Question

1. find $\angle gjk$

$g$ $(5x + 8)\degree$ $h$

$j$ $(7x - 16)\degree$ $i$

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Explanation:

Step1: Set angles equal (rectangle property)

In a rectangle, opposite sides are parallel, so alternate interior angles $\angle GHJ$ and $\angle JIG$ are equal:
$$5x + 8 = 7x - 16$$

Step2: Solve for $x$

Rearrange to isolate $x$:
$$8 + 16 = 7x - 5x$$
$$24 = 2x$$
$$x = \frac{24}{2} = 12$$

Step3: Find $\angle GHJ$

Substitute $x=12$ into $5x+8$:
$$\angle GHJ = 5(12) + 8 = 60 + 8 = 68^\circ$$

Step4: Identify $\triangle GJK$ properties

In rectangle diagonals, $GJ=IH$ and diagonals bisect each other, so $GK=JK$. Thus $\triangle GJK$ is isosceles, $\angle JGK = \angle GJK$. Also, $\angle JGK = \angle GHJ = 68^\circ$ (alternate interior angles).

Step5: Calculate $\angle GJK$

Sum of angles in a triangle is $180^\circ$. Let $\angle GJK = y$:
$$y + y + \angle GKJ = 180^\circ$$
First, find $\angle GKJ$: it is vertical to the angle formed by diagonals, and $\angle GHJ = 68^\circ$, so $\angle GKJ = 180^\circ - 2(68^\circ) = 44^\circ$ (from $\triangle GHK$).
$$2y + 44^\circ = 180^\circ$$
$$2y = 180^\circ - 44^\circ = 136^\circ$$
$$y = \frac{136^\circ}{2} = 68^\circ$$

Answer:

$68^\circ$