Question

given △mnp
prove m∠1 + m∠2 = m∠4
statements

1. △mnp
2. m∠1 + m∠2 + m∠3 = 180°

3.

1. ∠3 & ∠4 are supplementary
2. m∠3 + m∠4 = 180°

reasons

1. given
2. triangle sum theorem

3.

1. linear pair theorem

5.
note: use ctrl+o to drag the option via keyboard
△mnp.
definition of linear pair
triangle sum theorem
∠3 & ∠4 are supplementary
m∠1 + m∠2 = m∠4
m∠1 + m∠2 + m∠3 = 180°
∠3 & ∠4 are a linear pair
m∠1 + m∠2 + m∠3 = m∠3 + m∠4
definition of supplementary
subtraction property
substitution
linear pair theorem
m∠3 + m∠4 = 180°

Explanation:

Step1: Use Triangle Sum Theorem

In \(\triangle MNP\), by the Triangle Sum Theorem, \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\).

Step2: Use Linear Pair Theorem

\(\angle3\) and \(\angle4\) form a linear pair. By the Linear Pair Theorem, \(m\angle3 + m\angle4 = 180^{\circ}\).

Step3: Substitute

Since \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\) and \(m\angle3 + m\angle4 = 180^{\circ}\), we can substitute.
\(m\angle1 + m\angle2+m\angle3=m\angle3 + m\angle4\).

Step4: Use Subtraction Property

Subtract \(m\angle3\) from both sides of the equation \(m\angle1 + m\angle2+m\angle3=m\angle3 + m\angle4\).
By the Subtraction Property, \(m\angle1 + m\angle2=m\angle4\).

Answer:

1. \(m\angle1 + m\angle2 + m\angle3 = 180^{\circ}\) (Triangle Sum Theorem)
2. \(\angle3\) & \(\angle4\) are a linear pair (Definition of linear pair)
3. \(m\angle3 + m\angle4 = 180^{\circ}\) (Linear Pair Theorem)
4. \(m\angle1 + m\angle2+m\angle3=m\angle3 + m\angle4\) (Substitution)
5. \(m\angle1 + m\angle2=m\angle4\) (Subtraction Property)