Moise/Downs Geometry Chapter 9 Flashcards

Lines and Angles

• Skew Lines are non-coplanar and do not intersect.
• Parallel Lines are coplanar and do not intersect.
• Theorem 9-1 states two parallel lines lie within exactly one plane.
• Theorem 9-2 asserts that if two lines are both perpendicular to the same line in a plane, they are parallel.

Parallels and Transversals

• At least one line through point P exists that is parallel to line L if P is not on L.
• Transversal is a line intersecting two coplanar lines at two distinct points.
• Alternate Interior Angles (AIA) are formed when a transversal intersects two lines, creating angles that are on opposite sides of the transversal and inside the lines.
• Theorem 9-4 indicates that if a pair of alternate interior angles are congruent, then the other pair will also be congruent.
• If a pair of alternate interior angles are congruent, the lines cut by the transversal are parallel.

Angle Relationships

• Corresponding Angles - angles that occupy the same relative position at each intersection.
• Same-Side Interior Angles - interior angles on the same side of the transversal.
• CAAI suggests that if corresponding angles are congruent, then alternate interior angles are also congruent.
• CAP states if corresponding angles are congruent, the lines are parallel.
• SSSP indicates that if same-side interior angles are supplementary, the lines are parallel.

Postulates and Properties

• Parallel Postulate states through a given point outside a line, only one parallel line exists to that line.
• PAI states alternate interior angles of parallel lines cut by a transversal are congruent.
• PCA indicates correspondence in angles for parallel lines cut by a transversal.
• PSSS asserts that two pairs of same side interior angles in a parallelogram are supplementary.

Geometry of Parallelograms

• Theorem 9-10 declares if a line intersects one of two parallel lines at one point, it will intersect the other as well.
• Parallel Transitivity: If two lines are each parallel to a third line, they are parallel to each other.
• Perpendicular Transitivity: A line perpendicular to one of two parallel lines is perpendicular to the other.
• The Triangle Angle Sum states the total angle measures in any triangle equate to 180 degrees.

Quadrilaterals

• Convex Quadrilateral: no vertices lie on opposite sides of any line that extends a side.
• Parallelogram: quadrilateral where both pairs of opposite sides are parallel.
• Trapezoid: quadrilateral with one pair of parallel sides (the bases).
• Median of Trapezoid: segment connecting midpoints of non-parallel sides is parallel to the base and half its length.

Properties of Parallelograms

• Diagonals bisect each other, and opposite sides are congruent.
• Opposite angles within a parallelogram are congruent; consecutive angles are supplementary.
• If both pairs of opposite sides or angles are congruent, or if the diagonals bisect each other, the quadrilateral is a parallelogram.

Special Types of Quadrilaterals

• Rhombus: all sides congruent; diagonals perpendicular.
• Rectangle: all angles are right angles.
• Square: a rectangle with all sides congruent.
• If a parallelogram has one right angle, it contains four right angles.

Triangles and Angles

• Midline Theorem: mid-segment in a triangle is parallel to the base and half its length.
• Isosceles Trapezoid Theorem states the base angles of an isosceles trapezoid are congruent.
• In any triangle, the measure of an exterior angle is equal to the sum of the two remote interior angles.

Theorems Related to Parallel Lines

• Theorem 9-29: if three parallel lines intercept congruent segments on one transversal, they intercept congruent segments on any parallel transversal.
• Theorem 9-30: three parallel lines intercept congruent segments on one transversal will do so on any transversal.