Perpendicular LinesGeometry (Holt 3-4)

K.Santos

Perpendicular Bisector

Perpendicular bisector of a segment is a line perpendicular to a segment at the segment’s midpoint.

(could be a segment or ray)s

M t

Line s is perpendicular to line t at it’s midpoint M

Distance from a point to a line

The shortest segment from a point to a line is perpendicular to the line.

Distance form a point to a line is the length of the perpendicular segment from the point to the line.

Theorem (3-4-1)

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

Given: <1 and < 2 are a linear pair nThen: m ⊥ n 1 2 m

Perpendicular Transversal Theorem (3-4-2)

In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line.

a bGiven: a||b t ┴a tThen: t ┴b

Theorem (3-4-3) Theorem:

If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other.

Given: a ┴ t a b ┴tThen: a||b

b t

Proof of previous theoremGiven: a ┴ t b ┴t 1 aProve: a||b

2 b

t

Statements Reasons1. a ┴ t, b ┴t 1. Given2. < 1 and < 2 are right angles 2. Definition of perpendicular lines3. <1 <2 3. all right angles are congruent4. a||b 4. If corresponding angles are

congruent then the lines are parallel

Theorem

If two coplanar lines are parallel to the same line, then they are parallel to each other.

t

Given: a ||b a b ||c bThen: a || c

c

Example:Use the picture at the right to answer the questions below:

C P B

x – 8 12

Name the shortest segment from point A to .

Write and solve an inequality for x.AC > APx – 8 > 12x > 20

ExampleGiven the information below what can you conclude about lines a and d?

a || b ab ┴ cc || d ba ___ d?

c dDraw a picture with all the linein it and then make a conclusionabout lines a and d.

a ┴ d

ProofGiven: r||s t <1 <2 1 3 rProve: rs

2 s

Statements Reasons1. r||s 1. given2. <2 <3 2. Corresponding Angles postulate3. <1 <2 3. given4. <1 <3 4. Transitive Property (2, 3)5. rs 5. If two intersecting lines form a

then the lines are perpendicular