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