Proof #4: Two Perpendicular Lines Have Slopes That Are Opposite Reciprocals (Proof by Similar Triangles)

proof goal

Prove that two perpendicular lines have slopes that are opposite reciprocals

given

Two lines l and n intersect at a right angle

intersection labeling

Let A be the point where lines l and n intersect

construction step 1

Draw a horizontal line through point A

construction step 2

Draw vertical line segments from the horizontal line down to each of the two lines

triangle formation

Two right triangles, ΔABC and ΔEDA, are formed

labeling slope for line l

slope of l

labeling slope for line n

slope of n

negative slope reasoning

The negative sign is needed because line n decreases (falls to the right)

proof strategy

Prove the triangles ΔABC and ΔEDA are similar using angle relationships

angle sum theorem

The sum of a triangle’s interior angles equals 180°

angle equation for ΔABC

m∠ABC + m∠BCA + m∠CAB

right angle substitution

∠ABC

simplify equation

m∠BCA + m∠CAB

isolate one angle

m∠BCA

angle equation for ΔEDA

m∠EDA + m∠DAE + m∠EAD

right angle substitution for ΔEDA

∠EDA

simplify again

m∠DAE + m∠EAD

isolate one angle

m∠DAE

compare angle results

m∠BCA

angle equality

Since m∠CAB

congruent angles found

∠BCA ≅ ∠DAE

additional congruence

Both triangles have right angles, ∠ABC ≅ ∠EDA

angle-angle postulate

Triangles ΔABC and ΔEDA are similar by the AA similarity postulate

similar triangle consequence

Corresponding sides of similar triangles are proportional

set up side ratio

BC/AB

cross multiplication step

BC × DE

convert to slope terms

(BC/AB) × (DE/AD)

account for negative direction

(BC/AB) × (−DE/AD)

substitute slope expressions

slope₁ × slope₂

product meaning

The product of slopes of perpendicular lines is −1

final conclusion

Two perpendicular lines have slopes that are opposite reciprocals