Geometry Chapter 3 Review Flashcards

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Questions and Answers

What are parallel lines?

Are coplanar and do not intersect.

What defines perpendicular lines?

Intersect at 90° angles.

What are skew lines?

Are not coplanar, not parallel, and do not intersect.

What are parallel planes?

Are planes that do not intersect. Signup and view all the answers

What is a transversal?

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

What does the Corresponding Angles Postulate state?

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

What does the Alternate Interior Angles Theorem state?

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

What does the Alternate Exterior Angles Theorem state?

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

What does the Same Side Interior Angles Theorem state?

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

What is the Converse of the Corresponding Angles Postulate?

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

What is the Parallel Postulate?

Through a point P not on line L, there is exactly one line parallel to L. Signup and view all the answers

What does the Converse of the Alternate Interior Angles Theorem state?

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

What does the Converse of the Alternate Exterior Angles Theorem state?

If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the lines are parallel. Signup and view all the answers

What does the Converse of the Same Side Interior Angles Theorem state?

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

What is a perpendicular bisector?

A line perpendicular to a segment at its midpoint. Signup and view all the answers

What is the distance from a point to a line?

Length of the perpendicular segment from point to line. Signup and view all the answers

What does the theorem state about two intersecting lines and a linear pair of congruent angles?

If two intersecting lines form a linear pair of congruent angles then they are perpendicular. Signup and view all the answers

What does the Perpendicular Transversal Theorem state?

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

What does it mean if two coplanar lines are perpendicular to the same line?

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

What is the definition of rise in line geometry?

Difference in y-values of two points on a line. Signup and view all the answers

What is the definition of run in line geometry?

Difference in x values of two points on a line. Signup and view all the answers

What is slope in geometry?

Rise over run, or $m = \frac{y_2-y_1}{x_2-x_1}$. Signup and view all the answers

What is the difference between undefined slope and zero slope?

Undefined slope is vertical. Signup and view all the answers

What does the Parallel Lines Theorem state?

Two non-vertical lines are parallel if and only if they have the same slope. Signup and view all the answers

What does the Perpendicular Lines Theorem state?

Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. Signup and view all the answers

What is Point-Slope Form?

y - y₁ = m(x - x₁) where m is slope and (x₁, y₁) is a point on the line. Signup and view all the answers

What is Slope-Intercept Form?

y = mx + b. Signup and view all the answers

What is the equation of vertical lines?

x = x-intercept. Signup and view all the answers

What is the equation of horizontal lines?

y = y-intercept. Signup and view all the answers

What characterizes parallel lines?

Same slope, different y-intercept. Signup and view all the answers

What characterizes intersecting lines?

Different slopes. Signup and view all the answers

What characterizes coinciding lines?

Share slope and y-intercept. Signup and view all the answers

Study Notes

Lines and Their Relationships

Parallel Lines: Coplanar lines that never intersect (symbol: ||).
Perpendicular Lines: Lines that intersect at right angles (symbol: ⊥).
Skew Lines: Lines that are not coplanar, not parallel, and do not intersect.
Parallel Planes: Planes that do not intersect.

Transversals and Angles

Transversal: A line that intersects two coplanar lines at distinct points, creating eight angles.
Corresponding Angles Postulate: When two parallel lines are cut by a transversal, the corresponding angles are congruent.
Alternate Interior Angles Theorem: Indicates that alternate interior angles formed by a transversal with two parallel lines are congruent.
Alternate Exterior Angles Theorem: States that alternate exterior angles are also congruent when two parallel lines are cut by a transversal.
Same Side Interior Angles Theorem: The pairs of same side interior angles are supplementary when two parallel lines are intersected by a transversal.

Converse Statements

Converse of Corresponding Angles Postulate: If a pair of corresponding angles are congruent, then the lines are parallel.
Parallel Postulate: There is exactly one line parallel to a given line through a point not on that line.
Converse of Alternate Interior Angles Theorem: If alternate interior angles are congruent, the lines are parallel.
Converse of Alternate Exterior Angles Theorem: Congruent alternate exterior angles indicate that the lines are parallel.
Converse of Same Side Interior Angles Theorem: If same side interior angles are supplementary, the lines are parallel.

Special Line Segments

Perpendicular Bisector: A line that is perpendicular to a segment at its midpoint.
Distance from a Point to a Line: Defined as the length of the perpendicular segment drawn from the point to the line.

Perpendicular Lines Theorems

Intersecting Lines and Perpendicularity: If two lines intersect to form a linear pair of congruent angles, those lines are perpendicular.
Perpendicular Transversal Theorem: If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other.
Lines Perpendicular to the Same Line: If two coplanar lines are both perpendicular to the same line, they are parallel.

Slope Concepts

Rise: The difference in y-values between two points on a line.
Run: The difference in x-values between two points on a line.
Slope: The ratio of rise over run, represented as ( m = \frac{y_2 - y_1}{x_2 - x_1} ).
Undefined vs Zero Slope: Zero slope indicates a horizontal line, while undefined slope indicates a vertical line.

Line Relationships in Coordinate Geometry

Parallel Lines Theorem: In a coordinate plane, two non-vertical lines are parallel if they have the same slope; vertical lines are inherently parallel.
Perpendicular Lines Theorem: Two non-vertical lines are perpendicular if the product of their slopes equals -1; vertical and horizontal lines are also perpendicular.

Equation Forms

Point-Slope Form: Given as ( y - y_1 = m(x - x_1) ) where ( m ) is slope and ( (x_1, y_1) ) is a point on the line.
Slope-Intercept Form: Represented as ( y = mx + b ) where ( b ) is the y-intercept.
Equations of Vertical and Horizontal Lines: Vertical lines are represented as ( x = ) (x-intercept), while horizontal lines are represented as ( y = ) (y-intercept).

Line Properties

Parallel Lines: Have the same slope but different y-intercepts.
Intersecting Lines: Exhibit different slopes, confirming they cross at a point.
Coinciding Lines: Lines that overlay one another share both slope and y-intercept, meaning they are essentially the same line.

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Review the key concepts from Geometry Chapter 3 with these flashcards. Learn definitions of parallel lines, perpendicular lines, skew lines, and more to reinforce your understanding of geometric relationships. Perfect for students looking to prepare for exams or quizzes.