Question

12 points\
$\triangle lbz$ is a right triangle where $m\angle bzl = 90^\circ$, $bz = 24$, and $\angle b = 51$\

1. determine $m\angle lbz$ using a tangent ratio. show your work. if necessary, round your answer to the nearest thousandth.\

\\(\circ\\) 14.665\
\\(\circ\\) 28.601\
\\(\circ\\) 63.458\
\\(\circ\\) 32.451\
\\(\circ\\) 45.962\
\\(\circ\\) 56.134\
\\(\circ\\) 40\
\\(\circ\\) 48.888\
\\(\circ\\) 27\
\\(\circ\\) 61.928\
\\(\circ\\) 18\
\\(\circ\\) 15.549\
\\(\circ\\) 45\
\\(\circ\\) 12\
\\(\circ\\) 35

Explanation:

Step1: Identify tangent ratio sides

For $\angle LBZ$, $\tan(\angle LBZ) = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{LZ}{BZ}$

Step2: Calculate length of $LZ$

Use Pythagorean theorem: $LZ = \sqrt{LB^2 - BZ^2}$
$LZ = \sqrt{51^2 - 24^2} = \sqrt{2601 - 576} = \sqrt{2025} = 45$

Step3: Compute tangent value

$\tan(\angle LBZ) = \frac{45}{24} = 1.875$

Step4: Find angle via arctangent

$\angle LBZ = \arctan(1.875)$
$\angle LBZ \approx 61.928^\circ$

Answer:

$\boldsymbol{61.928}$