Question

List the angle measures of GHI in order from smallest to largest, assume S is a positive number

Answer 1

Angle G (43.4 degrees) is the smallest, followed by angle H (93 degrees), and lastly angle I (20.5 degrees), based on the Triangle Inequality and Law of Cosines.

Since HI is the longest side of triangle GHI, angle G will be the smallest angle, and angle I will be the largest angle, by the Angle-Side-Angle Inequality.

Triangle GHI with angle measures

We can use the Law of Cosines to find the measures of angles G and I. The Law of Cosines states that:

cos(C) =
`(a^2 + b^2 - c^2)` / 2ab

where C is the angle opposite side c, and a and b are the other two sides of the triangle.

In this case, we want to find angle G, so we will use the Law of Cosines to find cos(G). We know that HI = 42, GH = 69, and GI = 82.

cos(G) =
`(69^2 + 82^2 - 42^2)` / 2 * 69 * 82

cos(G) = 0.720

Using the inverse cosine function, we can find that the measure of angle G is approximately 43.4 degrees.

We can use the same method to find the measure of angle I.

cos(I) =
`(42^2 + 82^2 - 69^2)` / 2 * 42 * 82

cos(I) = 0.942

Using the inverse cosine function, we can find that the measure of angle I is approximately 20.5 degrees.

Therefore, the angle measures of GHI in order from smallest to largest are:

Angle G (43.4 degrees)

Angle H (93 degrees)

Angle I (20.5 degrees)