Answer:
To solve this problem, we'll use the properties of parallelograms.
a. Since JADE is a parallelogram, opposite sides are equal in length. Therefore, JA = ED and AD = JE. Setting up equations based on this:
1. ( JA = ED)
\( 3x + 4 = 4x - 10 \)
\( 3x - 4x = -10 - 4 \)
\( -x = -14 \)
\( x = 14 \)
b. Substitute the value of x into the expressions for ED and JE:
For ED:
\( ED = 4x - 10 = 4(14) - 10 = 56 - 10 = 46 \)
For JE:
\( JE = AD = 2x - 3 = 2(14) - 3 = 28 - 3 = 25 \)
So, ED = 46 and JE = 25.
c. Perimeter of JADE:
\( Perimeter = JA + AD + DE + JE \)
\( Perimeter = (3x + 4) + (2x - 3) + (4x - 10) + (2x - 3) \)
\( Perimeter = (3(14) + 4) + (2(14) - 3) + (4(14) - 10) + (2(14) - 3) \)
\( Perimeter = (42 + 4) + (28 - 3) + (56 - 10) + (28 - 3) \)
\( Perimeter = 46 + 25 + 46 + 25 \)
\( Perimeter = 142 \)
So, the perimeter of JADE is 142 units.
2. In a parallelogram, consecutive angles are supplementary, meaning they add up to 180 degrees. Therefore, we can set up an equation using the given angles:
\( (3x - 4) + (3x + 16) = 180 \)
Combine like terms:
\( 6x + 12 = 180 \)
Subtract 12 from both sides:
\( 6x = 168 \)
Divide both sides by 6:
\( x = 28 \)
Now, substitute the value of x back into the expressions for the angles:
For the first angle:
\( 3x - 4 = 3(28) - 4 = 84 - 4 = 80^\circ \)
For the second angle:
\( 3x + 16 = 3(28) + 16 = 84 + 16 = 100^\circ \)
So, the value of \( x \) is 28, and the measures of the angles are \( 80^\circ \) and \( 100^\circ \), respectively.
