Question

complete the following proof
given: \\( \triangle mnp \\)
prove: \\( m\angle 1 + m\angle 2 = m\angle 4 \\)

Explanation:

Step1: Use the triangle angle - sum theorem

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

Step2: Use the linear - pair postulate

\(\angle3\) and \(\angle4\) form a linear pair. So \(m\angle3 + m\angle4 = 180^{\circ}\) (Linear - Pair Postulate).

Step3: Set the two equations equal

Since \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\) and \(m\angle3 + m\angle4 = 180^{\circ}\), we can say \(m\angle1 + m\angle2 + m\angle3=m\angle3 + m\angle4\) (Substitution Property of Equality).

Step4: Subtract \(m\angle3\) from both sides

Subtract \(m\angle3\) from each side of the equation \(m\angle1 + m\angle2 + m\angle3=m\angle3 + m\angle4\). Using the Subtraction Property of Equality: \((m\angle1 + m\angle2 + m\angle3)-m\angle3=(m\angle3 + m\angle4)-m\angle3\).

Step5: Simplify

After simplification, we get \(m\angle1 + m\angle2=m\angle4\).

Answer:

The proof is completed as shown in the steps above, and \(m\angle1 + m\angle2=m\angle4\) is proved.