Questions: Find m angle E angle D=angle F 21x+1=22x-2 Find m angle K angle M=L h

Transcript text: Find $m \angle E$ $\angle D=\angle F$ $21 x+1=22 x-2$ Find $m \angle K$ $\angle M=L h$

Solution

Solution Steps

Step 1: Set up the equation for parallelogram FDEG

In a parallelogram, opposite sides are congruent. Therefore, $GD = FE$. We are given $GD = 21x + 1$ and $FE = 22x - 2$. So, $21x + 1 = 22x - 2$.

Step 2: Solve for x in parallelogram FDEG

Subtract $21x$ from both sides: $1 = x - 2$ Add 2 to both sides: $x = 3$

Step 3: Set up the equation for parallelogram JKLM

In a parallelogram, opposite sides are congruent. Therefore, $ML = JK$. We are given $ML = 15x + 5$ and $JK = 14x + 11$. So, $15x + 5 = 14x + 11$.

Step 4: Solve for x in parallelogram JKLM

Subtract $14x$ from both sides: $x + 5 = 11$ Subtract 5 from both sides: $x = 6$

Step 5: Find the measure of angle E

Consecutive angles in a parallelogram are supplementary, meaning they add up to 180 degrees. Thus, $m\angle E + m\angle F = 180^\circ$. $m\angle F = 22x - 2$ Substitute $x = 3$: $m\angle F = 22(3) - 2 = 66 - 2 = 64^\circ$ Therefore, $m\angle E = 180^\circ - 64^\circ = 116^\circ$

Step 6: Find the measure of angle K

Consecutive angles in a parallelogram are supplementary, meaning they add up to 180 degrees. $m\angle K + m\angle J = 180^{\circ}$. We have $m\angle J = 15x + 5$. Substitute $x = 6$: $m\angle J = 15(6) + 5 = 90 + 5 = 95^{\circ}$ Therefore, $m\angle K = 180^{\circ} - 95^{\circ} = 85^{\circ}$

Final Answer

$\boxed{m\angle E = 116^\circ}$ $\boxed{m\angle K = 85^\circ}$

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