Question

find $m\angle lon$ and $m\angle mln$.
$(4x + 7)^\circ$ $(7x - 35)^\circ$
$m\angle lon =$
number:
$m\angle mln =$
color:
if the perimeter of $rstu$ is 72 and $rv = 16,$

Explanation:

Step1: Set angles equal (rhombus property)

$4x + 7 = 7x - 35$

Step2: Solve for x

$7 + 35 = 7x - 4x$
$42 = 3x$
$x = \frac{42}{3} = 14$

Step3: Calculate $\angle LMN$

$m\angle LMN = (4(14)+7)^\circ + (7(14)-35)^\circ = 63^\circ + 63^\circ = 126^\circ$

Step4: Find $m\angle LON$ (rhombus opposite angles equal)

$m\angle LON = m\angle LMN = 126^\circ$

Step5: Find $m\angle MLN$ (rhombus consecutive angles supplementary, bisected)

First, $m\angle MLN = \frac{180^\circ - m\angle LMN}{2} = \frac{180^\circ - 126^\circ}{2} = 27^\circ$

Answer:

$m\angle LON = 126$
$m\angle MLN = 27$