Question

question 5 of 10

which of the following are necessary when proving that the opposite angles of a parallelogram are congruent? check all that apply.

a. corresponding parts of similar triangles are similar.
b. segment addition postulate.
c. angle addition postulate.
d. corresponding parts of congruent triangles are congruent.

Explanation:

Analyze the proof structure

Using the Geometric Proof Strategies knowledge point:
To prove that the opposite angles of a parallelogram are congruent, we draw a diagonal (e.g., \(BD\)) to divide the parallelogram \(ABCD\) into two triangles, \(\triangle ABD\) and \(\triangle CDB\).

Apply triangle congruence

Using the Triangle Congruence Postulates knowledge point:
We prove \(\triangle ABD \cong \triangle CDB\) using ASA (Angle-Side-Angle) based on parallel lines and the shared diagonal.

Identify CPCTC requirement

Using the Geometric Proof Strategies knowledge point

$$ \triangle ABD \cong \triangle CDB \implies \angle A \cong \angle C $$

This step relies directly on CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Thus, Option D is necessary.

Identify Angle Addition requirement

Using the Geometric Proof Strategies knowledge point
To prove the other pair of opposite angles are congruent (\(\angle B \cong \angle D\)), we write:

$$ \angle B = \angle ABD + \angle DBC \quad \text{and} \quad \angle D = \angle CDB + \angle BDA $$

This step requires the Angle Addition Postulate. Thus, Option C is necessary.

Answer:

• A. Corresponding parts of similar triangles are similar.
• B. Segment Addition Postulate.
• C. Angle Addition Postulate. (Correct answer)
• D. Corresponding parts of congruent triangles are congruent. (Correct answer)