Complete the proof of the Law of Sines/Cosines.

Given triangle ABC with altitude segment BD labeled x. Angles ADB and CDB are right angles by 1., making triangle ABD and triangle BCD right triangles. Using the trigonometric ratios sine of A equals x over c and sine of C equals x over a. Multiplying to isolate x in both equations gives x = 2. and x = a ⋅ sinC. We also know that x = x by the reflexive property. By the substitution property, 3.. Dividing each side of the equation by ac gives: sine of A over a equals sine of C over c.

1. definition of altitude

2. c ⋅ sinA

3. c ⋅ sinA = a ⋅ sinC

1. definition of right triangles

2. c ⋅ sinB

3. c ⋅ sinB = a ⋅ sinC

1. definition of right triangles

2. a ⋅ sinA

3. a ⋅ sinA = c ⋅ sinC

1. definition of altitude

2. c ⋅ sinA

3. a ⋅ sinA = c ⋅ sinC