Question

$\angle j$ and $\angle h$ are supplementary and $\angle h$ and $\angle i$ are supplementary. complete the proof that $\angle j \cong \angle i$.

	statement	reason
2	$\angle h$ and $\angle i$ are supplementary
3	$m\angle j + m\angle h = 180^\circ$
4	$m\angle h + m\angle i = 180^\circ$
5	$m\angle j + m\angle h = m\angle h + m\angle i$
6	$m\angle j = m\angle i$
7	$\angle j \cong \angle i$

Explanation:

Step1: Identify Given (Statement 1)

The problem states \( \angle J \) and \( \angle H \) are supplementary, so this is given.

Step2: Identify Given (Statement 2)

The problem states \( \angle H \) and \( \angle I \) are supplementary, so this is given.

Step3: Definition of Supplementary (Statement 3)

By the definition of supplementary angles, if two angles are supplementary, their measures sum to \( 180^\circ \). So \( m\angle J + m\angle H = 180^\circ \) (from Statement 1).

Step4: Definition of Supplementary (Statement 4)

By the definition of supplementary angles, \( m\angle H + m\angle I = 180^\circ \) (from Statement 2).

Step5: Transitive Property (Statement 5)

Since \( m\angle J + m\angle H = 180^\circ \) and \( m\angle H + m\angle I = 180^\circ \), we can set them equal: \( m\angle J + m\angle H = m\angle H + m\angle I \) (transitive property of equality, as both equal \( 180^\circ \)).

Step6: Subtraction Property (Statement 6)

Subtract \( m\angle H \) from both sides of \( m\angle J + m\angle H = m\angle H + m\angle I \). By the subtraction property of equality, \( m\angle J = m\angle I \).

Step7: Definition of Congruent (Statement 7)

If two angles have equal measures, they are congruent. So \( \angle J \cong \angle I \) (definition of congruent angles).

Answer:

Statement	Reason
2. \( \angle H \) and \( \angle I \) are supplementary	Given
3. \( m\angle J + m\angle H = 180^\circ \)	Definition of supplementary angles
4. \( m\angle H + m\angle I = 180^\circ \)	Definition of supplementary angles
5. \( m\angle J + m\angle H = m\angle H + m\angle I \)	Transitive property of equality (both equal \( 180^\circ \))
6. \( m\angle J = m\angle I \)	Subtraction property of equality (subtract \( m\angle H \) from both sides)
7. \( \angle J \cong \angle I \)	Definition of congruent angles (equal measures imply congruence)