Geometry Theorems Flashcards Section 3.5

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.