Question

3.4 cw
name

1. write a two - column proof:

given: ( mangle3 = 62^{circ},mparallel n )
prove: ( mangle5 = 118^{circ} )

1. write a two - column proof:

given: ( mparallel n,mangle6 = 47^{circ} ), and
prove: ( mangle8 = 133^{circ} )

1. finish the two - column proof below:

given: ( nparallel p,mangle5 = 62^{circ} )
prove: ( mangle7 = 118^{circ} )
statement

1. ( nparallel p,mangle5 = 62^{circ} )
2. ( angle5 suppangle7 )
3. ( mangle5 + mangle7 = 180 )

4.

1. ( mangle7 = 118^{circ} )

reason
1.

1. ( parallel ) lines ( \to )

3.

1. substitution

5.

Explanation:

1. For the first problem (Prove \(m\angle5 = 118^{\circ}\))

Step1: Use the property of corresponding angles

Since \(m\parallel n\), \(\angle3\) and \(\angle4\) are corresponding angles. So, \(m\angle4=m\angle3 = 62^{\circ}\) (Corresponding angles postulate).

Step2: Use the linear - pair property

\(\angle4\) and \(\angle5\) form a linear pair. By the linear - pair postulate, \(m\angle4 + m\angle5=180^{\circ}\).

Step3: Substitute and solve

Substitute \(m\angle4 = 62^{\circ}\) into \(m\angle4 + m\angle5=180^{\circ}\). We get \(62^{\circ}+m\angle5 = 180^{\circ}\). Then \(m\angle5=180^{\circ}-62^{\circ}=118^{\circ}\) (Subtraction property of equality).

2. For the second problem (Prove \(m\angle8 = 133^{\circ}\))

Step1: Use the property of corresponding angles

Since \(m\parallel n\), \(\angle6\) and \(\angle7\) are corresponding angles. So, \(m\angle7=m\angle6 = 47^{\circ}\) (Corresponding angles postulate).

Step2: Use the linear - pair property

\(\angle7\) and \(\angle8\) form a linear pair. By the linear - pair postulate, \(m\angle7 + m\angle8=180^{\circ}\).

Step3: Substitute and solve

Substitute \(m\angle7 = 47^{\circ}\) into \(m\angle7 + m\angle8=180^{\circ}\). We get \(47^{\circ}+m\angle8 = 180^{\circ}\). Then \(m\angle8=180^{\circ}-47^{\circ}=133^{\circ}\) (Subtraction property of equality).

3. For the third problem (Finish the proof)

Step1: Given

The first statement \(n\parallel p,m\angle5 = 62^{\circ}\) is given.

Step2: Consecutive interior angles

The reason for \(\angle5\) supp \(\angle7\) ( \(\angle5\) and \(\angle7\) are supplementary) is that if two parallel lines (\(n\parallel p\)) are cut by a transversal, then consecutive interior angles are supplementary.

Step3: Definition of supplementary angles

The reason for \(m\angle5 + m\angle7=180\) is the definition of supplementary angles (\(\angle A\) and \(\angle B\) are supplementary if \(m\angle A + m\angle B = 180^{\circ}\)).

Step4: Substitute

Substitute \(m\angle5 = 62^{\circ}\) into \(m\angle5 + m\angle7=180\). So, \(62^{\circ}+m\angle7 = 180^{\circ}\).

Step5: Solve for \(m\angle7\)

The reason for \(m\angle7 = 118^{\circ}\) is the subtraction property of equality (\(m\angle7=180^{\circ}-62^{\circ}\)).

Answer:

1.

Statement	Reason
\(m\angle4=m\angle3\)	Corresponding angles postulate
\(m\angle4 = 62^{\circ}\)	Substitution (\(m\angle3 = 62^{\circ}\))
\(m\angle4 + m\angle5=180^{\circ}\)	Linear - pair postulate
\(62^{\circ}+m\angle5 = 180^{\circ}\)	Substitution (\(m\angle4 = 62^{\circ}\))
\(m\angle5=118^{\circ}\)	Subtraction property of equality (\(m\angle5=180 - 62\))

2.

Statement	Reason
\(m\angle7=m\angle6\)	Corresponding angles postulate
\(m\angle7 = 47^{\circ}\)	Substitution (\(m\angle6 = 47^{\circ}\))
\(m\angle7 + m\angle8=180^{\circ}\)	Linear - pair postulate
\(47^{\circ}+m\angle8 = 180^{\circ}\)	Substitution (\(m\angle7 = 47^{\circ}\))
\(m\angle8=133^{\circ}\)	Subtraction property of equality (\(m\angle8=180 - 47\))

3.

Statement	Reason
\(\angle5\) supp \(\angle7\)	If two parallel lines are cut by a transversal, consecutive interior angles are supplementary
\(m\angle5 + m\angle7=180\)	Definition of supplementary angles
\(62^{\circ}+m\angle7 = 180^{\circ}\)	Substitution
\(m\angle7 = 118^{\circ}\)	Subtraction property of equality