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Theorems About Perpendicular Lines

Sep 12, 2022
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Key Concepts

  • Find the shortest distance between a point and a line.
  • Find the shortest distance between two parallel lines.
  • Define the perpendicular transversal theorem.
  • Define the lines perpendicular to a transversal theorem.
  • Theorem: If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
  • Theorem: If two lines are perpendicular, then they intersect to form four right angles.
  • Theorem: If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

Theorems About Perpendicular Lines

1. Shortest distance from a point to a line 

The distance from a point to a line is the length of the perpendicular segment from the point to the line. This perpendicular segment is the shortest distance between the point and the line. 

Shortest distance from a point to a line 

2. Shortest distance between two parallel lines 

The distance between two parallel lines is the length of any perpendicular segment joining the two lines. 

Shortest distance between two parallel lines 

3. Perpendicular transversal theorem 

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. 

Perpendicular transversal theorem 

4. Lines perpendicular to a transversal theorem 

In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. 

Lines perpendicular to a transversal theorem 

5. Theorem 

If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. 

parallel
Theorem 1

6. Theorem 

If two lines are perpendicular, then they intersect to form four right angles.  

Theorem 2

7. Theorem 

If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.  

Theorem 3

Exercise

  1. Find m∠1.
Find m∠1.
  1. In the diagram, , (FG) ⊥ (GH) Find the value of
In the diagram, , (FG) ⊥ (GH) Find the value of
  1. Determine which lines, if any, must be parallel. Explain your reasoning.
Determine which lines, if any, must be parallel. Explain your reasoning.
  1. Find all the unknown angle measures in the diagram at the right. Justify your reasoning for each angle measure.
Find all the unknown angle measures in the diagram at the right. Justify your reasoning for each angle measure.
  1. Find all the unknown angle measures in the diagram at the right. Justify your reasoning for each angle measure.
Find all the unknown angle measures in the diagram at the right. Justify your reasoning for each angle measure.

Concept Map

Concept Map: 

What we have learned

  • The distance from a point to a line is the length of the perpendicular segment from the point to the line.
  • The distance between two parallel lines is the length of any perpendicular segment joining the two lines.
  • Theorem: If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
  • Theorem: If two lines are perpendicular, then they intersect to form four right angles.
  • Theorem: If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
parallel

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