Complete the proof of the Law of Sines/Cosines. Given triangle ABC with altitude segment AD labeled x. Angles ADB and CDA are _1._ by the definition of altitudes, making triangle ABD and triangle CDA right triangles. Using the trigonometric ratios sine of B equals x over c and sine of C equals x over b. Multiplying to isolate x in both equations gives x = _2._ and x = b ⋅ sinC. We also know that x = x by the reflexive property. By the substitution property, _3._. Dividing each side of the equation by bc gives: sine of B over b equals sine of C over c.

1. altitudes
2. b ⋅ sinB
3. b ⋅ sinB = c ⋅ sinC

1. right angles
2. b ⋅ sinB
3. b ⋅ sinB =c ⋅ sinB

1. altitudes
2. c ⋅ sinB
3. c ⋅ sinB = b ⋅ sinC

1. right angles
2. c ⋅ sinB
3. c ⋅ sinB = b ⋅ sinC