In the figure, \(AD=CD\) and \(AB=CB\).

(i) Prove that \(\overline{BD}\) bisects angle \(ADC\) (i.e. that \(\overline{BD}\) cuts angle \(ADC\) into two equal angles).

(ii) Prove that \(\overline{AC}\) and \(\overline{DB}\) are perpendicular.

benjamingu22
Aug 17, 2017

#1**+1 **

AD is congruent to CD (given)

AB is congruent to CB (given)

DB is contruent to DB (identity)

Therefore, triangle(ADB) is congruent to triangle(DCB) (side-side-side)

and angle(ADB) is congruent to angle(CDB) (corresponding parts of congruent triangles are congruent)

Since angle(ADB) is congruent to angle(CDG), BD bisects angle(ADC).

AD is congruent to CD (given)

angle(ADB) is congruent to angleCDB) (proven above)

Therefore, triangle(ADE) is congruent to triangle(CDE) (side-angle-side)

and angle(AED) is congruent to angle(CED) (corresponding parts of congruent triangles are congruent)

Since angle(AED) and angle(CED) formf a linear pair (a straight line), angle(AED) and angle(CED) are right angles.

Thus, AC and DB are perpendicular.

geno3141
Aug 17, 2017

#2**+1 **

Here's another method to prove condition ii (\(\overline{AC}\perp\overline{DB}\)). I will utilize a two-column proof:

\(AD=CD\) \(AB=CB\) | Given |

\(\overline{AD}\cong\overline{CD}\) \(\overline{AB}\cong\overline{CB}\) | Definition of congruent segments |

Figure \(ABCD\) is a kite | Definition of a kite (If a quadrilateral has two unique pairs of sides that are congruent, then the figure is a kite) |

\(\overline{AC}\perp\overline{DB}\) | Property of a kite (Diagonals of a kite are perpendicular) |

TheXSquaredFactor
Aug 17, 2017