Question

the exterior angle is ∠4. the non - adjacent interior angles are ∠1 and ∠2. m∠3 + m∠4 = 180. m∠1 + m∠2 + m∠3 = 180. isolate m∠4 in the m∠1 + m∠2, m∠1 + m∠3, m∠2 + m∠3. m∠3 + m∠4 = m∠2 + m∠3. m∠4 =?

Explanation:

Step1: Recall angle - sum properties

We know that \(m\angle3 + m\angle4=180^{\circ}\) (linear - pair of angles) and \(m\angle1 + m\angle2 + m\angle3 = 180^{\circ}\) (sum of interior angles of a triangle).

Step2: Express \(m\angle3\) from the first equation

From \(m\angle3 + m\angle4 = 180^{\circ}\), we can get \(m\angle3=180 - m\angle4\). From \(m\angle1 + m\angle2 + m\angle3 = 180^{\circ}\), we can get \(m\angle3=180-(m\angle1 + m\angle2)\).

Step3: Equate the two expressions for \(m\angle3\)

\(180 - m\angle4=180-(m\angle1 + m\angle2)\).

Step4: Solve for \(m\angle4\)

Subtract 180 from both sides of the equation: \(-m\angle4=-(m\angle1 + m\angle2)\). Multiply both sides by - 1, we get \(m\angle4=m\angle1 + m\angle2\).

Answer:

\(m\angle1 + m\angle2\)