Question
Write a paragraph proof for the following.
Given:
mangleABCequals114
mangleDHEequals25
mangleEHFequals41
angleABC and angleGHF are supplementary.
Upper A
Upper B
Upper C
Upper D
Upper E
Upper F
Upper G
Upper H
Prove:
angleDHFcongruentangleGHF
Question content area bottom
Part 1
By the Angle Addition Postulate, mangleDHFequals
enter your response here. By the definition of supplementary angles, angleDHF is supplementary to
▼
angle EHF
angle DHE
angle DHG
angle ABC
because the sum of their measures is
enter your response here. Since
▼
angle EHF
angle GHF
angle DHG
angle DHE
is also supplementary to angleABC, angleDHFcongruentangleGHF by the Congruent
▼
Supplements
Complements
Theorem.
(Do not include the degree symbol in your answer.)
Given:
mangleABCequals114
mangleDHEequals25
mangleEHFequals41
angleABC and angleGHF are supplementary.
Upper A
Upper B
Upper C
Upper D
Upper E
Upper F
Upper G
Upper H
Prove:
angleDHFcongruentangleGHF
Question content area bottom
Part 1
By the Angle Addition Postulate, mangleDHFequals
enter your response here. By the definition of supplementary angles, angleDHF is supplementary to
▼
angle EHF
angle DHE
angle DHG
angle ABC
because the sum of their measures is
enter your response here. Since
▼
angle EHF
angle GHF
angle DHG
angle DHE
is also supplementary to angleABC, angleDHFcongruentangleGHF by the Congruent
▼
Supplements
Complements
Theorem.
(Do not include the degree symbol in your answer.)
Answers
GPT-4o mini
To prove that \( \angle DHF \cong \angle GHF \), we start with the given measures: \( m\angle ABC = 114^\circ \), \( m\angle DHE = 25^\circ \), and \( m\angle EHF = 41^\circ \). By the Angle Addition Postulate, we can find \( m\angle DHF \) as follows:
\[
m\angle DHF = m\angle DHE + m\angle EHF = 25 + 41 = 66^\circ.
\]
Next, since \( \angle ABC \) and \( \angle GHF \) are supplementary, we have:
\[
m\angle ABC + m\angle GHF = 180^\circ.
\]
Substituting the value of \( m\angle ABC \):
\[
114 + m\angle GHF = 180 \implies m\angle GHF = 180 - 114 = 66^\circ.
\]
Now we see that \( m\angle DHF = 66^\circ \) and \( m\angle GHF = 66^\circ \). Thus, we conclude that \( \angle DHF \) and \( \angle GHF \) have equal measures. By the Congruent Supplements Theorem, since both angles are supplementary to the same angle \( \angle ABC \), it follows that:
\[
\angle DHF \cong \angle GHF.
\]
Therefore, we have proven that \( \angle DHF \cong \angle GHF \).
\[
m\angle DHF = m\angle DHE + m\angle EHF = 25 + 41 = 66^\circ.
\]
Next, since \( \angle ABC \) and \( \angle GHF \) are supplementary, we have:
\[
m\angle ABC + m\angle GHF = 180^\circ.
\]
Substituting the value of \( m\angle ABC \):
\[
114 + m\angle GHF = 180 \implies m\angle GHF = 180 - 114 = 66^\circ.
\]
Now we see that \( m\angle DHF = 66^\circ \) and \( m\angle GHF = 66^\circ \). Thus, we conclude that \( \angle DHF \) and \( \angle GHF \) have equal measures. By the Congruent Supplements Theorem, since both angles are supplementary to the same angle \( \angle ABC \), it follows that:
\[
\angle DHF \cong \angle GHF.
\]
Therefore, we have proven that \( \angle DHF \cong \angle GHF \).