Questions and Answers
What does The Z-Theorem state?
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.
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What does the Linear Pair Theorem state?
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What does it mean for two angles to be supplementary?
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Define perpendicular lines.
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What does Theorem 3.5.9 state?
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What is a perpendicular bisector?
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What does the Existence and Uniqueness of Perpendicular Bisectors state?
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Define vertical pair.
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Vertical angles are congruent.
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What is the Continuity Axiom?
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What is the setting for the Continuity Axiom?
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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.