Geometry Theorems Flashcards Section 3.5

Questions and Answers

What does The Z-Theorem state?

Points A and D can be the same

AB ∩ DE is not defined

AB ∩ DE = ∅ if B and E are on opposite sides of the line l (correct)

AB ∩ DE = ∅ if B and E are on the same side of the line l

What is the Crossbar Theorem?

If ∆ABC is a triangle and point D is the interior of ∠BAC, then there is a point G such that point G lies on both rays AD and BC.

What does it mean for a point to be in the interior of an angle?

A point D is in the interior of the angle ∠BAC if the ray AD intersects the interior of the segment BC.

Define a linear pair.

Two angles ∠BAD and ∠DAC form a linear pair if rays AB and AC are opposite rays. Signup and view all the answers

What does the Linear Pair Theorem state?

If angles ∠BAD and ∠DAC form a linear pair, then u(∠BAD) + u(∠DAC) = 180°. Signup and view all the answers

What does it mean for two angles to be supplementary?

Two angles ∠BAC and ∠EDF are supplementary if u(∠BAC) + u(∠EDF) = 180°. Signup and view all the answers

Define perpendicular lines.

Two lines l and m are perpendicular if there exists a point A that lies on both l and m and there exist points B ∈ l and C ∈ m such that ∠BAC is a right angle. Signup and view all the answers

What does Theorem 3.5.9 state?

If l is a line and p is a point on l, then there exists exactly one line n such that p lies on m and m ⊥ l. Signup and view all the answers

What is a perpendicular bisector?

A perpendicular bisector of DE is a line n such that the midpoint of DE lies on n and n ⊥ line DE. Signup and view all the answers

What does the Existence and Uniqueness of Perpendicular Bisectors state?

If D and E are two distinct points, then there exists a unique perpendicular bisector for line segment DE. Signup and view all the answers

Define vertical pair.

Angles ∠BAC and ∠DAE form a vertical pair if rays AB and AE are opposite and rays AC & AD are opposite or if rays AB & AD are opposite and rays AC & AE are opposite. Signup and view all the answers

Vertical angles are congruent.

True Signup and view all the answers

What is the Continuity Axiom?

The function f is a continuous function, as is the inverse of f. Signup and view all the answers

What is the setting for the Continuity Axiom?

Let A, B, and C be three noncollinear points. For each point D on BC, there is an angle ∠CAD and there is a distance CD. Signup and view all the answers

Study Notes

The Z-Theorem

Defines the condition for intersection involving a line and points on either side that cannot converge.
Specifically states if A and D are on line l, and B and E are on opposite sides of l, then rays AB and DE do not intersect.

The Crossbar Theorem

Relates to triangles where a point lies within an angle.
Guarantees that in triangle ∆ABC, if D is inside ∠BAC, there exists point G that lies on both ray AD and line segment BC.

Theorem 3.5.3 (Interior)

Establishes criteria for whether a point is within an angle.
A point D is considered in the interior of angle ∠BAC if ray AD crosses segment BC.

Linear Pair Definition

Two angles, ∠BAD and ∠DAC, create a linear pair when their rays (AB and AC) are opposite.
This configuration implies a straight line formation.

Linear Pair Theorem

States that if angles ∠BAD and ∠DAC create a linear pair, the sum of their measures equals 180 degrees.

Supplementary Angles Definition

Two angles, ∠BAC and ∠EDF, are deemed supplementary when the sum of their measures also equals 180 degrees.

Perpendicular Definition

Describes two lines l and m as perpendicular if they intersect at point A, forming a right angle (∠BAC), with points B on l and C on m.

Theorem 3.5.9 (Perpendicular Line Existence)

Affirms that for any line l and a point p on l, there is exactly one line n through p that is perpendicular to l.

Perpendicular Bisector Definition

A perpendicular bisector of segment DE is defined as line n where the midpoint of DE is on the line, and n meets DE at a right angle.

Existence and Uniqueness of Perpendicular Bisectors

Confirms that for two distinct points D and E, there is one unique perpendicular bisector to segment DE.

Vertical Pair Definition

Angles ∠BAC and ∠DAE form a vertical pair if their rays alternate in oppositional arrangement.
This includes scenarios where rays AB and AE or rays AC and AD are opposite.

Vertical Angles Theorem

States that vertical angles, formed by intersecting lines, are congruent.

The Continuity Axiom

Describes continuity in function f, with its inverse also being continuous, indicating a smooth, unbroken curve or line.

Setting for the Continuity Axiom

Involves three noncollinear points A, B, and C where point D exists on line BC, thus defining both an angle ∠CAD and a distance CD.

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Description

Explore essential theorems in geometry with this set of flashcards focused on Section 3.5. Learn about the Crossbar Theorem, the Z-Theorem, and more through concise definitions. Ideal for quick study and review of key geometric concepts.