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,

$\frac{AB}{RS}=\frac{AC}{RT}=\frac{BC}{ST}$

thus,

$\frac{AC}{RT}=\frac{AB}{RS}$

hence proof,