Question

rowena is proving that \\( \overline { a d } \cong \overline { e b } \\). which statement does the \\( \diamond \\) represent in her proof? statements reasons 1. \\( \overline { a b } \cong \overline { e d } ; \overline { b c } \cong \overline { d c } \\) 1. given 2. \\( \angle c a d \cong \angle c e b \\) 2. given 3. \\( a b = e d ; b c = d c \\) 3. def. of \\( \cong \\) segments 4. \\( a b + b c = a c \\); \\( e d + d c = c e \\) 4. segment addition postulate 5. \\( e d + d c = a c \\); \\( e d + d c = c e \\) 5. substitution property 6. \\( a c = c e \\) 6. substitution property 7. \\( \overline { a c } \cong \overline { c e } \\) 7. def. of \\( \cong \\) segments 8. \\( \angle c \cong \angle c \\) 8. reflexive property 9. \\( \diamond \\) 9. asa

Explanation:

Step1: Recall ASA (Angle - Side - Angle) congruence criterion

ASA states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Step2: Identify the triangles and their corresponding parts

From the given statements:

• We have \(\angle CAD\cong\angle CEB\) (given as statement 2)
• \(\overline{AC}\cong\overline{CE}\) (from statement 7)
• \(\angle C\cong\angle C\) (reflexive property, statement 8)

These correspond to two angles (\(\angle CAD\) and \(\angle C\) in \(\triangle ACD\), \(\angle CEB\) and \(\angle C\) in \(\triangle ECB\)) and the included side (\(\overline{AC}\) in \(\triangle ACD\) and \(\overline{CE}\) in \(\triangle ECB\))

Answer:

\(\triangle ACD\cong\triangle ECB\)