Question

in the rectangle below, $ei=4x+6$, $fh=5x+24$, and $m\angle ifg=57^\circ$.
find the value of $x$ and $m\angle igh$.
$x=\square$
$m\angle igh=\square^\circ$

Explanation:

Step1: Use rectangle diagonal property

In a rectangle, diagonals are equal and bisect each other, so $FH = 2 \times EI$.
$$5x + 24 = 2(4x + 6)$$

Step2: Solve for x

Expand and simplify the equation.
$$5x + 24 = 8x + 12$$
$$24 - 12 = 8x - 5x$$
$$12 = 3x$$
$$x = \frac{12}{3} = 4$$

Step3: Find $\angle IGH$

In rectangle, $FG \parallel EH$, $\angle IFG = \angle IHG = 57^\circ$. $\triangle IGH$ is isosceles ($IG=IH$), so base angles are equal. The sum of angles in a triangle is $180^\circ$.
$$m\angle IGH = \frac{180^\circ - 90^\circ}{2}? \text{No, correct: } \angle HGF=90^\circ, \angle IFG=\angle HGI=57^\circ? \text{No, correct: } \angle IFG = \angle EHG=57^\circ, \angle IGH = 90^\circ - 57^\circ = 33^\circ$$
Corrected Step3: In rectangle, $\angle FGH=90^\circ$. $\triangle IFG$ is isosceles, so $\angle IFG=\angle IGF=57^\circ$. Thus $\angle IGH = 90^\circ - 57^\circ = 33^\circ$

Answer:

$x = 4$
$m\angle IGH = 33^\circ$