Question

1. given: \\(\overline{gh}\parallel\overline{ji}\\), \\(i\\) is the midpoint of \\(hk\\) and \\(\overline{gh}\cong\overline{ji}\\)

prove: \\(\angle g\cong\angle j\\)
statements | reasons
--- | ---

1. \\(\overline{gh}\parallel\overline{ji}\\) | 1.
2. \\(i\\) is the midpoint of \\(\overline{hk}\\) | 2.
3. | 3. given
4. \\(\overline{hi}\cong\overline{ik}\\) | 4.
5. | 5. corresponding
6. | 6. sas
7. \\(\angle g\cong\angle j\\) | 7.

1. given: \\(\overline{mp}\cong\overline{np}\\), \\(\overline{mn}\perp\overline{op}\\)

prove: \\(\overline{mo}\cong\overline{on}\\)
statements | reasons
--- | ---

1. | 1. given
2. \\(\overline{mn}\perp\overline{op}\\) | 2.
3. \\(\overline{op}\cong\overline{op}\\) | 3.
4. \\(\triangle mop\cong\triangle nop\\) | 4.
5. | 5.

Explanation:

Step1: Analyze Problem 8

We are given \( \overline{GH} \parallel \overline{JI} \), \( I \) is the midpoint of \( HK \), and \( \overline{GH} \cong \overline{JI} \). We need to prove \( \angle G \cong \angle J \).

Step 1.1: State the reason for \( \overline{GH} \parallel \overline{JI} \)

The reason for \( \overline{GH} \parallel \overline{JI} \) is "Given".

Step 1.2: State the definition of midpoint

The reason for \( I \) being the midpoint of \( HK \) is "Definition of Midpoint".

Step 1.3: Write the given congruence

Statement 3: \( \overline{GH} \cong \overline{JI} \) (Given).

Step 1.4: State the reason for \( \overline{HI} \cong \overline{IK} \)

The reason for \( \overline{HI} \cong \overline{IK} \) is "Definition of Midpoint" (since \( I \) is the midpoint, it divides \( HK \) into two congruent segments).

Step 1.5: Identify the corresponding angles

Since \( \overline{GH} \parallel \overline{JI} \) and \( HK \) is a transversal, \( \angle GHI \cong \angle JIK \) (Corresponding Angles Postulate). So statement 5: \( \angle GHI \cong \angle JIK \).

Step 1.6: Prove triangle congruence

We have \( \overline{GH} \cong \overline{JI} \), \( \angle GHI \cong \angle JIK \), and \( \overline{HI} \cong \overline{IK} \), so by SAS, \( \triangle GHI \cong \triangle JIK \) (statement 6: \( \triangle GHI \cong \triangle JIK \)).

Step 1.7: State the reason for \( \angle G \cong \angle J \)

Corresponding parts of congruent triangles are congruent (CPCTC), so \( \angle G \cong \angle J \) (reason: CPCTC).

Step2: Analyze Problem 9

We are given \( \overline{MP} \cong \overline{NP} \) and \( \overline{MN} \perp \overline{OP} \). We need to prove \( \overline{MO} \cong \overline{ON} \).

Step 2.1: Write the given congruence

Statement 1: \( \overline{MP} \cong \overline{NP} \) (Given).

Step 2.2: State the definition of perpendicular lines

The reason for \( \overline{MN} \perp \overline{OP} \) is "Definition of Perpendicular Lines" (which means \( \angle MOP \) and \( \angle NOP \) are right angles, so \( \angle MOP \cong \angle NOP \)).

Step 2.3: State the reason for \( \overline{OP} \cong \overline{OP} \)

The reason for \( \overline{OP} \cong \overline{OP} \) is "Reflexive Property of Congruence".

Step 2.4: Prove triangle congruence

We have \( \overline{MP} \cong \overline{NP} \), \( \angle MOP \cong \angle NOP \), and \( \overline{OP} \cong \overline{OP} \), so by SAS, \( \triangle MOP \cong \triangle NOP \) (reason: SAS).

Step 2.5: State the corresponding parts

Since \( \triangle MOP \cong \triangle NOP \), by CPCTC, \( \overline{MO} \cong \overline{ON} \) (statement 5: \( \overline{MO} \cong \overline{ON} \), reason: CPCTC).

Answer:

Problem 8:

Statements	Reasons
2. \( I \) is the midpoint of \( \overline{HK} \)	2. Definition of Midpoint
3. \( \overline{GH} \cong \overline{JI} \)	3. Given
4. \( \overline{HI} \cong \overline{IK} \)	4. Definition of Midpoint
5. \( \angle GHI \cong \angle JIK \)	5. Corresponding Angles Postulate
6. \( \triangle GHI \cong \triangle JIK \)	6. SAS
7. \( \angle G \cong \angle J \)	7. CPCTC

Problem 9:

Statements	Reasons
2. \( \overline{MN} \perp \overline{OP} \)	2. Given (or Definition of Perpendicular Lines)
3. \( \overline{OP} \cong \overline{OP} \)	3. Reflexive Property of Congruence
4. \( \triangle MOP \cong \triangle NOP \)	4. SAS ( \( \overline{MP} \cong \overline{NP} \), \( \angle MOP \cong \angle NOP \), \( \overline{OP} \cong \overline{OP} \))
5. \( \overline{MO} \cong \overline{ON} \)	5. CPCTC