Respuesta :
Statement Reason
AB and BC are congruent
CPCTC
DB and FB are congruent
CPCTC
Angle EBD congruent to angle EFB
CPCTC
Angle ABD congruent to angle EBD
Alternate angle theorem
Angle FBC congruent to angle EFB
Alternate angle theorem
ABD CBF
SAS theorem
Two angles, sides or figure are said to be congruent when they are equal
1. Question: The given parameters in the question are resented as follows;
[tex]\overline{AC} \parallel \overline{DF}[/tex]
The midpoint of the line [tex]\overline{DF}[/tex], is the point E
The midpoint of the line [tex]\overline{AC}[/tex], is the point B
The line [tex]\overline{EB} \perp \overline{AC}[/tex]
1. Required:
To complete the table from the question
Solution:
Engineer [tex]{}[/tex] Conjecture
Natalie
SAS [tex]{}[/tex] ΔABD ≅ ΔCBF
Summary: Natalie's statement is that ΔABD is congruent to ΔCBF by Side Angle Side rule of congruency
Emma
SSS [tex]{}[/tex] ΔABD ≅ ΔCBF
Summary: Emma's statement is that ΔABD is congruent to ΔCBF by Side Side Side rule of congruency
2. Required:
Analysis of the two conjectures
Solution:
Given that the information on ΔABD and ΔCBF are based on the relative lengths of the height and base of both triangles, and that both refer to the same triangle, both are correct. However, there are no direct information with regards to the sides [tex]\overline{AD}[/tex], and [tex]\overline{CF}[/tex], Therefore, Natalie's conjecture is based directly on the given information
3. Required:
To prove that ΔBED ≅ ΔBEF
The two column proof is presented as follows;
Statement [tex]{}[/tex] Reason
1. E is the point [tex]\overline{DF}[/tex] [tex]{}[/tex] 1. Given
2. [tex]\overline{DE}[/tex] = [tex]\overline{EF}[/tex] [tex]{}[/tex] 2. By definition of midpoint of [tex]\overline{DF}[/tex]
3. [tex]\overline{EB} \perp \overline{AC}[/tex] [tex]{}[/tex] 3. Given
4. ∠BED = ∠BEF = 90° [tex]{}[/tex] 4. By definition of [tex]\overline{EB} \perp \overline{AC}[/tex]
5. [tex]\overline{EB} \cong\overline{EB}[/tex] [tex]{}[/tex] 5. Reflexive property
6. ΔBED ≅ ΔBEF [tex]{}[/tex] 6. Side Angle Side, (SAS) rule of congruency
4. Required:
To state how ΔBDF is an isosceles triangle
Solution:
Triangle ΔBDF is an isosceles triangle because by Congruent Parts of Congruent Triangles are Congruent, CPCTC, [tex]\mathbf{\overline{BD}=\overline{BF}}[/tex], therefore, given that two sides of ΔBDF are equal, ΔBDF is an isosceles triangle by definition
5. Required:
To fill out the proof that ΔABD ≅ ΔCBF
Solution;
Statement [tex]{}[/tex] Reason
1. B is the point [tex]\overline{AC}[/tex] [tex]{}[/tex] 1. Given
2. [tex]\overline{BC}[/tex] = [tex]\overline{AB}[/tex] [tex]{}[/tex] 2. By definition of midpoint of [tex]\overline{AC}[/tex]
3. [tex]\overline{BD}=\overline{BF}[/tex] [tex]{}[/tex] 3. ΔBED ≅ ΔBEF (or ΔBDF is an isosceles Δ)
4. ∠BDE = ∠BFE [tex]{}[/tex] 4. Base angles of isosceles ΔBDF
5. ∠ABD = ∠BDE, ∠CBF = ∠BFE [tex]{}[/tex] [tex]{}[/tex] 5. Alternate angle theorem
6. ∠ABD = ∠CBF [tex]{}[/tex] 6. Transitive property
7. ΔABD ≅ ΔCBF [tex]{}[/tex] 7. Side Angle Side, (SAS) rule of congruency
Learn more about two column proofs here:
https://brainly.com/question/13969040
