Question

7 given: ∠1 ≅ ∠4; ∠4 and ∠5 form a linear pair
prove: ∠1 and ∠5 are supplementary

statements\treasons

1. ∠1 ≅ ∠4\t1.
2. \t2. definition of congruence
3. \t3. given
4. ∠4 and ∠5 are supplementary\t4.
5. \t5. definition of supplementary angles
6. \t6. substitution
7. ∠1 and ∠5 are supplementary\t7.

Explanation:

Step1: Fill Reason for ∠1 ≅ ∠4

The first statement \( \angle 1 \cong \angle 4 \) is given, so the reason is "Given".

Step2: Use Congruence Definition

By the definition of congruence, if \( \angle 1 \cong \angle 4 \), then \( m\angle 1 = m\angle 4 \). So the statement for step 2 is \( m\angle 1 = m\angle 4 \).

Step3: State Given about Linear Pair

The given information is that \( \angle 4 \) and \( \angle 5 \) form a linear pair, so the statement for step 3 is \( \angle 4 \) and \( \angle 5 \) form a linear pair.

Step4: Reason for Supplementary ∠4 and ∠5

If two angles form a linear pair, they are supplementary (by the Linear Pair Postulate). So the reason for \( \angle 4 \) and \( \angle 5 \) being supplementary is "Linear Pair Postulate".

Step5: Definition of Supplementary Angles

By the definition of supplementary angles, if \( \angle 4 \) and \( \angle 5 \) are supplementary, then \( m\angle 4 + m\angle 5 = 180^\circ \). So the statement for step 5 is \( m\angle 4 + m\angle 5 = 180^\circ \).

Step6: Substitution

Substitute \( m\angle 1 \) for \( m\angle 4 \) (from step 2) into \( m\angle 4 + m\angle 5 = 180^\circ \). So the statement for step 6 is \( m\angle 1 + m\angle 5 = 180^\circ \).

Step7: Definition of Supplementary Angles

If \( m\angle 1 + m\angle 5 = 180^\circ \), then by the definition of supplementary angles, \( \angle 1 \) and \( \angle 5 \) are supplementary. So the reason for step 7 is "Definition of Supplementary Angles".

Filling the table:

Statements	Reasons
2. \( m\angle 1 = m\angle 4 \)	2. Definition of Congruence
3. \( \angle 4 \) and \( \angle 5 \) form a linear pair	3. Given
4. \( \angle 4 \) and \( \angle 5 \) are supplementary	4. Linear Pair Postulate
5. \( m\angle 4 + m\angle 5 = 180^\circ \)	5. Definition of Supplementary Angles
6. \( m\angle 1 + m\angle 5 = 180^\circ \)	6. Substitution
7. \( \angle 1 \) and \( \angle 5 \) are supplementary	7. Definition of Supplementary Angles

Answer:

The completed table has the following entries (key steps filled as above):

1. Reason for 1: Given
2. Statement for 2: \( m\angle 1 = m\angle 4 \)
3. Statement for 3: \( \angle 4 \) and \( \angle 5 \) form a linear pair
4. Reason for 4: Linear Pair Postulate
5. Statement for 5: \( m\angle 4 + m\angle 5 = 180^\circ \)
6. Statement for 6: \( m\angle 1 + m\angle 5 = 180^\circ \)
7. Reason for 7: Definition of Supplementary Angles