Complete the proof of the Law of Sines/Cosines.Given triangle ABC with altitude segment AD labeled x. Angles ADB and CDB are _____1._____ by the definition of altitudes, making triangle ABD and triangle BCD 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.Answer choices: 1. altitudes2. b ⋅ sinB3. b ⋅ sinB = c ⋅ sinC 1. right angles2. b ⋅ sinB3. b ⋅ sinB =c ⋅ sinB 1. altitudes2. c ⋅ sinB3. c ⋅ sinB = b ⋅ sinC 1. right angles2. c ⋅ sinB3. c ⋅ sinB = b ⋅ sinC

To complete the proof, we need to fill in the blanks correctly. The correct answer is:

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

Using these answers, we can write out the full completed proof of the Law of Sines as applied to this situation:

Given triangle ABC with altitude segment AD labeled x. Angles ADB and CDB are right angles by the definition of altitudes, making triangle ABD and triangle BCD 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 = c ⋅ sinB and x = b ⋅ sinC. We also know that x = x by the reflexive property. By the substitution property, c ⋅ sinB = b ⋅ sinC. Dividing each side of the equation by bc gives: sine of B over b equals sine of C over c. This is consistent with the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

Extra: The Law of Sines is a fundamental relationship in trigonometry that applies to all triangles. It states that the ratio of the length of a side of a triangle to the sine of its opposite angle is equal for all three sides of the triangle. The formula can be written as:

a/sinA = b/sinB = c/sinC

Where, a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the angles opposite those sides, respectively.

This law is very useful for finding unknown side lengths or angles in a triangle when we have some of this information initially given. The Law of Sines can be especially useful for solving oblique triangles, which are triangles that do not have a right angle.

Remember, the Law of Cosines is a separate but related rule that can be used to find a side or angle in a triangle when the Law of Sines cannot be applied directly, typically in cases when two sides and the included angle or three sides are known.

The Law of Cosines states:

c^2 = a^2 + b^2 - 2ab * cosC

By using these two laws, we can solve virtually any triangle problem that does not have a right angle and thereby understand the relationships between the angles and sides of different types of triangles.