Question

given: \\( \angle 1 \cong \angle 2 \\)
prove: rs \\( \
eq \\) st (using the
indirect method)

1. \\( \angle 1 \cong \angle t \\)
2. click here to insert
3. base angles theorem
4. statement 4 contra-

dicts the exterior an-
gle inequality theo-
rem. the assumption
made in statement 2
must be false. by
elimination, statement
5 must be true.

Explanation:

Step1: Assume the opposite

Assume \( RS = ST \).

Step2: Use the Base Angles Theorem

If \( RS = ST \), then by the Base Angles Theorem, \( \angle R=\angle T \).

Step3: Use the Exterior Angle Inequality Theorem

\( \angle1\) is an exterior angle of \(\triangle SWT\). By the Exterior Angle Inequality Theorem, \( \angle1>\angle T\). But we are given \( \angle1\cong\angle2\) and if \( \angle R = \angle T\) (from step 2) and \( \angle2=\angle R\) (base angles of \(\triangle RSW\) if we assume \( RS = ST\) and consider the triangle properties in the figure), then \( \angle1=\angle T\) which contradicts the Exterior Angle Inequality Theorem.

Answer:

By the indirect method (proof by contradiction), since assuming \( RS = ST\) leads to a contradiction (as shown in the steps above), \( RS
eq ST\).