1 Answer

Tetsuya Yamamoto · Answer 1 · 2023-02-10T22:22:18+0000

1. Firstly, statement 1 is given in the question.

$\bar{PL}\left|\right|\bar{MT}\text{ }\rightarrow\text{ 1. Given}$

2. When two lines cut through two parallel lines, the alternate interior angles are congruent(equal)

So;

$\measuredangle P\cong\measuredangle T\text{ }\rightarrow\text{ Alternate interior angles are equal}$

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.

$PK=KT\text{ }\rightarrow\text{ Since K is the midpoint of }\bar{PT}$

5. Vertically opposite angles are equal.

So;

$\measuredangle PKL=\measuredangle TKM\text{ }\rightarrow\text{ Vertical angles Theorem}$

6. when two corresponding angles and the included side are respectively equal, a triangle is said to be congruent.

$\Delta PKL\cong\Delta TKM\text{ }\rightarrow\text{ Congruent triangles (SAS- two sides and one including side are equal)}$

7. The Triangle Inequality theorem states that the sum of two sides of a triangle is greater than the third side.

$PK+KL>PL\text{ }\rightarrow\text{ Triangle Inequality Theorem}$

8. CPCTC means that Corresponding Parts of Congruent Triangles are Congruent.

The corresponding sides of the two congruent triangles are;

$\begin{gathered} \bar{KL}=\bar{KM} \\ \bar{PK}=\bar{KT} \\ \bar{PL}=\bar{MT} \end{gathered}$

So;

$\begin{gathered} \bar{KL}=\bar{KM}\text{ .} \\ \bar{PK}=\bar{KT}\text{ }\rightarrow\text{ CPCTC} \\ \bar{PL}=\bar{MT}\text{ .} \end{gathered}$

9. The final one is a conbination of CPCTC and Triangle Inequality theorem

$\bar{PK}+\bar{KM}>\bar{PL}\text{ }\rightarrow\text{ since }PK+KL>PL\text{ and }\bar{KL}=\bar{KM}$

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