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 dhf cong angle 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 dhf cong angle ghf ) by the congruent ( square ) theorem.
(do not include the degree symbol in your answer.)

Explanation:

Step1: Calculate m∠DHF

By the Angle Addition Postulate, \( m\angle DHF = m\angle DHE + m\angle EHF \). Given \( m\angle DHE = 25 \) and \( m\angle EHF = 41 \), we substitute these values: \( m\angle DHF = 25 + 41 = 66 \).

Step2: Determine supplementary angle to ∠DHF

We know that supplementary angles sum to 180. Let's find what angle is supplementary to ∠DHF. First, find the supplement of ∠DHF: \( 180 - 66 = 114 \). Notice that \( m\angle ABC = 114 \), so ∠DHF is supplementary to ∠ABC (since \( 66 + 114 = 180 \)).

Step3: Use Congruent Supplements Theorem

Given that ∠ABC and ∠GHF are supplementary (so \( m\angle ABC + m\angle GHF = 180 \)), and we just found that ∠DHF is also supplementary to ∠ABC ( \( m\angle DHF + m\angle ABC = 180 \) ). By the Congruent Supplements Theorem, if two angles are supplementary to the same angle, then they are congruent. So ∠DHF ≅ ∠GHF.

Filling in the blanks:

• First blank (m∠DHF): 66 (from Step1)
• Second blank (angle supplementary to ∠DHF): ∠ABC (from Step2)
• Third blank (sum of measures): 180 (since supplementary angles sum to 180)
• Fourth blank (angle supplementary to ∠ABC): ∠GHF (given)
• Fifth blank (Theorem): Supplements (Congruent Supplements Theorem)

Answer:

First blank: 66
Second blank: ∠ABC
Third blank: 180
Fourth blank: ∠GHF
Fifth blank: Supplements