Complete the proof of the Law of Sines/Cosines. Triangle ABC with side c between points A and B, side a between points B and C. Segment drawn from point B to point D where D is between points A and C, segment BD is labeled x. 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