Question

write a paragraph proof for the following
given: ( mangle abc = 114 )
( mangle dhe = 25 )
( mangle ehf = 41 )
( angle abc ) and ( angle ghf ) are supplementary.
prove: ( angle dhfcongangle ghf )
by the angle addition postulate, ( mangle dhf=square ). by the definition of supplementary angles, ( angle dhf ) is supplementary to ( square ) because the sum of their measures is ( square ). since ( square ) is also supplementary to ( angle abc ), ( angle dhfcongangle ghf ) by the congruent ( square ) theorem
(do not include the degree symbol in your answer)

Explanation:

Step1: Calculate \(m\angle DHF\)

By the Angle Addition Postulate, \(m\angle DHF=m\angle DHE + m\angle EHF\). Given \(m\angle DHE = 25\) and \(m\angle EHF=41\), so \(m\angle DHF=25 + 41=66\).

Step2: Use the definition of supplementary angles

Since \(\angle ABC\) and \(\angle GHF\) are supplementary, \(m\angle ABC+m\angle GHF = 180\). Given \(m\angle ABC = 114\), then \(m\angle GHF=180 - 114 = 66\). Also, \(m\angle DHF+m\angle ABC=66 + 114=180\), so \(\angle DHF\) is supplementary to \(\angle ABC\)

Step3: Apply the Congruent Supplements Theorem

Since \(\angle DHF\) and \(\angle GHF\) are both supplementary to \(\angle ABC\), by the Congruent Supplements Theorem, \(\angle DHF\cong\angle GHF\)

Answer:

By the Angle Addition Postulate, \(m\angle DHF = 66\). By the definition of supplementary angles, \(\angle DHF\) is supplementary to \(\angle ABC\) because the sum of their measures is \(180\). Since \(\angle GHF\) is also supplementary to \(\angle ABC\), \(\angle DHF\cong\angle GHF\) by the Congruent Supplements Theorem.