Question

1. find the stated lengths and angle measures of rhombus fghj. round to the nearest tenth if necessary. gj = __ jk = ∠hfj = ∠hjf = __

Explanation:

Step1: Solve for x (diagonal bisects)

In a rhombus, diagonals bisect each other, so $3x + 1 = x + 9$
Subtract $x$ from both sides: $2x + 1 = 9$
Subtract 1: $2x = 8$
Divide by 2: $x = 4$

Step2: Calculate JK length

Substitute $x=4$ into $x+9$:
$JK = x + 9 = 4 + 9 = 13$

Step3: Calculate GJ length

GJ is twice JK (diagonals bisect):
$GJ = 2 \times JK = 2 \times 13 = 26$

Step4: Solve for y (alt. int. angles)

$\angle G = \angle H$ (alternate interior angles, since $FG \parallel HJ$), so $y + 4 = 4y - 8$
Subtract $y$: $4 = 3y - 8$
Add 8: $12 = 3y$
Divide by 3: $y = 4$

Step5: Find $\angle HJF$

Substitute $y=4$ into $y+4$:
$\angle HJF = (y + 4)^\circ = (4 + 4)^\circ = 8^\circ$

Step6: Find $\angle HFJ$

In $\triangle HJF$, $\angle H = 90^\circ$ (given right angle), so:
$\angle HFJ = 180^\circ - 90^\circ - 8^\circ = 82^\circ$

Answer:

$GJ = 26$
$JK = 13$
$\angle HFJ = 82^\circ$
$\angle HJF = 8^\circ$