An isosceles triangle have two sides and two angles equal, therefore,
ΔBDC is an isosceles Δ when [tex]\overline{BD} \cong \overline{DC}[/tex].
The correct option is the second option;
1. [tex]\overline{AB} \cong \overline{AC}[/tex] [tex]{}[/tex](Given)
2. [tex]\overline{AD}[/tex] bisects ∠BAC [tex]{}[/tex] (Given)
3. ∠BAD ≅ ∠CAD (Def. of ∠bisector)
4. [tex]\overline{AD} \cong \overline{AD}[/tex] [tex]{}[/tex] (Reflex Prop of ≅)
5. ΔBAD ≅ ΔCAD (SAS Steps 1, 3, 4)
6. [tex]\overline{BD} \cong \overline{DC}[/tex] [tex]{}[/tex](CPCTC)
7. ΔBDC is isosceles (Def of isosceles Δ)
Reasons;
The proof that ΔBDC is an isosceles triangle can be found using the
correct option, presented in the following two column proof;
Statement [tex]{}[/tex] Reason
1. [tex]\overline{AB} \cong \overline{AC}[/tex] [tex]{}[/tex] 2. Given
2. [tex]\overline{AD}[/tex] bisects ∠BAC [tex]{}[/tex] 2. Given
3. ∠BAD ≅ ∠CAD [tex]{}[/tex] 3. Definition of angle bisector
4. [tex]\overline{AD} \cong \overline{AD}[/tex] [tex]{}[/tex] [tex]{}[/tex] 4. Reflexive property of congruency
5. ΔBAD ≅ ΔCAD [tex]{}[/tex] 5. SAS rule rule of congruency
6. [tex]\overline{BD} \cong \overline{DC}[/tex] [tex]{}[/tex] 6. CPCTC
7. ΔBDC is isosceles [tex]{}[/tex] 7. By definition of isosceles triangle
The acronyms in the correct proof are;
SAS stands for Side Angle Side congruency postulate, which state that
two triangles are congruent where two sides and an included angle in one
triangle are congruent to the corresponding two sides and included angle
in the other triangle.
CPTCT stands for Congruent Parts of Congruent Triangle are Congruent,
which means that two triangles that are said to be congruent have equal
lengths of corresponding sides and equal measures of corresponding
angles.
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