## 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.

#### 2. Shortest distance between two parallel lines

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

#### 3. Perpendicular transversal theorem

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

#### 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.

#### 5. Theorem

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

#### 6. Theorem

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

#### 7. Theorem

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

## Exercise

- Find m∠1.

- In the diagram, , (FG) ⊥ (GH) Find the value of

- Determine which lines, if any, must be parallel. Explain your reasoning.

- 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

## 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.

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