Question

∠1与∠3

1. proof write a proof.

given: \\(\overline{cd} \cong \overline{cg}\\)
\\(\overline{de} \cong \overline{gf}\\)
prove: \\(\overline{ce} \cong \overline{cf}\\)

1. proof write a proof.

given: \\(\overline{de} \parallel \overline{bc}\\)
\\(\angle 1 \cong \angle 2\\)
prove: \\(\overline{ab} \cong \overline{ac}\\)

Explanation:

Step1: List given info

$\overline{CD} \cong \overline{CG}$, $\overline{DE} \cong \overline{GF}$, $\angle D \cong \angle G$ (right angles, implied by diagram)

Step2: Prove $\triangle CDE \cong \triangle CGF$

By SAS congruence: $\overline{CD} \cong \overline{CG}$, $\angle D \cong \angle G$, $\overline{DE} \cong \overline{GF}$

Step3: Conclude $\overline{CE} \cong \overline{CF}$

Corresponding parts of congruent triangles are congruent (CPCTC)

Step1: List given info

$\overline{DE} \parallel \overline{BC}$, $\angle 1 \cong \angle 2$

Step2: Relate angles from parallel lines

$\angle 2 \cong \angle 3$ (corresponding angles, $\overline{DE} \parallel \overline{BC}$)

Step3: Transitive property for angles

$\angle 1 \cong \angle 3$ (since $\angle 1 \cong \angle 2$ and $\angle 2 \cong \angle 3$)

Step4: Conclude $\overline{AB} \cong \overline{AC}$

If base angles $\angle 1 \cong \angle 3$, then $\triangle ABC$ is isosceles, so $\overline{AB} \cong \overline{AC}$

Answer:

$\overline{CE} \cong \overline{CF}$ is proven via SAS congruence and CPCTC.