Question

Fill in the missing reasons to correctly complete the proof.

Answer 1

For the first question, how do you know that

`\begin{gathered} \bar{AB}\parallel\bar{CD}; \\ \bar{AD}\parallel\bar{BC} \\ \text{ and these segments are crossed by cross-segment }BD \\ \end{gathered}`

Then, then angles

`\begin{gathered} \angle ABD \\ \text{and} \\ \angle CDB \end{gathered}`

satisfy the definition of alternate interior angles

The same goes for angles

`\begin{gathered} \angle ADB \\ \text{and} \\ \angle CBD \end{gathered}`

Therefore, for the first question, the correct answer is Alternate interior angles are congruent.

For the second question, you know that

`\begin{gathered} \angle ABD\cong\angle CDB \\ \angle ADB\cong\angle CBD \\ \text{and} \\ BD\cong BD \end{gathered}`

And this satisfies the triangle congruence theorem ASA (Angle-side-Angle).

Therefore, for the second question, the correct answer is ASA Triangle Congruence Theorem.

For the third question, since you already know that the triangles ABD and CDB are congruent, then the respective segments that make up the triangles will also be congruent.

Therefore, for the third question, the correct answer is Corresponding parts of congruent triangles are congruent.