Cumulative Assessment
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Recall the definitions of parallelograms and congruent angles. Also, use the Alternate Interior Angles Theorem, the Angle-Side-Angle (ASA) Congruence Theorem, and the Angle Addition Postulate.
See solution.
We are given a diagram where ABCD is a parallelogram and BD is one of its diagonals.
Next, let's write the first two statements of the given flowchart.
Next, we will complete the flowchart by filling one blank at a time.
Statement 3: & AB∥ CD, BC∥ AD The reason for which we obtain these two pairs of parallel lines is because of the definition of parallelograms, which says that the opposite sides are parallel.
Statement 4: & ∠ ABD ≅ ∠ CDB & ∠ ADB ≅ ∠ CBD The reason for which we obtain these two pairs of congruent angles is because of the Alternate Interior Angles Theorem.
Statement 5 : & BD≅ DB This is true due to the Reflexive Property of Congruence. Statement 6 : & △ ABD ≅ △ CDB In statement 4, we obtained two pairs of congruent angles and in statement 5 we got that the included sides are congruent. Therefore, the two triangles above are congruent because of the Angle-Side-Angle (ASA) Congruence Theorem.
Statement 7 : & ∠A≅ ∠C Since △ ABD ≅ △ CDB, by definition of congruent triangles, we have that the corresponding angles are congruent. This is the reason for the statement above.
Statement 8: & m∠ ABD = m∠ CDB & m∠ ADB = m∠ CBD In statement 4, we got two pairs of congruent angles. Then, by definition of congruent angles, the measures of these pairs of angles must be the same. This is the reason for this statement.
Statement 9: & m∠ B = m∠ ABD + m∠ CBD & m∠ D = m∠ ADB + m∠ CDB The reason why we can rewrite m∠ B and m∠ D as above is because of the Angle Addition Postulate.
Statement 10: & m∠ B = m∠ D By substituting statement 8 into statement 9, we obtain the statement written above.
Statement 11: & ∠ B ≅ ∠ D In the previous statement, we got that these two angles have the same measure, which by definition of congruent angles, implies that they are congruent.
That way, we've filled in all the blanks in the given flowchart.