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 segment lengths

First, find $JK = x + 9 = 4 + 9 = 13$
$GJ = 2 \times JK = 2 \times 13 = 26$

Step3: Solve for y (consecutive angles supplementary)

Consecutive angles in a rhombus are supplementary: $(y + 4) + (4y - 8) = 180$
Simplify: $5y - 4 = 180$
Add 4: $5y = 184$
Divide by 5: $y = 36.8$

Step4: Find $\angle HFJ$ (alternate interior angles)

$\angle HFJ = \angle HGF = y + 4 = 36.8 + 4 = 40.8^\circ$

Step5: Find $\angle HJF$ (triangle angle sum)

In $\triangle HJF$, $\angle JHF = 90^\circ$ (diagonals of rhombus are perpendicular)
$\angle HJF = 180 - 90 - 40.8 = 49.2^\circ$

Answer:

$GJ = 26$
$JK = 13$
$\angle HFJ = 40.8^\circ$
$\angle HJF = 49.2^\circ$