Question

han wrote a proof that triangle bcd is congruent to triangle dab. hans proof is incomplete. fill in the blanks to make the proof work.
dc || ab
line ab is parallel to line dc and cut by transversal db. so
angles cdb and abd are ____ because theyre the ____
side db is congruent to side bd because ______
angle a is congruent to angle c because theyre both right angles.
by the ______ triangle bcd is congruent to triangle dab.

Explanation:

Step1: Alternate Interior Angles

Since \(DC\parallel AB\) and \(DB\) is a transversal, \(\angle CDB\cong\angle ABD\) (alternate - interior angles theorem).

Step2: Hypotenuse - Leg (HL) Congruence

We know that \(\angle C\) and \(\angle A\) are right angles (\(\angle C\cong\angle A = 90^{\circ}\)), \(DB = DB\) (common side), and \(\angle CDB\cong\angle ABD\). But for right - triangles, the Hypotenuse - Leg (HL) Congruence Theorem states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, the triangles are congruent. Here, \(DB\) is the hypotenuse for both right - triangles \(\triangle BCD\) and \(\triangle DAB\), and we can consider the fact that from the parallel lines and transversal we have a pair of congruent angles which helps in establishing the congruence.

Answer:

By the Hypotenuse - Leg (\(HL\)) Congruence Theorem, triangle \(BCD\) is congruent to triangle \(DAB\).