Question

solve for the following using the theorems/properties of a isosceles trapezoid.
11 m

ne=

m
if jn = 31 m, then oe=

m
if m∠jon = 57°, then m∠one=

°, and m∠nej=

Explanation:

Step1: Use the property of non - parallel sides of isosceles trapezoid

In an isosceles trapezoid \(OJEN\), the non - parallel sides \(OJ\) and \(NE\) are equal. Given \(OJ = 11m\), so \(NE=11m\).

Step2: Use the property of diagonals of isosceles trapezoid

In an isosceles trapezoid, the diagonals are equal. So \(JN = OE\). Given \(JN = 31m\), then \(OE = 31m\).

Step3: Use the property of base angles and supplementary angles

Since \(OJEN\) is an isosceles trapezoid, \(ON\parallel JE\). \(\angle JON\) and \(\angle OJE\) are supplementary (\(\angle JON+\angle OJE = 180^{\circ}\)), but we use the property of triangle \(ONJ\) (isosceles triangle as \(OJ = NE\) and \(OJEN\) is isosceles trapezoid, \(ON\) is common side). In \(\triangle ONJ\), \(\angle JON = 57^{\circ}\), and \(\triangle ONJ\cong\triangle NEO\) (by SSS, \(OJ = NE\), \(ON=ON\), \(JN = OE\)). \(\angle ONE=\angle OJN\). In \(\triangle OJN\), \(OJ = NE\) (isosceles trapezoid non - parallel sides), let's use the property of parallel lines \(ON\parallel JE\). \(\angle JON\) and \(\angle OJE\) are supplementary. Also, in \(\triangle OJN\), \(OJ = NE\) (isosceles trapezoid non - parallel sides). Since \(OJEN\) is isosceles trapezoid, \(\angle ONE=\frac{180^{\circ}-\angle JON}{2}\) (base angles of isosceles triangle \(\triangle OJN\) where \(OJ = NE\) and \(OJEN\) is isosceles trapezoid). \(\angle ONE = 61.5^{\circ}\). And \(\angle NEJ = 180^{\circ}-\angle JON=123^{\circ}\) (because \(ON\parallel JE\), consecutive interior angles are supplementary).

Answer:

\(NE = 11m\), \(OE = 31m\), \(m\angle ONE=61.5^{\circ}\), \(m\angle NEJ = 123^{\circ}\)