Question

question 6 of 8
select the correct answer from each drop - down menu.
given: \\( \overline { a c } \cong \overline { x z } , \overline { b c } \cong \overline { x y } \\)
prove: \\( \overline { a b } \cong \overline { y z } \\)
complete the proof.
suppose \\( \overline { a b } \
ot \cong \overline { y z } \\). by the definition of congruence, \\( a b \
eq y z \\). by the addition property of equality, \\( a b + b c \
eq y z + b c \\). using the given, \\( b c = x y \\) because of the definition of congruence. applying the substitution property of equality, \\( a b + b c \
eq y z + x y \\). applying segment addition to \\( a b + b c \
eq y z + x y \\). however, we are given that \\( \overline { a c } \cong \overline { x z } \\), which implies \\( a c = x z \\) by the definition of congruenc

Explanation:

Step1: Segment Addition Postulate

By segment addition postulate, \(AC = AB + BC\) and \(XZ=YZ + XY\).

Step2: Substitution

Since \(AC\cong XZ\) (given, so \(AC = XZ\)) and \(BC\cong XY\) (given, so \(BC = XY\)), if we assume \(AB
eq YZ\), then \(AB + BC
eq YZ + XY\) (by addition property of inequality). But by segment addition postulate \(AC=AB + BC\) and \(XZ = YZ+XY\). Since \(AC = XZ\) (from \(\overline{AC}\cong\overline{XZ}\)), the contradiction comes from the wrong assumption.

Answer:

\(AC\) equals \(XZ\)