Question

use the diagram to prove the same - side exterior angles theorem. complete the paragraph proof with each correct angle number.
given: ( a parallel b ) and ( t ) is a transversal.
prove: ( angle 1 ) and ( angle 4 ) are supplementary angles.
(diagram: two parallel lines ( a ) and ( b ), cut by transversal ( t ). angles at the intersections are numbered 1, 2 (on line ( a ) and transversal ( t )), 3, 4 (on line ( b ) and transversal ( t )))
enter the answer in each space provided. use numbers instead of words.

Explanation:

Step1: Identify Linear Pair

$\angle1$ and $\angle2$ form a linear pair, so $\angle1 + \angle2 = 180^\circ$ (linear pair postulate).

Step2: Identify Corresponding Angles

Since $a \parallel b$ and $t$ is a transversal, $\angle2$ and $\angle4$ are corresponding angles, so $\angle2 = \angle4$ (corresponding angles postulate for parallel lines).

Step3: Substitute and Prove

Substitute $\angle2$ with $\angle4$ in $\angle1 + \angle2 = 180^\circ$. We get $\angle1 + \angle4 = 180^\circ$, which means $\angle1$ and $\angle4$ are supplementary.

Answer:

$\angle1$ and $\angle4$ are supplementary because $\angle1$ and $\angle2$ are supplementary (linear pair), $\angle2 = \angle4$ (corresponding angles, $a \parallel b$), so substituting gives $\angle1 + \angle4 = 180^\circ$.