Question

13) Complete the proof. Given: A RST, RT > ST Prove: measure of angle RST > measure of angle R

Proof: Since RT > ST, there exists a point Q on RT, where QT is the same length as ST. By the Isosceles Triangle Theorem, angle 2 is congruent to angle 3, and by definition of congruent angles, the measure of angle 2 = the measure of angle 3.​

Answer 1

To prove that the measure of angle RST is greater than the measure of angle R, we can use the following steps:

By the definition of a triangle, angle RST is supplementary to angles R, S and T.

Since RT > ST, we can conclude that angle R is less than angle S.

Since angle RST is supplementary to angles R and S, the measure of angle RST is greater than the measure of angle R.

This can be written as measure of angle RST = measure of angle R + measure of angle S > measure of angle R

So, we've proven that if RT > ST in triangle RST, then measure of angle RST > measure of angle R.

Proof by contradiction could also be used to prove that if RT > ST in triangle RST, then measure of angle RST > measure of angle R.

Assume that the measure of angle RST <= measure of angle R. By the triangle inequality theorem, we know that RT + ST > R. But we are given that RT > ST, so RT + ST > RT > R which is a contradiction. Therefore, measure of angle RST must be greater than measure of angle R.