1. The Parallel Postulate & Special Angles
Section 2.1
The Parallel Postulate &
Special Angles
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2. Construction 6 Constructing a line perpendicular to a given line from a point not on the given line.
(Figure 2.1 p. 72)
Steps:
From point P, using your compass, swipe an arc that intersects line l in two places.
Placing the compass point where the swipe intersects the line, open the compass past halfway and swipe below the line.
Repeat step 2 from the other swipe to form an x
With a straightedge connect point P and the x.
Label the new line.
Theorem 2.1.1: From a point not on a given line, there is exactly one line perpendicular to the given line.
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3. Parallel Lines Lines in the same plane that never intersect
Notation: l || m
Line l
Line m
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4. Several Applications of the Word Parallel Fig. 2.3 p. 73
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5. It follows from definition of parallel lines that:
Portions (segments or rays) of parallel lines are parallel.
Extensions of parallel segments or rays are parallel.
Postulate 10: (Parallel Postulate): Through a point not on a line, exactly one line is parallel to the given line.
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6. Transversal
A line that intersects 2 or more lines in different points. 1 2 4 3 6 5 7 8 t
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7. Transversal continued
At each point of intersection 4 angles are formed.
Angles that lie in the interior region – angles 3, 4, 5, & 6 – are interior angles.
Angles 1, 2, 7, & 8 lie in the exterior region are called exterior angles.
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8. Types of Angle Pairs Alternate interior angles Corresponding angles
(4 & 5, 3 & 6)
Angles are interior angles.
Angles are on opposite of the transversal.
Angles do not have the same vertex.
Hint: look for z oriented in any way.
Corresponding angles
(4 & 8, 3 & 7, 2 & 6, 1 & 5)
One angle is an interior angle; the other is an exterior angle.
Angles are on the same side of the transversal.
Angles do not have the same vertex.
Hint: Look for F, oriented any way.
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9. || lines & transversal Corresponding angles are congruent. Ex. 1 p. 75
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10. || lines & transversal Alternate interior angles are congruent.
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11. || lines & transversal
Interior angles on the same side of the transversal are supplementary. a b
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12. || lines & transversal Alternate exterior angles are congruent.
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13. Parallel Lines & Transversal
Postulate 11: If 2 parallel lines are cut by a
transversal, then the corresponding angles
are congruent.
Theorem 2.1.2: If two parallel lines are cut by
a transversal, then the alternate interior angles
are congruent. Proof p. 75
Theorem 2.1.3: If two parallel lines are cut by
a transversal, then the alternate exterior angles
Theorem 2.1.4: If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary. Proof p. 76
Theorem 2.1.5: If two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary.
Ex. 2,3 p.77-8
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