Chapter 3.6 Notes: Prove Theorems about Perpendicular

Lines

Goal: You will find the distance between a point and a line.

• Theorem 3.8:

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

• Theorem 3.9

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

Ex.1: In the diagram below, . What can you conclude about and .

• Theorem 3.10:

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

AB BC���������������������������������������� ���

1 2

Ex.2: Prove that if two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

Given:

Prove: and are complementary.

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7 8

Ex.3: Given that , what can you conclude about and ? Explain how you know.

• Theorem 3.11 Perpendicular Transversal Theorem:

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

ABC ABD 3 4

• Theorem 3.12 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.

Ex.4: Determine which lines, if any, must be parallel in the diagram. Explain your reasoning.

Ex.5: Use the diagram below.

a. Is ? Explain your reasoning.

b. Is ? Explain your reasoning.

b ab c

Distance form 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 a line.

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

In the diagram, . Find the value of .RS ST x���������������������������������������� ���

1.

2x + 18 + 36 = 90

2x + 54 = 90

2x = 36

x = 18°

In the diagram, . Find the value of .RS ST x���������������������������������������� ���

1.

3x – 11 + 38 = 90

3x + 27 = 90

3x = 63

x = 21°

38°

Ex.6: The sculpture below is drawn on a graph where units are measured in inches. What is the approximate length of , the depth of a seat? SR

Ex.7: In the figure, and are congruent. What can you conclude about ?

a

1 b

2

1 22m

Ex.8: Use the graph below for (a) and (b).

a. What is the distance from point A to line c?

b. What is the distance from line c to line d?