Question

in △abc, point d is the midpoint of ac. fill in the blanks to prove the isosceles triangle theorem. given: ab ≅ bc prove: ∠a ≅ ∠c it is __ that ab ≅ bc. by the reflexive property, segment bd is congruent to . since d is the midpoint of ac, ≅ dc. △abd ≅ △cbd by . therefore, ∠a ≅ ∠c by __.

Explanation:

Step1: Identify Given Information

Given \( AB \cong BC \) (from the problem's "Given" section) and \( D \) is the midpoint of \( AC \), so \( AD \cong CD \) (by definition of midpoint: a midpoint divides a segment into two congruent segments). Also, \( BD \cong BD \) (Reflexive Property of Congruence).

Step2: Determine Triangle Congruence Criterion

We have three sides: \( AB \cong BC \), \( AD \cong CD \), and \( BD \cong BD \). So the SSS (Side - Side - Side) congruence criterion applies to \( \triangle ABD \) and \( \triangle CBD \). Thus, \( \triangle ABD \cong \triangle CBD \) by SSS.

Step3: Use CPCTC for Angle Congruence

After proving \( \triangle ABD \cong \triangle CBD \) by SSS, we use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to conclude that \( \angle A \cong \angle C \).

Filling the Blanks:

1. First blank (reason for \( AB \cong BC \)): "given"
2. Blank for the segment congruent to \( BD \) (by Reflexive Property): "segment BD" (since \( BD \cong BD \))
3. Blank for the midpoint conclusion: "segment AD" \( \cong \) "segment CD" (because \( D \) is the midpoint of \( AC \))
4. Blank for triangle congruence: "SSS" (since we used three sides for congruence)
5. Blank for angle congruence reason: "CPCTC" (Corresponding Parts of Congruent Triangles are Congruent)

Answer:

• First blank: given
• Segment congruent to \( BD \): segment BD
• Midpoint conclusion: segment AD \( \cong \) segment CD
• Triangle congruence: SSS
• Angle congruence reason: CPCTC