Question

statement reason 1 \\( \overline { d e } \\) given 2 if two parallel lines are cut by a transversal, the corresponding angles are congruent 3 \\( \triangle a b c - \triangle d b e \\) aa criterion for similarity 4 \\( \frac { a d } { d b } = \frac { c e } { e b } \\) corresponding sides of similar triangles are proportional 5 \\( a b = a d + d b \\) \\( c b = c e + e b \\) segment addition 6 \\( \frac { a d + d b } { d b } = \frac { c e + e b } { e b } \\) substitution property of equality 7 \\( \frac { a d } { d b } + 1 = \frac { c e } { e b } + 1 \\) division 8 \\( \frac { a d } { d b } = \frac { c e } { e b } \\) subtraction property of equality 15 what is the missing step in this proof? \\( \bigcirc \\) a. \\( \angle c a b \cong \angle a c b , \angle e d b \cong \angle d e b \\) \\( \bigcirc \\) b. \\( \angle a d e \cong \angle d b e , \angle c e d \cong \angle e b d \\) \\( \bigcirc \\) c. \\( \angle c a d \cong \angle a c e , \angle a d e \cong \angle c e d \\) \\( \bigcirc \\) d. \\( \angle c a b \cong \angle e d b , \angle a c b \cong \angle d e b \\)

Explanation:

• For two triangles \( \triangle ABC\) and \( \triangle DBE\) to be similar by the AA (Angle - Angle) criterion, we need two pairs of congruent angles.
• If \( \overline{DE}\parallel\overline{AC}\) (implied by the proof structure as we are using the parallel - transversal angle congruence property), then \( \angle CAB\cong\angle EDB\) (corresponding angles) and \( \angle ACB\cong\angle DEB\) (corresponding angles).

Answer:

D. \( \angle CAB\cong\angle EDB,\angle ACB\cong\angle DEB\)