Solve triangle ABC given the following conditions:

1. C = 135.77 deg, b = 10cm, a = 12cm
2. a = 11.5m, b = 10m, c = 16.94m

in each case, use law of cosines to get c or C:

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

Now, having all 3 sides, and angle C, use law of sines to get A and B:

sinA/a = sinB/b = sinC/c

To solve triangle ABC, we will use the Law of Sines and the Law of Cosines. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides of a triangle. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.

Let's solve the first triangle with the given conditions:

1. C = 135.77 deg, b = 10 cm, a = 12 cm

First, we can use the Law of Sines to find angle A. The Law of Sines states:

a / sin(A) = b / sin(B) = c / sin(C)

In this case, we have:

12 cm / sin(A) = 10 cm / sin(B)

We can rearrange the equation to solve for sin(A):

sin(A) = (12 cm * sin(B)) / 10 cm

Next, we can use the Law of Cosines to find side c. The Law of Cosines states:

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

In this case, we have:

c^2 = 12 cm^2 + 10 cm^2 - 2 * 12 cm * 10 cm * cos(135.77 deg)

We can solve this equation to find c.

Now, let's solve the second triangle with the given conditions:

2. a = 11.5 m, b = 10 m, c = 16.94 m

First, let's use the Law of Cosines to find angle C. The Law of Cosines states:

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

In this case, we have:

cos(C) = (11.5 m^2 + 10 m^2 - 16.94 m^2) / (2 * 11.5 m * 10 m)

We can solve this equation to find cos(C).

Next, we can use the Law of Sines to find angle A. We can use the same formula as in the previous triangle and solve it for sin(A):

sin(A) = (a * sin(C)) / c

In this case, we have:

sin(A) = (11.5 m * sin(C)) / 16.94 m

We can solve this equation to find sin(A).

I hope this explanation helps you understand how to solve triangles using the given conditions!