Question

complete the following proof using definitions and/or properties. (2 pts per blank) given: a is the mid - point of cr r is the mid - point of as prove: ca = rs statements reasons a is the mid - point of cr r is the mid - point of as given ca≅ar;ar≅rs ca≅rs ca = rs

Explanation:

Step1: Apply mid - point definition

By the definition of a mid - point, if \(A\) is the mid - point of \(\overline{CR}\), then \(CA = AR\), and if \(R\) is the mid - point of \(\overline{AS}\), then \(AR=RS\). In terms of congruence, \(\overline{CA}\cong\overline{AR}\) and \(\overline{AR}\cong\overline{RS}\) because congruent segments have equal lengths. The reason for this step is the mid - point definition.

Step2: Use transitive property of congruence

Since \(\overline{CA}\cong\overline{AR}\) and \(\overline{AR}\cong\overline{RS}\), by the transitive property of congruence (if \(a\cong b\) and \(b\cong c\), then \(a\cong c\)), we can say that \(\overline{CA}\cong\overline{RS}\). The reason for this step is the transitive property of congruence.

Step3: Convert congruence to equality

Congruent segments have equal lengths. So, if \(\overline{CA}\cong\overline{RS}\), then \(CA = RS\). The reason for this step is the definition of congruent segments.

Answer:

Statements	Reasons
\(\overline{CA}\cong\overline{AR}\); \(\overline{AR}\cong\overline{RS}\)	Definition of mid - point
\(\overline{CA}\cong\overline{RS}\)	Transitive property of congruence
\(CA = RS\)	Definition of congruent segments