First, we should assign an angle to each side. A, B, and C will be the angles opposite the sides in increasing order. That means a = 10, b = 11, and c = 12. We'll find A first. a2 = b2 + c2 – 2bc cos A To minimize confusion, we'll rearrange the equation before plugging things in. ![](https://media1.shmoop.com/images/geometry/geo_6_sec6_latek_8.png)
Wonderful. Now we're ready to plug in our sides. ![](https://media1.shmoop.com/images/geometry/geo_6_sec6_latek_9.png)
Our calculator can handle the rest. A ≈ 51.3° Noice. Now we have an angle-side ratio, so we'll make use of the Law of Sines for the rest. ![](https://media1.shmoop.com/images/geometry/geo_6_sec6_latek_10.png)
We just calculated A and we want to calculate B. ![](https://media1.shmoop.com/images/geometry/geo_6_sec6_latek_11.png)
Rearrange the equation to solve for B. ![](https://media1.shmoop.com/images/geometry/geo_6_sec6_latek_12.png)
Calculate the angle. B ≈ 59.1° Sweet. Two down, one to go. A + B + C = 180° 51.3° + 59.1° + C = 180° For the last angle, we get: C = 69.6° That means the angles of the triangle are 51.3°, 59.1°, and 69.6°. |