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Questions and Answers
What are perpendicular lines?
What are perpendicular lines?
- Two lines that never meet
- Two lines that intersect to form right angles (correct)
- Two lines that are equal in length
- Two lines that are parallel to each other
What happens if one angle formed by two intersecting lines is a right angle?
What happens if one angle formed by two intersecting lines is a right angle?
The lines are perpendicular.
If Line JK is perpendicular to Line MN, then each of the numbered angles is a right angle.
If Line JK is perpendicular to Line MN, then each of the numbered angles is a right angle.
True (A)
What is Theorem 2-4 about?
What is Theorem 2-4 about?
What do you conclude if two lines form congruent adjacent angles?
What do you conclude if two lines form congruent adjacent angles?
What does it mean if the exterior sides of two adjacent acute angles are perpendicular?
What does it mean if the exterior sides of two adjacent acute angles are perpendicular?
According to notes, what definition is given for perpendicular lines?
According to notes, what definition is given for perpendicular lines?
What are the two theorems that are converses of each other?
What are the two theorems that are converses of each other?
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Study Notes
Perpendicular Lines Overview
- Perpendicular lines intersect to form right angles (90 degrees).
- If one of the angles formed by two intersecting lines is a right angle, all angles formed are right angles.
Key Definitions
- Perpendicular Lines: Lines l and m are considered perpendicular if they intersect to create a right angle.
- Congruent Adjacent Angles: If two lines are perpendicular, adjacent angles formed are congruent.
Theorems Related to Perpendicular Lines
- Theorem 2-4: If two lines are perpendicular, then they create congruent adjacent angles.
- Theorem 2-5: If lines create congruent adjacent angles, then the lines are perpendicular.
- Theorem 2-6: If the exterior sides of two adjacent acute angles are perpendicular, those angles are complementary.
Biconditional Theorem
- Two lines are perpendicular if and only if they form congruent adjacent angles (Theorems 2.4 and 2.5).
Proofs Involving Perpendicular Lines
- Proof of Congruent Angles: Given two perpendicular lines, one angle is shown to be a right angle, leading to the conclusion that the adjacent angle is also 90 degrees, hence congruent.
- Proof of Perpendicularity from Congruent Angles: When two lines create congruent angles, it's proved that each angle is 90 degrees, confirming the lines are perpendicular.
- Proof of Complementary Angles: If two adjacent acute angles have perpendicular exterior sides, they are proven to be complementary, summing to 90 degrees.
Conclusion
- The definition and properties of perpendicular lines link closely to angles that are right, congruent, and complementary, facilitating geometric understanding and applications.
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