Question

solve the triangle. write each answer as an integer or as a decimal rounded to the nearest tenth. m∠x =

°

m∠y =

°

m∠z =

°

Explanation:

Step1: Use the Law of Cosines to find an angle

The Law of Cosines formula is $c^{2}=a^{2}+b^{2}-2ab\cos C$. To find $\angle X$, we have $z^{2}=x^{2}+y^{2}-2xy\cos X$. Substituting $x = 11$, $y = 13$, $z = 12$:
$12^{2}=11^{2}+13^{2}-2\times11\times13\times\cos X$
$144 = 121+169 - 286\cos X$
$144=290 - 286\cos X$
$286\cos X=290 - 144$
$286\cos X = 146$
$\cos X=\frac{146}{286}\approx0.5105$
$X=\cos^{-1}(0.5105)\approx59.3^{\circ}$

Step2: Use the Law of Cosines to find another angle

To find $\angle Y$, use the Law of Cosines: $x^{2}=y^{2}+z^{2}-2yz\cos Y$. Substituting values:
$11^{2}=13^{2}+12^{2}-2\times13\times12\times\cos Y$
$121 = 169+144 - 312\cos Y$
$121=313 - 312\cos Y$
$312\cos Y=313 - 121$
$312\cos Y = 192$
$\cos Y=\frac{192}{312}\approx0.6154$
$Y=\cos^{-1}(0.6154)\approx52.0^{\circ}$

Step3: Use the angle - sum property of a triangle

Since the sum of the interior angles of a triangle is $180^{\circ}$, we find $\angle Z$ as $Z=180^{\circ}-X - Y$.
$Z = 180-(59.3 + 52.0)$
$Z=68.7^{\circ}$

Answer:

$m\angle X\approx59.3^{\circ}$
$m\angle Y\approx52.0^{\circ}$
$m\angle Z\approx68.7^{\circ}$