Question

def. of ≅
def. of ≅
def. of ≅
def. of ≅
def. of perpendicular
def. of complementary ∠s
def. of complementary ∠s
def. of supplementary ∠s
def. of supplementary ∠s
def. of supplementary ∠s
def. of supplementary ∠s
supplement thm.
complement thm.
congruent complements thm.
angle addition postulate
subtraction property
subtraction property
substitution
substitution
substitution
transitive property
transitive property
m\angle1 = m\angle4
m\angle2 = m\angle3
m\angle2 = m\angle3
m\angle2 = m\angle4
m\angle1 = m\angle2;
m\angle3 = m\angle4
m\angle1 + m\angle2 = 90^\circ
m\angle3 + m\angle4 = 90^\circ
m\angle1 + m\angle2 = 180^\circ
m\angle1 + m\angle2 = 180^\circ
m\angle1 + m\angle2 = 180^\circ
m\angle aec = 90^\circ;
m\angle bed = 90^\circ
m\angle aec = m\angle bed
\angle1 and \angle2 are complementary
\angle1 and \angle2 are supplementary
given: \angle1 and \angle2 form a linear pair;
m\angle1 = m\angle3
prove: \angle2 and \angle3 are supplementary

1. \angle1 and \angle2 form a linear pair;

m\angle1 = m\angle3

1. \angle1 and \angle2 are supplementary

3.

1. m\angle3 + m\angle2 = 180^\circ
2. \angle2 and \angle3 are supplementary
3. given

2.

1. defn. of supplementary \angle s

4.
5.
given: \angle1 and \angle2 a linear pair;\angle2, \angle3, and \angle4 supplementary;
prove: m\angle1 = m\angle3 + m\angle4

1. \angle1 and \angle2 a linear pair;\angle2, \angle3, and \angle4 supplementary;

2.
3.

1. m\angle2 + m\angle3 + m\angle4 = 180^\circ
2. m\angle1 + m\angle2 = m\angle2 + m\angle3 + m\angle4
3. m\angle1 = m\angle3 + m\angle4
4. given
5. supplement theorem

3.
4.
5.
6.

Explanation:

First Proof:

Step1: Reason for Statement 2

Linear - pair angles are supplementary. So, the reason for “\(\angle1\) and \(\angle2\) are supplementary” is “Linear - Pair Postulate”.

Step2: Reason for Statement 4

Since \(m\angle1=m\angle3\) (given) and \(m\angle1 + m\angle2=180^{\circ}\) (from the definition of supplementary angles for \(\angle1\) and \(\angle2\)), by substitution (if \(a = b\) and \(a + c=180^{\circ}\), then \(b + c = 180^{\circ}\)), we get \(m\angle3+m\angle2 = 180^{\circ}\). So the reason for \(m\angle3+m\angle2=180^{\circ}\) is “Substitution”.

Step3: Reason for Statement 5

If \(m\angle3+m\angle2 = 180^{\circ}\), then by the definition of supplementary angles (two angles whose measures add up to \(180^{\circ}\) are supplementary), \(\angle2\) and \(\angle3\) are supplementary. So the reason for “\(\angle2\) and \(\angle3\) are supplementary” is “Definition of Supplementary \(\angle\)s”.

Second Proof:

Step1: Reason for Statement 2

Since \(\angle1\) and \(\angle2\) form a linear pair, by the Supplement Theorem (if two angles form a linear pair, then they are supplementary), \(m\angle1+m\angle2 = 180^{\circ}\).

Step2: Reason for Statement 4

Since \(\angle2,\angle3,\) and \(\angle4\) are supplementary, by the definition of supplementary angles (the sum of the measures of supplementary angles is \(180^{\circ}\)), \(m\angle2+m\angle3+m\angle4=180^{\circ}\).

Step3: Reason for Statement 5

Since \(m\angle1+m\angle2 = 180^{\circ}\) and \(m\angle2+m\angle3+m\angle4=180^{\circ}\), by the substitution property (if \(a=b\) and \(b = c\), then \(a=c\)), \(m\angle1+m\angle2=m\angle2+m\angle3+m\angle4\).

Step4: Reason for Statement 6

Subtract \(m\angle2\) from both sides of the equation \(m\angle1+m\angle2=m\angle2+m\angle3+m\angle4\) using the subtraction property of equality (\(a + b=b + c\) implies \(a=c\) when we subtract \(b\) from both sides). So the reason for \(m\angle1=m\angle3+m\angle4\) is “Subtraction Property of Equality”.

Answer:

First Proof:

1. Given
2. Linear - Pair Postulate
3. \(m\angle1 + m\angle2=180^{\circ}\)
4. Substitution
5. Definition of Supplementary \(\angle\)s

Second Proof:

1. Given
2. \(m\angle1+m\angle2 = 180^{\circ}\)
3. \(m\angle2+m\angle3+m\angle4=180^{\circ}\) (Definition of Supplementary)
4. Definition of Supplementary
5. Substitution
6. Subtraction Property of Equality