Question

given: △mnp
prove: ( mangle1 + mangle2 = mangle4 )
statements

1. ( \triangle mnp )
2. ( mangle1 + mangle2 + mangle3 = 180^circ )

3.

1. ( angle3 & angle4 ) are supplementary
2. ( mangle3 + mangle4 = 180^circ )

6.
reasons

1. given
2. triangle sum theorem

3.

1. linear pair theorem

5.
6.
subtraction property
( mangle1 + mangle2 = mangle4 )
( angle3 & angle4 ) are supplementary
( mangle1 + mangle2 + mangle3 = 180^circ )
linear pair theorem
( angle3 & angle4 ) are a linear pair
definition of linear pair
( mangle1 + mangle2 + mangle3 = mangle3 + m )
definition of supplementary
triangle sum theorem
( \triangle mnp )
( mangle3 + mangle4 = 180^circ )
note: use ctrl+d to drag the opti

Explanation:

Step1: Recall the Triangle Sum Theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, for \(\triangle MNP\), \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\) (Statement 2).

Step2: Use the Linear Pair Theorem

If \(\angle3\) and \(\angle4\) are a linear pair (by the definition of a linear pair), then by the Linear Pair Theorem (Statement 4), \(m\angle3 + m\angle4 = 180^{\circ}\) (Statement 5).

Step3: Apply the Subtraction Property

Since \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\) and \(m\angle3 + m\angle4 = 180^{\circ}\), we can set \(m\angle1 + m\angle2 + m\angle3=m\angle3 + m\angle4\). Subtracting \(m\angle3\) from both sides (Subtraction Property), we get \(m\angle1 + m\angle2=m\angle4\).

Answer:

The missing reason for Statement 3 is \(m\angle1 + m\angle2 + m\angle3=m\angle3 + m\angle4\) (from equating the two expressions equal to \(180^{\circ}\)). The reason for Statement 5 is Definition of supplementary (since \(\angle3\) and \(\angle4\) are supplementary, \(m\angle3 + m\angle4 = 180^{\circ}\)). The reason for Statement 6 (the final step \(m\angle1 + m\angle2=m\angle4\)) is Subtraction Property.