Moise/Downs Geometry Chapter 9 Flashcards

Questions and Answers

What are skew lines?

Lines that are parallel

Lines that intersect

Lines that are coplanar

Lines that don't lie in the same plane (correct)

Parallel lines must always intersect.

False

What is Theorem 9-1?

Two parallel lines lie in exactly one plane.

What does Theorem 9-2 state?

If two lines are both perpendicular to the same line, then they are parallel. Signup and view all the answers

What is the existence of parallels?

Let L be a line and let P be a point not on L. There is at least one line through P, parallel to L. Signup and view all the answers

What is a transversal?

A line which intersects two coplanar lines at two different points. Signup and view all the answers

What are alternate interior angles?

Angles defined by the figure and positioning of lines cut by a transversal. Signup and view all the answers

What is Theorem 9-4?

If two lines are cut by a transversal, and one pair of alternate interior angles are congruent, then the other pair of alternate interior angles are also congruent. Signup and view all the answers

What does AIP stand for?

Given two lines cut by a transversal. If a pair of alternate interior angles are congruent, then the lines are parallel. Signup and view all the answers

What are corresponding angles?

[Answer not provided in content] Signup and view all the answers

What are same-side interior angles?

[Answer not provided in content] Signup and view all the answers

What does CAAI stand for?

Given two lines cut by a transversal. If a pair of corresponding angles are congruent, then a pair of alternate interior angles are congruent. Signup and view all the answers

What does CAP stand for?

Given two lines cut by a transversal. If a pair of corresponding angles are congruent, then the lines are parallel. Signup and view all the answers

What does SSSP stand for?

Given two lines cut by a transversal. If a pair of same-side interior angles are supplementary, then the lines are parallel. Signup and view all the answers

What is the Parallel Postulate?

Through a given external point, there is only one parallel to a given line. Signup and view all the answers

What does PAI state?

If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Signup and view all the answers

What does PCA state?

If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent. Signup and view all the answers

What does PSSS state?

If two parallel lines are cut by a transversal, the two pairs of same-side interior angles are supplementary. Signup and view all the answers

What is Theorem 9-10?

In a plane, if a line intersects one of two parallel lines in only one point, then it intersects the other. Signup and view all the answers

What is Parallel Transitivity?

In a plane, if two lines are each parallel to a third line, then they are parallel to each other. Signup and view all the answers

What is Perpendicular Transitivity?

In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other. Signup and view all the answers

What is Triangle Angle Sum?

For every triangle, the sum of the measures of the angles is 180. Signup and view all the answers

What does Convex Quadrilateral describe?

A quadrilateral where no two of its vertices lie on opposite sides of the line containing a side of the quadrilateral. Signup and view all the answers

What is a Parallelogram?

A quadrilateral where both pairs of opposite sides are parallel. Signup and view all the answers

What is a Trapezoid?

A quadrilateral where only one pair of opposite sides are parallel. Signup and view all the answers

What are the bases of a trapezoid?

The parallel sides of a trapezoid. Signup and view all the answers

What is the median of a trapezoid?

Segment formed by the midpoints of the two non-parallel sides of a trapezoid. Signup and view all the answers

What does the Midline Theorem state?

The segment between midpoints of two sides of a triangle is parallel to the third side and half as long. Signup and view all the answers

What is the Quadrilateral Midpoint Theorem?

The midpoints of the sides of any quadrilateral form a parallelogram. Signup and view all the answers

What does the Isosceles Trapezoid Theorem state?

The base angles of an isosceles trapezoid are congruent. Signup and view all the answers

What is a Rhombus?

A parallelogram all of whose sides are congruent. Signup and view all the answers

What is a Rectangle?

A parallelogram all of whose angles are right angles. Signup and view all the answers

What is a Square?

A rectangle all of whose sides are congruent. Signup and view all the answers

What does it mean if a parallelogram has one right angle?

Then it has four right angles, and the parallelogram is a rectangle. Signup and view all the answers

What is true about the diagonals of a rhombus?

The diagonals are perpendicular to one another. Signup and view all the answers

What can be concluded if the diagonals of a quadrilateral bisect each other and are perpendicular?

Then the quadrilateral is a rhombus. Signup and view all the answers

What is the median to the hypotenuse of a right triangle?

Half as long as the hypotenuse. Signup and view all the answers

What does the converse of 30-60-90 imply?

If one leg of a right triangle is half as long as the hypotenuse, then the opposite angle has measure 30. Signup and view all the answers

What does Theorem 9-29 state?

If three parallel lines intercept congruent segments on transversal T, then they intercept congruent segments on any transversal T' parallel to T. Signup and view all the answers

What does Theorem 9-30 state?

If three parallel lines intercept congruent segments on one transversal, then they intercept congruent segments on any other transversal. Signup and view all the answers

What does TMPT state?

If three or more parallel lines intercept congruent segments on one transversal, then they intercept congruent segments on any other transversal. Signup and view all the answers

Study Notes

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.

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.

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Test your knowledge on the key concepts from Chapter 9 of Moise and Downs' Geometry. This quiz covers essential definitions and theorems regarding lines, including skew and parallel lines, and their properties in a plane. Perfect for students looking to reinforce their understanding of geometric principles.