We are to write two columns proofs for the given geometry problem.
There are two ways to express the proof to the given problem. We will investigate each of them separately:
A two coloumn proof is categorized into " STATEMENTS " and " REASONS ".
Where, statements gives us the relations between angles, sides, and/or lines with the help of some standard mathematical symbols. Reasons gives us the explanantion behind the statements made.
Method 1:
[tex]\text{\textcolor{#FF7968}{Statement: }}\text{m }\angle\text{ 1 =}\text{\textcolor{#FF7968}{ }}\text{62}\text{\textcolor{#FF7968}{ }}\text{degrees}[/tex][tex]\text{\textcolor{#FF7968}{Reason: }}\text{Given}[/tex]Now, we have to recall laws of intersecting lines. When two lines intersect ( l and t ) they always make up 4 angles with two sets of complementary angles.
Amoung these 4 angles ( 1 , 2 , 3 , 4 ) there are two set of vertically opposite angles that are equal in magnitude.
Using the given image, we see that the sets of vertically opposite angles are 1 : 4 AND 2 : 3.
So on the basis of law of angles for two intersecting lines we can make our next statement as follows:
[tex]\text{\textcolor{#FF7968}{Statement:}}\text{ }\angle1\text{ = }\angle4\text{ = 62 degrees}[/tex][tex]\text{\textcolor{#FF7968}{Reason:}}\text{ Vertically opposite angles}[/tex]Method 2:
[tex]\text{\textcolor{#FF7968}{Statement: }}\text{m }\angle\text{ 1 =}\text{\textcolor{#FF7968}{ }}\text{62}\text{\textcolor{#FF7968}{ }}\text{degrees}[/tex][tex]\text{\textcolor{#FF7968}{Reason}}\text{: Given}[/tex]For this we will recall the sum of of supplementary angles for a line is given as:
[tex]\begin{gathered} LineL\colon180\text{ = }\angle\text{ 1 + }\angle\text{ 2} \\ AND \\ LineL\colon180\text{ = }\angle\text{ 3 + }\angle\text{ 4} \\ AND \\ t\colon\text{ 180 = }\angle\text{2 + }\angle4 \\ \text{AND} \\ t\colon\text{ 180 = }\angle\text{1 + }\angle3 \end{gathered}[/tex]Hence, the statement 2 would be:
[tex]\begin{gathered} LineL\colon\angle2\text{ = 180 - }\angle1=118degrees\text{ } \\ OR \\ Linet\colon\angle3\text{ = 180 - }\angle1\text{ = 118 degrees} \end{gathered}[/tex][tex]\text{\textcolor{#FF7968}{Reason:}}\text{ Sum of supplementary angles is always 180 degrees for two intersecting lines.}[/tex]Now, we appl
