Question

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

Explanation:

Step1: Triangle Sum Theorem

$$m\angle1 + m\angle2 + m\angle3=180^{\circ}$$

Step2: Definition of linear pair

$\angle3$ and $\angle4$ are a linear pair.

Step3: Linear Pair Theorem

$$m\angle3 + m\angle4 = 180^{\circ}$$

Step4: Substitution

Substitute $180^{\circ}$ in the first equation with $m\angle3 + m\angle4$.
$$m\angle1 + m\angle2 + m\angle3=m\angle3 + m\angle4$$

Step5: Subtraction Property of Equality

Subtract $m\angle3$ from both sides.
$$m\angle1 + m\angle2=m\angle4$$

Answer:

The proof is completed as above, showing that \(m\angle1 + m\angle2 = m\angle4\) using the Triangle Sum Theorem, properties of linear pairs, substitution, and the subtraction property of equality.