Question

1. complete the proof of the consecutive angles theorem.

given: abcd is a parallelogram
prove: \\(m\angle 1 + m\angle 2 = 180^\circ\\)

statements | reasons
abcd is a parallelogram | given
\\(m\angle 1 = m\angle 3\\) and \\(m\angle 2 = m\angle 4\\) |
\\(m\angle 1 + m\angle 2 + m\angle 3 + m\angle 4 = 360^\circ\\) |
\\(m\angle 1 + m\angle 2 + m\angle 1 + m\angle 2 = 360^\circ\\) |
\\(2(m\angle 1 + m\angle 2) = 360^\circ\\) |
| division

Explanation:

Identify the missing reasons in the proof

Using the Two-Column Proof knowledge point
We need to complete the missing reasons for the statements in the given proof.

Analyze Statement 2

Using the Parallelogram Opposite Angles knowledge point

• Statement: \(m\angle 1 = m\angle 3\) and \(m\angle 2 = m\angle 4\)
• Reason: Opposite angles of a parallelogram are equal in measure.

Analyze Statement 3

• Statement: \(m\angle 1 + m\angle 2 + m\angle 3 + m\angle 4 = 360^\circ\)
• Reason: The sum of the interior angles of a quadrilateral is \(360^\circ\).

Analyze Statement 4

• Statement: \(m\angle 1 + m\angle 2 + m\angle 1 + m\angle 2 = 360^\circ\)
• Reason: Substitution Property of Equality (substituting \(m\angle 1\) for \(m\angle 3\) and \(m\angle 2\) for \(m\angle 4\)).

Analyze Statement 5

• Statement: \(2(m\angle 1 + m\angle 2) = 360^\circ\)
• Reason: Simplify / Distributive Property (or combining like terms).

Answer:

The completed proof of the Consecutive Angles Theorem is:

STATEMENTS	REASONS
\(m\angle 1 = m\angle 3\) and \(m\angle 2 = m\angle 4\)	<blank>Opposite angles of a parallelogram are equal</blank>
\(m\angle 1 + m\angle 2 + m\angle 3 + m\angle 4 = 360^\circ\)	<blank>The sum of interior angles of a quadrilateral is \(360^\circ\)</blank>
\(m\angle 1 + m\angle 2 + m\angle 1 + m\angle 2 = 360^\circ\)	<blank>Substitution Property of Equality</blank>
\(2(m\angle 1 + m\angle 2) = 360^\circ\)	<blank>Simplify / Combine like terms</blank>
\(m\angle 1 + m\angle 2 = 180^\circ\)	Division