1. Firstly, statement 1 is given in the question.
[tex]\bar{PL}\left|\right|\bar{MT}\text{ }\rightarrow\text{ 1. Given}[/tex]
2. When two lines cut through two parallel lines, the alternate interior angles are congruent(equal)
So;
[tex]\measuredangle P\cong\measuredangle T\text{ }\rightarrow\text{ Alternate interior angles are equal}[/tex]
3. Given
4. PK = KT
Since K is the mid point of PT as stated in the question, then PK will be of equal length as KT.
[tex]PK=KT\text{ }\rightarrow\text{ Since K is the midpoint of }\bar{PT}[/tex]
5. Vertically opposite angles are equal.
So;
[tex]\measuredangle PKL=\measuredangle TKM\text{ }\rightarrow\text{ Vertical angles Theorem}[/tex]
6. when two corresponding angles and the included side are respectively equal, a triangle is said to be congruent.
[tex]\Delta PKL\cong\Delta TKM\text{ }\rightarrow\text{ Congruent triangles (SAS- two sides and one including side are equal)}[/tex]
7. The Triangle Inequality theorem states that the sum of two sides of a triangle is greater than the third side.
[tex]PK+KL>PL\text{ }\rightarrow\text{ Triangle Inequality Theorem}[/tex]
8. CPCTC means that Corresponding Parts of Congruent Triangles are Congruent.
The corresponding sides of the two congruent triangles are;
[tex]\begin{gathered} \bar{KL}=\bar{KM} \\ \bar{PK}=\bar{KT} \\ \bar{PL}=\bar{MT} \end{gathered}[/tex]
So;
[tex]\begin{gathered} \bar{KL}=\bar{KM}\text{ .} \\ \bar{PK}=\bar{KT}\text{ }\rightarrow\text{ CPCTC} \\ \bar{PL}=\bar{MT}\text{ .} \end{gathered}[/tex]
9. The final one is a conbination of CPCTC and Triangle Inequality theorem
[tex]\bar{PK}+\bar{KM}>\bar{PL}\text{ }\rightarrow\text{ since }PK+KL>PL\text{ and }\bar{KL}=\bar{KM}[/tex]
