Question

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

Answer 1

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,