Consider the incomplete paragraph proof shown. Given: Isosceles AABC with AB BC. Prove: LA LC A D C Let D be the midpoint of AC and draw in BD. From the given statements and the definition of isosceles triangle, AB BC. Also, since D is the midpoint of AC By the ,it is also known that BD BD Therefore, AABD ACBD by the Therefore, LA LC because corresponding parts of congruent triangles are congruent. What are the missing parts of the proof? A AD CD Reflexive Property, SAS Congruence Postulate B AD CD. Reflexive Property, SSS Congruence Postulate C C AC BD. Symmetric Property, SAS Congruence Postulate D AC BD Symmetric Property, SSS Congruence Postulate
The Correct Answer and Explanation is :
The correct answer is A: ( AD \cong CD ), Reflexive Property, SAS Congruence Postulate.
Explanation:
We are given an isosceles triangle ( \triangle ABC ) with ( AB \cong BC ), and we need to prove that ( \angle A \cong \angle C ).
- Identifying Key Given Information
- Since ( \triangle ABC ) is isosceles with ( AB \cong BC ), we know that two sides are congruent.
- ( D ) is the midpoint of ( AC ), meaning ( AD \cong CD ).
- Defining BD and Using Reflexive Property
- Since we drew segment ( BD ), we note that ( BD \cong BD ) by the Reflexive Property (a segment is always congruent to itself).
- Proving Triangle Congruence Using SAS Postulate
- We have two pairs of congruent sides:
- ( AB \cong BC ) (given).
- ( AD \cong CD ) (since ( D ) is the midpoint of ( AC )).
- The included angle ( \angle ABD \cong \angle CBD ) (because they are opposite each other along the common segment ( BD )). By the SAS (Side-Angle-Side) Congruence Postulate, we conclude that ( \triangle ABD \cong \triangle CBD ).
- Using CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
- Since ( \triangle ABD \cong \triangle CBD ), their corresponding angles are also congruent.
- This means ( \angle A \cong \angle C ), proving the statement.
Why Other Options Are Incorrect:
- Option B: Incorrect because it states the SSS Postulate, but we do not have three pairs of congruent sides.
- Option C & D: Incorrect because they state the Symmetric Property, which is not necessary here. Instead, the Reflexive Property is needed for ( BD \cong BD ).
Thus, the correct missing parts are ( AD \cong CD ), Reflexive Property, SAS Congruence Postulate.