Question

practice 1 *

1. complete the proof.

statements reasons

1. ( hi = 8 ) 1.
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.

given: ( hi = 8, ij = 8, overline{ij} cong overline{jh} )
prove: ( overline{hi} cong overline{jh} )
triangle diagram with vertices h, i, j
reason options: given, def. of congruent, seg. add. post., angle add. post., add. prop, subt. prop, relex. prop, symm. prop, trans. prop, ... with circular selection buttons for 1–4 (and more below)

Explanation:

Step1: Analyze Statement 1

The statement \( HI = 8 \) is given in the problem, so the reason is "given".

Step2: Analyze Statement 2

The statement \( IJ = 8 \) is also given in the problem, so the reason is "given".

Step3: Analyze Statement 3

Since \( HI = 8 \) and \( IJ = 8 \), by the transitive property of equality (if \( a = b \) and \( b = c \), then \( a = c \), here \( a = HI \), \( b = 8 \), \( c = IJ \)), we get \( HI = IJ \). Wait, actually, since both \( HI \) and \( IJ \) equal 8, we can say \( HI = IJ \) by substitution (or transitive property of equality). But more accurately, since \( HI = 8 \) (statement 1) and \( IJ = 8 \) (statement 2), then \( HI = IJ \) by the transitive property of equality (or substitution, as both equal 8).

Step4: Analyze Statement 4

If \( HI = IJ \), then by the definition of congruent segments (segments with equal length are congruent), \( \overline{HI} \cong \overline{IJ} \).

Step5: Analyze Statement 5

We are given that \( \overline{IJ} \cong \overline{JH} \), so this is a given? Wait, no, the given is \( IJ = 8 \), \( HI = 8 \), and \( \overline{IJ} \cong \overline{JH} \)? Wait, the original given is "Given: \( HI = 8 \), \( IJ = 8 \), \( \overline{IJ} \cong \overline{JH} \)". So statement 5: \( \overline{IJ} \cong \overline{JH} \) is given.

Step6: Analyze Statement 6

Since \( \overline{HI} \cong \overline{IJ} \) (statement 4) and \( \overline{IJ} \cong \overline{JH} \) (statement 5), by the transitive property of congruence (if \( \overline{a} \cong \overline{b} \) and \( \overline{b} \cong \overline{c} \), then \( \overline{a} \cong \overline{c} \)), we get \( \overline{HI} \cong \overline{JH} \).

But let's focus on the first few steps as per the question (maybe up to step 4 as the image shows up to 4 in the multiple-choice).

For step 1 (statement \( HI = 8 \)):

Step1: Identify Reason for \( HI = 8 \)

The value \( HI = 8 \) is provided in the "Given" section, so the reason is "given".
<Reason: given>

For step 2 (statement \( IJ = 8 \)):

Step2: Identify Reason for \( IJ = 8 \)

The value \( IJ = 8 \) is also provided in the "Given" section, so the reason is "given".
<Reason: given>

For step 3 (statement \( HI = IJ \)):

Step3: Identify Reason for \( HI = IJ \)

Since \( HI = 8 \) (from step 1) and \( IJ = 8 \) (from step 2), by the transitive property of equality (or substitution, as both equal 8), we conclude \( HI = IJ \). The transitive property of equality states that if \( a = b \) and \( b = c \), then \( a = c \). Here \( a = HI \), \( b = 8 \), \( c = IJ \), so \( HI = IJ \).
<Reason: trans. prop (of equality) or substitution>

For step 4 (statement \( \overline{HI} \cong \overline{IJ} \)):

Step4: Identify Reason for \( \overline{HI} \cong \overline{IJ} \)

If two segments have equal length (\( HI = IJ \)), then by the definition of congruent segments, the segments are congruent. The definition of congruent segments is that two segments are congruent if and only if their lengths are equal.
<Reason: def. of congruent (segments)>

Answer:

1. Reason for \( HI = 8 \): given
2. Reason for \( IJ = 8 \): given
3. Reason for \( HI = IJ \): transitive property of equality (or substitution)
4. Reason for \( \overline{HI} \cong \overline{IJ} \): definition of congruent segments

(For the multiple - choice part in the image, for step 1, the "given" option should be selected; for step 2, "given" should be selected; for step 3, "trans. prop" (transitive property) or related equality property; for step 4, "def. of congruent".)