Question

prove: ∠1 ≅ ∠4
diagram of angles 1,2,3,4 with arrows
statements

1. ∠1 and ∠2 are supplementary; ∠3 and ∠4 are supplementary; ∠2 ≅ ∠3
2. ( mangle1 + mangle2 = 180^circ ), ( mangle3 + mangle4 = 180^circ )
3. ( mangle2 = mangle3 )
4. ( mangle2 + mangle4 = 180^circ )
5. ( mangle1 + mangle2 = mangle2 + mangle4 )
6. blank
7. ( mangle1 = mangle4 )
8. blank

reasons

1. given
2. def. of supplementary ∠ s
3. def. of ≅ ∠ s
4. blank
5. substitution
6. reflexive prop. of =
7. blank
8. def. of ≅ ∠ s

options:
a. def. of supplementary ∠s
b. ( mangle2 = mangle2 )
c. substitution
d. ( angle1 ≅ angle4 )
e. subtr. prop. of =

Explanation:

Step 1: Analyze Step 4

We know that \( m\angle3 + m\angle4 = 180^\circ \) (from step 2) and \( m\angle2 = m\angle3 \) (from step 3). So we can substitute \( m\angle2 \) for \( m\angle3 \) in the equation \( m\angle3 + m\angle4 = 180^\circ \), which gives \( m\angle2 + m\angle4 = 180^\circ \). The reason for this is substitution (option c).

Step 2: Analyze Step 6

The reflexive property of equality states that a quantity is equal to itself. So \( m\angle2 = m\angle2 \) (option b) is the statement for step 6.

Step 3: Analyze Step 7

From step 5: \( m\angle1 + m\angle2 = m\angle2 + m\angle4 \). Using the subtraction property of equality, we can subtract \( m\angle2 \) from both sides to get \( m\angle1 = m\angle4 \). The reason is the subtraction property of equality (option e).

Step 4: Analyze Step 8

If \( m\angle1 = m\angle4 \), then by the definition of congruent angles, \( \angle1 \cong \angle4 \) (option d).

Answer:

1. c
2. b
3. e
4. d