Question

given: \\( \triangle mnp \\) prove \\( m \angle 1 + m \angle 2 = m \angle 4 \\) statements 1. \\( \triangle mnp \\) 2. \\( m \angle 1 + m \angle 2 + m \angle 3 = 180 ^ { \circ } \\) 3. 4. 5. \\( m \angle 3 + m \angle 4 = 180 ^ { \circ } \\) reasons 1. given 2. triangle sum theorem 3. 4. 5. note: use ctrl+o to drag the option definition of supplementary triangle sum theorem \\( \angle 3 \\& \angle 4 \\) are a linear pair subtraction property substitution \\( m \angle 1 + m \angle 2 + m \angle 3 = m \angle 3 + m \angle 4 \\) \\( \triangle mnp \\) \\( m \angle 3 + m \angle 4 = 180 ^ { \circ } \\) \\( m \angle 1 + m \angle 2 = m \angle 4 \\) linear pair theorem definition of linear pair

Explanation:

Step1: Analyze the relationship between ∠3 and ∠4

Since ∠3 and ∠4 are a linear pair (by the definition of a linear pair), we know that \(m\angle3 + m\angle4=180^{\circ}\) (Linear Pair Theorem).

Step2: Use the Triangle Sum Theorem

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

Step3: Substitute the value of \(m\angle3 + m\angle4\)

From \(m\angle3 + m\angle4 = 180^{\circ}\) (Linear Pair Theorem) and \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\) (Triangle Sum Theorem), we can use the Subtraction Property.
If \(m\angle3 + m\angle4=m\angle1 + m\angle2 + m\angle3\), then subtracting \(m\angle3\) from both sides (Subtraction Property) gives \(m\angle1 + m\angle2=m\angle4\) (Substitution).

Answer:

1. \(m\angle1 + m\angle2=m\angle4\): Subtraction Property
2. \(m\angle3 + m\angle4 = 180^{\circ}\): Linear Pair Theorem
3. \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\): Triangle Sum Theorem