Question

$▱\mathrm{DECK}$is a parallelogram. Complete.

If$\mathrm{m}\angle 1=3\mathrm{x},\mathrm{m}\angle 2=4\mathrm{x}$ and $\mathrm{m}\angle 3={\mathrm{x}}^2-70$then $\mathrm{x}=?$and$\mathrm{m}\angle \mathrm{CED}=?$ (numerical answers).

Short answer

The value of x and$\angle CED$ is 10 and $70°$respectively.

Step-by-step solution

Step 1. Apply property of parallelogram.

Alternate interior angles of a parallelogram are equal.

From the figure, it can be observed that,

$\begin{array}{c}3x=x^2-70\\ x^2-3x-70=0;\\ 4x=\angle DEK\end{array}$

Step 2. Description of step.

Solve the obtained quadratic equation in step 1 to find the value of x.

$\begin{array}{l}{x}^2-3x-70=0\\ x^2-10x+7x-70=0\\ x\left(x-10\right)+7\left(x-10\right)=0\\ x-10=0\text{or}x+7=0\\ x=10\text{orx}=-7\end{array}$

Step 3. Solve the quadratic equation.

Solving equation $x^2-3x-70=0$gives the roots as, $x=10;x=-7$

Only positive values are to be considered, therefore the acceptable value of x is 10.

Step 4. Solve the quadratic equation.

$\angle CED$can be expressed as $\angle CED=\angle DEK+x^2-70$.

Substitute 4x for $\angle DEK$into $\angle CED=\angle DEK+x^2-70$.

$\angle CED=4x+x^2-70$

Step 5. Description of step.

Substitute 10 for x into $\angle CED=4x+x^2-70$.

$\begin{array}{c}\angle CED=4\left(10\right)+{\left(10\right)}^2-70\\ =40+100-70\\ =70°\end{array}$

Therefore, the value of x and$\angle CED$ is 10 and $70°$respectively.