Question

Given: dd is the midpoint of start overline, a, c, end overline, comma ac , start overline, e, a, end overline, cong, start overline, f, c, end overline ea ≅ fc and start overline, b, a, end overline, cong, start overline, b, c, end overline, . ba ≅ bc . prove: start overline, d, e, end overline, cong, start overline, d, f, end overline de ≅ df .

Answer 1

Triangles DE and DF are congruent using the properties of similar triangles.

Step-by-step explanation:

To prove that de ≅ df, we can use the given information that triangles BAC and B₁A₁C are similar. From this, we know that triangles NOF and B₁A₁F are similar as well. Since ea ≅ fc and ba ≅ bc, we can conclude that AB = A₁B₁. Therefore, using the properties of similar triangles, we can find that triangles DEF and DFG are also similar. By the transitive property, we can then conclude that de ≅ df.

Answer 2

To prove DE ≅ DF, given D is the midpoint of AC and EA ≅ FC as well as BA ≅ BC, one can show that triangles BAE and BCF are congruent using the SSS criteria, and thus DE ≅ DF by CPCTC.

Step-by-step explanation:

To prove that DE ≡ DF, given that D is the midpoint of AC, and EA ≡ FC and BA ≡ BC, we can use triangle congruence criteria.

Firstly, we know that D is the midpoint of AC, so AD ≡ DC. Since EA ≡ FC, adding AD to EA and DC to FC gives DE ≡ DF because of the Segment Addition Postulate. Lastly, BA ≡ BC confirms that ∤ BAE and ∤ BCF are congruent by Side-Side-Side (SSS) congruence.

Since ∤ BAE and ∤ BCF are congruent, ∤ BDE and ∤ BDF are also congruent by Corresponding Parts of Congruent Triangles are Congruent (CPCTC), which implies that DE ≡ DF.