Question

1. complete the proof.

given: ( hi = 8 ), ( ij = 8 ), ( overline{ij} cong overline{jh} )
prove: ( overline{hi} cong overline{jh} )

statements	reasons
2. ( ij = 8 )	2.
3. ( hi = ij )	3.
4. ( overline{hi} cong overline{ij} )	4.
5. ( overline{ij} cong overline{jh} )	5.
6. ( overline{hi} cong overline{jh} )	6.

triangle diagram with vertices h, i, j

Explanation:

Step1: Identify Given (HI = 8)

The first statement \( HI = 8 \) is given in the problem. So the reason is "Given".

Step2: Identify Given (IJ = 8)

The second statement \( IJ = 8 \) is also given. Reason: "Given".

Step3: Transitive Property of Equality

Since \( HI = 8 \) and \( IJ = 8 \), by the transitive property of equality (if \( a = c \) and \( b = c \), then \( a = b \)), we get \( HI = IJ \). Reason: "Transitive Property of Equality".

Step4: Definition of Congruent Segments

If two segments have equal length (\( HI = IJ \)), then they are congruent. So \( \overline{HI} \cong \overline{IJ} \). Reason: "Definition of Congruent Segments (if \( AB = CD \), then \( \overline{AB} \cong \overline{CD} \))".

Step5: Given (\( \overline{IJ} \cong \overline{JH} \))

The problem states \( \overline{IJ} \cong \overline{JH} \) (from "Given: \( IJ = 8, \overline{IJ} \cong \overline{JH} \)"). Reason: "Given".

Step6: Transitive Property of Congruence

Since \( \overline{HI} \cong \overline{IJ} \) and \( \overline{IJ} \cong \overline{JH} \), by the transitive property of congruence (if \( \overline{AB} \cong \overline{CD} \) and \( \overline{CD} \cong \overline{EF} \), then \( \overline{AB} \cong \overline{EF} \)), we get \( \overline{HI} \cong \overline{JH} \). Reason: "Transitive Property of Congruence".

Answer:

1. Reason: Given
2. Reason: Given
3. Reason: Transitive Property of Equality
4. Reason: Definition of Congruent Segments
5. Reason: Given
6. Reason: Transitive Property of Congruence