Question

a parallel ruler, shown to the right, can be used to draw parallel lines. the distance between the parallel segments ab and dc can vary. the ruler is constructed so that the distance between a and b equals the distance between d and c. the distance between a and c is the same as the distance between b and d. explain why ab and dc are always parallel. complete the sentences below. when the parallel ruler is open at any setting, the distance bc = . it is given that ab = dc and ac = bd so △abc ≅ △dcb by sss. hence, ∠abc ≅ ∠dcb by cpctc. because these are alternate interior angles formed by ab and cd with transversal line bc, ab || dc.

Explanation:

Step1: Identify equal - side conditions

We know that \(AB = DC\), \(AC = BD\), and \(BC\) is common to \(\triangle ABC\) and \(\triangle DCB\).

Step2: Prove triangle congruence

By the Side - Side - Side (SSS) congruence criterion, \(\triangle ABC\cong\triangle DCB\) since \(AB = DC\), \(AC = BD\), and \(BC=CB\).

Step3: Find equal angles

By Corresponding Parts of Congruent Triangles are Congruent (CPCTC), \(\angle ABC=\angle DCB\).

Step4: Use angle - relationship for parallel lines

\(\angle ABC\) and \(\angle DCB\) are alternate interior angles formed by lines \(AB\) and \(DC\) with transversal \(BC\). Since alternate interior angles are equal, \(AB\parallel DC\).

Answer:

When the parallel ruler is open at any setting, the distance \(BC = CB\). It is given that \(AB = DC\) and \(AC = BD\), so \(\triangle ABC\cong\triangle DCB\) by \(SSS\). Hence, \(\angle ABC=\angle DCB\) by \(CPCTC\). Because these are alternate interior angles formed by \(AB\) and \(CD\) with transversal line \(BC\), \(AB\parallel DC\).