Question

5.prove for kite abcd above that BD bisects both ABC and ADC

Answer 1

The kite is a quadrilateral that has two pairs of equal adjacent sides and

one of equal opposite angles.

5. The two column proof is presented as follows;

Statement ${}$                            Reason

$\overline{AB}$ = $\overline{BC}$     ${}$                           Given

$\overline{AB}$ ≅ $\overline{BC}$     ${}$                          Definition of congruency

$\overline{AD}$ = $\overline{CD}$     ${}$                           Given

$\overline{AD}$ ≅ $\overline{CD}$     ${}$                          Definition of congruency

$\overline{BD}$ ≅ $\overline{BD}$     ${}$                          By reflexive property

ΔABD ≅ ΔBDC    ${}$                   By Side-Side-Side, SSS congruency postulate

∠ABC = ∠ABD + ∠CBD      ${}$    Angle addition postulate

∠ABD ≅ ∠CBD     ${}$                  CPCTC

∠ABD = ∠CBD     ${}$                   Definition of congruency

$\overline{BD}$ is a bisector of  ∠ABC    ${}$ Definition of angle bisector

∠ADC = ∠ABD + ∠CBD      ${}$     Angle addition postulate

∠ABD ≅ ∠CBD     ${}$                  CPCTC

∠ABD = ∠CBD     ${}$                   Definition of congruency

Therefore;

$\overline{BD}$ is a bisector of ∠ADC     ${}$ Definition of angle bisector

CPCTC is the acronym for Congruent Parts of Congruent Triangles are Congruent

6. The two column proof is presented as follows;

Statement ${}$                          Reason

∠ABD ≅ ∠CBD  ${}$           CPCTC (question 5)

$\overline{AB}$ = $\overline{BC}$     ${}$                  Given

$\overline{BM}$ ≅ $\overline{BM}$      ${}$              Reflexive property

ΔABM ≅ ΔCBM      ${}$      According to Side-Angle-Side, SAS congruency rule

$\overline{AM}$ ≅ $\overline{CM}$      ${}$               CPCTC

$\overline{AM}$ = $\overline{CM}$    ${}$                  Definition of congruency

$\overline{AC}$ = $\overline{AM}$ + $\overline{CM}$    ${}$        Segment addition postulate

$\overline{AC}$ = $\overline{AM}$ + $\overline{AM}$ =  2·$\overline{AM}$ ${}$ Substitution property of equality

$\overline{AM}$ = 0.5·$\overline{AC}$    ${}$        Division property of equality

M divides $\overline{AC}$ to two   ${}$  Definition

M is midpoint of $\overline{AC}$   ${}$   Definition of midpoint

Learn more here:

brainly.com/question/20644467