Question

given: ab = cb
prove: ad ≠ de
(using the indirect
method).

1. click here to insert
2. either ad = de or

ad ≠ de. assume
ad = de.

1. click here to insert
2. m∠2 > m∠c
3. given
4. a statement is either

true or false.

1. base angles theorem
2. click here to insert

Explanation:

Step1: State the given

\(AB = CB\)

Step2: Assume the opposite

Assume \(AD=DE\). If \(AD = DE\), then \(\angle1=\angle2\) (Base - Angles Theorem for \(\triangle ADE\)).

Step3: Use triangle - exterior angle property

In \(\triangle BDC\), \(\angle BDC\) is an exterior angle. So, \(\angle BDC>\angle C\). Also, \(\angle BDC = \angle1+\angle ABD\). Since \(\angle1=\angle2\), and \(AB = CB\) (given), if we assume \(AD = DE\), we get a contradiction. Because if \(AD = DE\), and considering the relationships in the triangles (using properties of triangle angles and side - angle relationships), we violate the basic triangle inequality and angle - side principles.

Answer:

By the indirect method (proof by contradiction), since assuming \(AD = DE\) leads to a contradiction with the given \(AB = CB\) and triangle properties, we conclude \(AD
eq DE\).