Question

complete the proof that $overleftrightarrow{ab}paralleloverleftrightarrow{ce}$. step statement reason 1 $ac = ad=bc = bd$ the three given circles have radii of the same length. 2 $cd = dc$ theyre lengths of the same segment. 3 side - side - side congruence (1, 1, 2) 4 corresponding parts of congruent triangles are congruent (3). 5 $overleftrightarrow{ab}paralleloverleftrightarrow{ce}$ lines with congruent alternate interior angles are parallel.

Explanation:

Step1: Identify congruent triangles

Since $AC = AD=BC = BD$ and $CD = DC$, by the Side - Side - Side (SSS) congruence criterion, $\triangle ACD\cong\triangle BDC$.

Step2: Find congruent angles

Because $\triangle ACD\cong\triangle BDC$, corresponding parts of congruent triangles are congruent. Let $\angle ACD$ and $\angle BDC$ be a pair of corresponding angles. So, $\angle ACD=\angle BDC$.

Step3: Apply parallel - line theorem

Since $\angle ACD$ and $\angle BDC$ are alternate interior angles and they are congruent, by the theorem that lines with congruent alternate interior angles are parallel, $\overleftrightarrow{AB}\parallel\overleftrightarrow{CE}$.

Answer:

1. $\triangle ACD\cong\triangle BDC$; 4. $\angle ACD=\angle BDC$