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

6.Given from what we now know from 5, prove that M must be the midpoint of AC​

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 [tex]{}[/tex]                            Reason

[tex]\overline{AB}[/tex] = [tex]\overline{BC}[/tex]     [tex]{}[/tex]                           Given

[tex]\overline{AB}[/tex] ≅ [tex]\overline{BC}[/tex]     [tex]{}[/tex]                          Definition of congruency

[tex]\overline{AD}[/tex] = [tex]\overline{CD}[/tex]     [tex]{}[/tex]                           Given

[tex]\overline{AD}[/tex] ≅ [tex]\overline{CD}[/tex]     [tex]{}[/tex]                          Definition of congruency

[tex]\overline{BD}[/tex] ≅ [tex]\overline{BD}[/tex]     [tex]{}[/tex]                          By reflexive property

ΔABD ≅ ΔBDC    [tex]{}[/tex]                   By Side-Side-Side, SSS congruency postulate

∠ABC = ∠ABD + ∠CBD      [tex]{}[/tex]    Angle addition postulate

∠ABD ≅ ∠CBD     [tex]{}[/tex]                  CPCTC

∠ABD = ∠CBD     [tex]{}[/tex]                   Definition of congruency

[tex]\overline{BD}[/tex] is a bisector of  ∠ABC    [tex]{}[/tex] Definition of angle bisector

∠ADC = ∠ABD + ∠CBD      [tex]{}[/tex]     Angle addition postulate

∠ABD ≅ ∠CBD     [tex]{}[/tex]                  CPCTC

∠ABD = ∠CBD     [tex]{}[/tex]                   Definition of congruency

Therefore;

[tex]\overline{BD}[/tex] is a bisector of ∠ADC     [tex]{}[/tex] 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 [tex]{}[/tex]                          Reason

∠ABD ≅ ∠CBD  [tex]{}[/tex]           CPCTC (question 5)

[tex]\overline{AB}[/tex] = [tex]\overline{BC}[/tex]     [tex]{}[/tex]                  Given

[tex]\overline{BM}[/tex] ≅ [tex]\overline{BM}[/tex]      [tex]{}[/tex]              Reflexive property

ΔABM ≅ ΔCBM      [tex]{}[/tex]      According to Side-Angle-Side, SAS congruency rule

[tex]\overline{AM}[/tex] ≅ [tex]\overline{CM}[/tex]      [tex]{}[/tex]               CPCTC

[tex]\overline{AM}[/tex] = [tex]\overline{CM}[/tex]    [tex]{}[/tex]                  Definition of congruency

[tex]\overline{AC}[/tex] = [tex]\overline{AM}[/tex] + [tex]\overline{CM}[/tex]    [tex]{}[/tex]        Segment addition postulate

[tex]\overline{AC}[/tex] = [tex]\overline{AM}[/tex] + [tex]\overline{AM}[/tex] =  2·[tex]\overline{AM}[/tex] [tex]{}[/tex] Substitution property of equality

[tex]\overline{AM}[/tex] = 0.5·[tex]\overline{AC}[/tex]    [tex]{}[/tex]        Division property of equality

M divides [tex]\overline{AC}[/tex] to two   [tex]{}[/tex]  Definition

M is midpoint of [tex]\overline{AC}[/tex]   [tex]{}[/tex]   Definition of midpoint

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