Question

prove the triangle exterior angle theorem
given: δabc
prove: ( mangle 4 = mangle 1 + mangle 2 )
(there is a table with statements and reasons columns, and a triangle diagram with vertices a, b, c, angles labeled 1, 2, 3, 4)

Explanation:

Step1: Identify angle sum on a line

$\angle 3$ and $\angle 4$ are supplementary, so $m\angle 3 + m\angle 4 = 180^\circ$ (Linear Pair Postulate).

Step2: Identify triangle angle sum

In $\triangle ABC$, $m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$ (Triangle Angle - Sum Theorem).

Step3: Substitute and simplify

Since $m\angle 3 + m\angle 4 = m\angle 1 + m\angle 2 + m\angle 3$ (both equal to $180^\circ$), subtract $m\angle 3$ from both sides. We get $m\angle 4 = m\angle 1 + m\angle 2$.

Answer:

By using the Linear Pair Postulate (supplementary angles on a line sum to $180^\circ$) and the Triangle Angle - Sum Theorem (sum of interior angles of a triangle is $180^\circ$), we can show that $m\angle 4=m\angle 1 + m\angle 2$, proving the Triangle Exterior Angle Theorem.