Question

prove: pl = an
statements reasons

1. 1.
2. pl + la = pa 2.
3. la + an = ln 3.
4. pl + la = la + an 4.
5. 5.
6. given: ∠2 ≅ ∠4

prove: ∠6 ≅ ∠8
statements reasons

1. ∠2 ≅ ∠4 1.
2. if two angles are congruent, then they have the same measure. (definition of congruent angles) 2.
3. definition of linear pair 3.
4. ∠2 and ∠6 are supplementary; ∠4 and ∠8 are supplementary 4.
5. m∠2 + m∠6 = 180°; m∠4 + m∠8 = 180° 5. if two angles are supplementary, then their measures add to 180°. (definition of supplementary)
6. m∠4 + m∠6 = 180° 6.
7. m∠4 + m∠6 = m∠4 + m∠8 7.
8. subtraction property of equality 8.
9. ∠6 ≅ ∠8 9.

Explanation:

Step 1: First proof - Statements and Reasons

1. Statements: Assume some initial given information (not shown in the problem - could be related to the line - segment relationships). Reasons: Given (not explicitly stated in the problem - assumed).
2. Statements: $PL + LA=PA$. Reasons: Segment - Addition Postulate (if a point $L$ lies on a line - segment $\overline{PA}$, then the sum of the lengths of the two sub - segments $\overline{PL}$ and $\overline{LA}$ is equal to the length of the whole segment $\overline{PA}$).
3. Statements: $LA + AN=LN$. Reasons: Segment - Addition Postulate (if a point $A$ lies on a line - segment $\overline{LN}$, then the sum of the lengths of the two sub - segments $\overline{LA}$ and $\overline{AN}$ is equal to the length of the whole segment $\overline{LN}$).
4. Statements: $PL + LA=LA + AN$. Reasons: Transitive Property of Equality (if $a = b$ and $b = c$, then $a = c$. Here, assume some equalities that lead to this step, for example, if $PA=LN$ and using the previous two segment - addition equations).
5. Statements: $PL = AN$. Reasons: Subtraction Property of Equality (subtract $LA$ from both sides of the equation $PL + LA=LA + AN$).

Step 2: Second proof - Statements and Reasons

1. Statements: $\angle2\cong\angle4$. Reasons: Given.
2. Statements: $m\angle2 = m\angle4$. Reasons: If two angles are congruent, then they have the same measure (Definition of congruent angles).
3. Statements: $\angle2$ and $\angle6$ form a linear pair; $\angle4$ and $\angle8$ form a linear pair. Reasons: Definition of linear pair (two adjacent angles whose non - common sides are opposite rays).
4. Statements: $\angle2$ and $\angle6$ are supplementary; $\angle4$ and $\angle8$ are supplementary. Reasons: If two angles form a linear pair, then they are supplementary.
5. Statements: $m\angle2 + m\angle6=180^{\circ}$; $m\angle4 + m\angle8 = 180^{\circ}$. Reasons: If two angles are supplementary, then their measures add to $180^{\circ}$ (Definition of supplementary angles).
6. Statements: $m\angle4 + m\angle6=180^{\circ}$. Reasons: Substitution Property of Equality (since $m\angle2 = m\angle4$, substitute $m\angle2$ with $m\angle4$ in $m\angle2 + m\angle6=180^{\circ}$).
7. Statements: $m\angle4 + m\angle6=m\angle4 + m\angle8$. Reasons: Transitive Property of Equality (since $m\angle4 + m\angle6=180^{\circ}$ and $m\angle4 + m\angle8 = 180^{\circ}$).
8. Statements: $m\angle6=m\angle8$. Reasons: Subtraction Property of Equality (subtract $m\angle4$ from both sides of $m\angle4 + m\angle6=m\angle4 + m\angle8$).
9. Statements: $\angle6\cong\angle8$. Reasons: If two angles have the same measure, then they are congruent (Definition of congruent angles).

Answer:

1. For the first proof:

• Statements:
• 1. (Initial given, not shown)
• 2. $PL + LA=PA$
• 3. $LA + AN=LN$
• 4. $PL + LA=LA + AN$
• 5. $PL = AN$
• Reasons:
• 1. Given (assumed)
• 2. Segment - Addition Postulate
• 3. Segment - Addition Postulate
• 4. Transitive Property of Equality
• 5. Subtraction Property of Equality

1. For the second proof:

• Statements:
• 1. $\angle2\cong\angle4$
• 2. $m\angle2 = m\angle4$
• 3. $\angle2$ and $\angle6$ form a linear pair; $\angle4$ and $\angle8$ form a linear pair
• 4. $\angle2$ and $\angle6$ are supplementary; $\angle4$ and $\angle8$ are supplementary
• 5. $m\angle2 + m\angle6=180^{\circ}$; $m\angle4 + m\angle8 = 180^{\circ}$
• 6. $m\angle4 + m\angle6=180^{\circ}$
• 7. $m\angle4 + m\angle6=m\angle4 + m\angle8$
• 8. $m\angle6=m\angle8$
• 9. $\angle6\cong\angle8$
• Reasons:
• 1. Given
• 2. Definition of congruent angles
• 3. Definition of linear pair
• 4. If two angles form a linear pair, then they are supplementary
• 5. Definition of supplementary angles
• 6. Substitution Property of Equality
• 7. Transitive Property of Equality
• 8. Subtraction Property of Equality
• 9. Definition of congruent angles