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Write a paragraph proof to prove that AC:RT as AB: RS based on the information and figure shown below.

Write a paragraph proof to prove that AC:RT as AB: RS based on the information and-example-1

1 Answer

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Given figure has two triangle, TRS & CAB

where, RS is parallel to AB

RT is parallel to AC

we can proof AC/RT = AB/RS by applying similar triangle properties by AAA method

Since,

AB is parallel to RS and TS is the transversal line,

so,


\angle ABC=\angle RST\text{ (Corresponding angles)}

AC is parallel to RTand TS is transversal line,

so,


\angle BCA=\angle RTS\text{ ( Corresponding angles)}

The sum of all angles in a triangle is 180 degree,

then In triangle ABC.


\begin{gathered} \angle ABC+\angle BCA+\angle CAB=180 \\ \text{where, we have, }\angle ABC=\angle RST\text{ and }\angle BCA=\angle RTS \\ \angle RST+\angle RTS+\angle CAB=180 \\ \angle RST+\angle RTS=180-\angle CAB \\ \text{ similarly, In triangle, RST,} \\ \angle RST+\angle STR+\angle TRS=189 \\ 180-\angle CAB+\angle TRS=180 \\ \angle CAB=\angle TRS \end{gathered}

So, we have now,


\begin{gathered} \angle ABC=\angle RST \\ \angle BCA=\angle RTS \\ \angle CAB=\angle TRS \end{gathered}

so, by Angle Angle Angle methodology, traingle ABC and RST are similarl.


\Delta ABC\approx\Delta RST

In similarl triangle, The ratio of corresponding sides is the same,

so,


(AB)/(RS)=(AC)/(RT)=(BC)/(ST)

thus,


(AC)/(RT)=(AB)/(RS)

hence proof,

User Akarsakov
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