Question

1. find the measurement indicated.

find ( mangle dec )
image of a quadrilateral-like figure with points d, c, f, e. angle at d is ( 75^circ ), angle at e (in triangle dec) is ( 7x - 5 ), angle at e (in quadrilateral efc...) is ( 15x ).

Explanation:

Step1: Identify parallel sides property

In parallelogram \(DECF\), \(DF \parallel EC\) and \(DE \parallel FC\), so consecutive angles are supplementary, and \(DE \parallel FC\) implies \(\angle D + \angle DEC + \angle ECF = 180^\circ\)? No, correct: \(EF \parallel DC\), so alternate interior angles? Wait, no: \(DECF\) is a parallelogram, so \(DE \parallel FC\), \(EF \parallel DC\). So \(\angle D\) and \(\angle DEF\) are supplementary:
\(\angle D + \angle DEF = 180^\circ\), where \(\angle DEF = 15x + 7x - 5\)

Step2: Set up supplementary angle equation

\(75 + (15x + 7x - 5) = 180\)
Simplify left side: \(75 -5 +22x = 180\) → \(70 +22x =180\)

Step3: Solve for \(x\)

\(22x = 180 -70 = 110\)
\(x = \frac{110}{22} = 5\)

Step4: Calculate \(m\angle DEC\)

Substitute \(x=5\) into \(7x-5\):
\(7(5)-5 = 35-5=30\)

Answer:

\(30^\circ\)