4. In \( \triangle \mathrm{MNO}, m=820 \mathrm{~cm}, n=360 \mathrm{~cm} \) and \( o=770 \mathrm{~cm} \). Find the measure of \( \angle \mathrm{M} \) to the nearest 10th of a degree.

We are given triangle MNO with sides of lengths m = 820 cm, n = 360 cm, and o = 770 cm. In the usual notation, side m is opposite angle M, side n is opposite angle N, and side o is opposite angle O. To find the measure of angle M, we use the Law of Cosines:   m² = n² + o² − 2·n·o·cos(M) Rearranging for cos(M), we have:   cos(M) = (n² + o² − m²) / (2 · n · o) Step 1. Calculate the squares of the sides:   n² = 360² = 129600   o² = 770² = 592900   m² = 820² = 672400 Step 2. Substitute into the formula:   cos(M) = (129600 + 592900 − 672400) / (2 · 360 · 770) Step 3. Simplify the numerator:   129600 + 592900 = 722500   Numerator = 722500 − 672400 = 50100 Step 4. Calculate the denominator:   2 · 360 · 770 = 554400 Step 5. Compute cos(M):   cos(M) = 50100 / 554400 ≈ 0.0904 Step 6. Find angle M by taking the inverse cosine:   M ≈ cos⁻¹(0.0904) Using a calculator to find the inverse cosine gives:   M ≈ 84.8° (to the nearest tenth of a degree) Thus, the measure of angle M is approximately 84.8°.