Question

a. 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.

1. dc || ab

reflexive property
line ab is parallel to line dc and cut by transversal db. so
angles cdb and abd are
and must be congruent.
side db is congruent to side bd because theyre the
angle a is congruent to angle c because theyre both right angles.
by the
triangle congruence theorem
triangle bcd is congruent to triangle dab.

Explanation:

Step1: Identify the alternate interior angles

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

Step2: Analyze the right - angle property

\(\angle C\) and \(\angle A\) are right angles (\(90^{\circ}\)).

Step3: Use the Hypotenuse - Leg (HL) Congruence Theorem

In right - triangles \(\triangle BCD\) and \(\triangle DAB\), we have \(DB = DB\) (common side, hypotenuse) and we can use the properties of parallel lines (for the non - hypotenuse sides related to the alternate interior angles). But if we consider the right - triangle congruence theorem, for right - triangles \(\triangle BCD\) and \(\triangle DAB\), we have two right angles (\(\angle C=\angle A = 90^{\circ}\)), a common side \(DB\) (hypotenuse). Also, because \(DC\parallel AB\), the non - hypotenuse sides (legs) are congruent (by the properties of parallel lines and the fact that the distance between parallel lines is constant in a parallelogram - like setting, but more precisely for right - triangles, if we use the Hypotenuse - Leg (HL) Congruence Theorem.

Answer:

Hypotenuse - Leg (HL)