Question

Two congruent angles of an isosceles trapezoid have measures $3\mathrm{x}+10$and $5\mathrm{x}-10$. Find the value of x and then give the measures of all angles of the trapezoid.

Short answer

The value of x is 10 and the measure of other angles is $\mathbf{40}\mathbf{,}\mathbf{}\mathbf{40}\mathbf{,}\mathbf{}\mathbf{140}$and $\mathbf{140}$.

Step-by-step solution

Step 1. Draw and label the diagram.

The figure shows, isosceles trapezoid. Let the measure of angle B be $3x+10$.

Step 2. Apply the property of isosceles trapezoid.

The base angles of an isosceles trapezoid are congruent.

Consider an isosceles trapezoid ABCD such that,

$\begin{array}{l}\angle A\cong \angle D;\\ \angle B\cong \angle C\end{array}$

Therefore$m\angle B=m\angle C$

Step 3. Description of step.

Substitute $3x+10$for $m\angle B$and $5x-10$for $m\angle C$into $m\angle B=m\angle C$.

$\begin{array}{c}3x+10=5x-10\\ 2x=20\\ x=10\end{array}$

Step 4. Description of step.

Substitute 10 for x into $m\angle B=3x+10$

$\begin{array}{c}m\angle B=3\left(10\right)+10\\ =30+10\\ =40\end{array}$

Therefore,$m\angle B=m\angle C=40$

Step 5. Description of step.

The sum of the same side interior angles is 180. Therefore,

$\begin{array}{c}m\angle A+m\angle B=180\\ m\angle A+40=180\\ m\angle A=180-40\\ =140\end{array}$

Step 6. Description of step.

Substitute 140 for $m\angle A$into $m\angle A=m\angle D$to find the value of $m\angle D$.

$m\angle D=140$

Therefore, the value of x is 10 and the measure of other angles is $40,40,140$and $140$.