Pre-Cal 30S December 18, 2008
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This document discusses conditional statements and their various forms. It defines a conditional statement as one that can be expressed in the form "if...then". It provides examples of conditional statements and rewrites them in conditional, converse, inverse, and contrapositive forms. For example, one statement is rewritten in conditional form as "If each angle in a triangle is 60°, then you have an equilateral triangle".
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Proving lines are parallel
The document discusses several theorems and properties for proving when two lines are parallel, including:
1) If corresponding angles formed by a transversal intersecting two lines are congruent, then the lines are parallel.
2) If alternate interior angles or alternate exterior angles formed by a transversal are congruent, then the lines are parallel.
3) If consecutive interior angles formed by a transversal are supplementary, then the lines are parallel.
It also provides an example of using these properties to prove that if two boats sail at 45 degree angles to the constant wind, their paths will not cross since their paths are parallel.
Proving Lines Parallel
This document discusses different ways to prove that two lines are parallel using a transversal. It states that if two lines are cut by a transversal and their corresponding angles, alternate interior angles, consecutive interior angles, or consecutive exterior angles are congruent or supplementary, then the lines are parallel. It provides examples of proving lines parallel by finding values of x that make the lines satisfy one of these conditions. Finally, it lists the different ways one can prove two lines are parallel.
Proving lines are perpendicular
This document discusses properties and theorems related to perpendicular lines:
- Two lines are perpendicular if the product of their slopes is -1. Vertical and horizontal lines are also perpendicular.
- If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
- If two lines are each perpendicular to the same line, then they are parallel to each other.
- It provides a proof that if two coplanar lines are each perpendicular to the same line, then they are parallel to each other.
- Examples are given to determine if given lines are perpendicular or parallel based on their slopes or equations. Homework exercises are assigned from the textbook.
3-3 part 1 properties of parallel lines
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This document discusses different ways to prove that two lines are parallel using properties of parallel lines cut by a transversal. It introduces the converse theorems for corresponding angles, alternate interior angles, consecutive interior angles, and alternate exterior angles being congruent to show lines are parallel. It also presents two additional theorems: if two lines are parallel to the same line then they are parallel to each other, and if two lines are perpendicular to the same line then they are parallel to each other. The document provides example problems and homework assignments to practice these concepts.
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3) Rewriting conditional statements as biconditional statements using "if and only if".
4) Identifying whether biconditional statements are true based on the truth of both the conditional statement and its converse.
3.2 theorems about perpendicular lines
The document discusses different types of proofs used to prove theorems about perpendicular lines. It presents three theorems: 1) if two lines intersect to form congruent angles, they are perpendicular, 2) if two adjacent acute angles have perpendicular sides, they are complementary, and 3) if two lines are perpendicular, they intersect to form four right angles. It also provides an example proof of theorem 3.2 using a two-column format.
Lines and angles
Two angles are complementary if their measures sum to 90 degrees. Two angles are supplementary if their measures sum to 180 degrees. Adjacent angles have a common vertex and one common side but no interior points in common, such as the angles formed by scissors. Vertically opposite angles are formed without a common side when two lines intersect. A linear pair is formed when the non-common sides of adjacent angles lie in a straight line. A transversal is a line that intersects two or more other lines at distinct points, and intersecting lines cut each other at a common point.
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1) If corresponding angles formed by a transversal intersecting two lines are congruent, then the lines are parallel.
2) If alternate interior angles or alternate exterior angles formed by a transversal are congruent, then the lines are parallel.
3) If consecutive interior angles formed by a transversal are supplementary, then the lines are parallel.
It also provides an example of using these properties to prove that if two boats sail at 45 degree angles to the constant wind, their paths will not cross since their paths are parallel.
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This document discusses different ways to prove that two lines are parallel using a transversal. It states that if two lines are cut by a transversal and their corresponding angles, alternate interior angles, consecutive interior angles, or consecutive exterior angles are congruent or supplementary, then the lines are parallel. It provides examples of proving lines parallel by finding values of x that make the lines satisfy one of these conditions. Finally, it lists the different ways one can prove two lines are parallel.
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- If two lines are each perpendicular to the same line, then they are parallel to each other.
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- A pair of consecutive interior angles is supplementary
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Contrapositive: If two angles are not supplementary, then they do not form a linear pair.
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- A conditional statement has two parts: a hypothesis and conclusion connected by "if...then"
- The converse switches the hypothesis and conclusion
- The inverse negates both the hypothesis and conclusion
- A statement is equivalent to its contrapositive
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Pre-Cal 30S December 18, 2008
- 1. What If? whatif tagged photos by flickr user psychofish
- 2. Conditionals A quot;conditional statementquot; is a statement that can be expressed in the form quot;if...thenquot;. Hypothesis: This is the part of the sentence that follows the word quot;If.... ,quot; Conclusion: This is the part of the sentence that follows the word quot;then ... .quot;
- 4. Conditionals A quot;conditional statementquot; is a statement that can be expressed in the form quot;if...thenquot;. Rewrite each of the following in conditional form. Each angle of an equilateral triangle is 60°. If each angle in a triangle is 60° then you have an equilateral triangle. Biconditonal If a triangle is an equilateral triangle then each angle must equal to 60°. Parallel lines do not intersect. Inscribed angles in a circle subtended by the same arc are congruent.
- 5. Conditionals A quot;conditional statementquot; is a statement that can be expressed in the form quot;if...thenquot;. Rewrite each of the following in conditional form. Parallel lines do not intersect. If two lines are parallel then the lines do not intersect.
- 6. Conditionals A quot;conditional statementquot; is a statement that can be expressed in the form quot;if...thenquot;. Rewrite each of the following in conditional form. Inscribed angles in a circle subtended by the same arc are congruent. If 2 or more inscribed angles in a circle are subtended by the same arc then they're all congruent.
- 7. Related Conditionals: Converse, Inverse, Contrapositive
- 8. Related Conditionals: Converse Rewrite each of the following in converse form. If each angle in a triangle is 60° then you have an equilateral triangle. If you have an equilateral triangle then each angle is 60°. If two lines are parallel then the lines do not intersect. If two lines do not intersect then the lines are parallel. If 2 or more inscribed angles in a circle are subtended by the same arc then they're all congruent. If 2 or more inscribed angles in a circle are congruent then they're subtended by the same arc.
- 9. Related Conditionals: Converse, Inverse, Contrapositive
- 10. Related Conditionals: Inverse Rewrite each of the following in inverse form. If each angle in a triangle is not 60° then you do not have an equilateral triangle. If two lines are not parallel then the lines do intersect. If 2 or more inscribed angles in a circle are not subtended by the same arc then they're not all congruent.
- 11. Related Conditionals: Converse, Inverse, Contrapositive
- 12. Related Conditionals: Contrapositive Rewrite each of the following in contrapositive form. If you do not have an equilateral triangle then each angle is not 60°. If two lines do intersect then the lines are not parallel. DO THIS ONE FOR HOMEWORK If 2 or more inscribed angles in a circle are subtended by the same arc then they're all congruent. http://regentsprep.org/Regents/math/math-topic.cfm?TopicCode=relcond