Determine the area of the following, in some cases leave the answer in terms of x: (DO NOT MEASURE THE DIAGRAM)

Here are the solutions for the given geometry problem:

1.1 Determine the area of the following:

1.1.1 ABJH ABJH is a square with side length $x$. Step 1: Identify the shape and its dimensions. ABJH is a square with side $s = x$ units. Step 2: Apply the area formula for a square. Area $= s^2 = x \times x = x^2$ units$^2$. $\boxed{*x^2 units^2*}$

1.1.2 BCDJ BCDJ is a rectangle. Step 1: Identify the shape and its dimensions. BCDJ is a rectangle with length $BC = 6$ units and width $BJ = x$ units (since ABJH is a square, $BJ = AB = x$). Step 2: Apply the area formula for a rectangle. Area $= length \times width = BC \times BJ = 6 \times x = 6x$ units$^2$. $\boxed{*6x units^2*}$

1.1.3 DEFJ DEFJ is a rectangle. Step 1: Identify the shape and its dimensions. DEFJ is a rectangle. From the diagram, $JD = BC = 6$ units. The total height $CE = AG = AH + HG = x + 4$ units. Also, $CE = CD + DE$. Since $CD = BJ = x$ units, we have $x + 4 = x + DE$, which means $DE = 4$ units. Step 2: Apply the area formula for a rectangle. Area $= length \times width = JD \times DE = 6 \times 4 = 24$ units$^2$. $\boxed{*24 units^2*}$

1.1.4 HJFG HJFG is a rectangle. Step 1: Identify the shape and its dimensions. HJFG is a rectangle with length $JH = x$ units (since ABJH is a square, $JH = AH = x$) and width $HG = 4$ units. Step 2: Apply the area formula for a rectangle. Area $= length \times width = JH \times HG = x \times 4 = 4x$ units$^2$. $\boxed{*4x units^2*}$

1.2 Write in terms of $x$, the length of:

1.2.1 AC Step 1: Identify the segments that make up AC. The length AC is the sum of segments AB and BC. Step 2: Substitute the given lengths. $AC = AB + BC = x + 6$ units. $\boxed{*(x + 6) units*}$

1.2.2 AG Step 1: Identify the segments that make up AG. The length AG is the sum of segments AH and HG. Step 2: Substitute the given lengths. $AG = AH + HG = x + 4$ units. $\boxed{*(x + 4) units*}$

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