Question

reasons bank definition of congruent angles given transitive property of equality making an argument 44) in the figure, $overline{sr} cong overline{cb}$ and $overline{ac} cong overline{qr}$. your friend claims that, because of this, $overline{cb} cong overline{ac}$ by the transitive property of segment congruence. is your friend correct? explain your reasoning.

Explanation:

The Transitive Property of Segment Congruence states that if segment \(a \cong\) segment \(b\) and segment \(b \cong\) segment \(c\), then segment \(a \cong\) segment \(c\). For the transitive property to apply, the middle segment (the one congruent to both) must be the same. Here, we have \(\overline{SR} \cong \overline{CB}\) and \(\overline{AC} \cong \overline{QR}\). The segments involved in the first congruence are \(\overline{SR}\) and \(\overline{CB}\), and in the second are \(\overline{AC}\) and \(\overline{QR}\). There is no common segment (like a segment that is congruent to both \(\overline{CB}\) and \(\overline{AC}\)) to apply the transitive property. So we can't conclude \(\overline{CB} \cong \overline{AC}\) using the Transitive Property of Segment Congruence.

Answer:

Your friend is not correct. The Transitive Property of Segment Congruence requires a common segment (a segment congruent to both of the other two segments) to apply. Here, \(\overline{SR} \cong \overline{CB}\) and \(\overline{AC} \cong \overline{QR}\), and there is no common segment between \(\overline{CB}\) and \(\overline{AC}\) (i.e., no segment that is congruent to both \(\overline{CB}\) and \(\overline{AC}\)) to use the transitive property. Thus, we cannot conclude \(\overline{CB} \cong \overline{AC}\) by the Transitive Property of Segment Congruence.