Question

trapezoid abcd is isosceles, where ( ab = cd ), ( mangle abc = (4x)^circ ), and ( mangle adc = (x + 10)^circ ). find ( mangle dab ).
( mangle dab = )

Explanation:

Step1: Use isosceles trapezoid angle property

In an isosceles trapezoid, base angles are equal, so $\angle ABC = \angle BCD$ and $\angle DAB = \angle ADC$. Also, consecutive angles between the bases are supplementary: $\angle ABC + \angle DAB = 180^\circ$. Substitute $\angle DAB = (x+10)^\circ$ and $\angle ABC = (4x)^\circ$:
$$4x + (x + 10) = 180$$

Step2: Solve for $x$

Combine like terms and isolate $x$:
$$5x + 10 = 180$$
$$5x = 170$$
$$x = 34$$

Step3: Calculate $\angle DAB$

Substitute $x=34$ into $\angle DAB = (x+10)^\circ$:
$$m\angle DAB = 34 + 10$$

Answer:

$44^\circ$