelogram
  are supplementary.

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Corollaries

• 4.1.5 Consecutive angles of a parallelogram
  are supplementary.
• Why?

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Corollaries

• 4.1.5 Consecutive angles of a parallelogram
  are supplementary.
• Why?

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Corollaries

• 4.1.5 Consecutive angles of a parallelogram
  are supplementary.
• Why?

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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).

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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).

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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,

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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,

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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.

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Observation

• If two non-right angles are supplementary

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Observation

• If two non-right angles are supplementary
• Can they both be acute?

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Observation

• If two non-right angles are supplementary
• Can they both be acute?
• Can they both be obtuse?

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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.

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Theorem 4.1.7

• In a parallelogram with unequal pairs of
  consecutive angles, the longer diagonal lies
  opposite the obtuse angle.

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Theorem 4.1.7

• In a parallelogram with unequal pairs of
  consecutive angles, the longer diagonal lies
  opposite the obtuse angle.

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Theorem 4.1.7

• In a parallelogram with unequal pairs of
  consecutive angles, the longer diagonal lies
  opposite the obtuse angle.