Segments
AB
and
CD
intersect (see picture to the right). Prove that if segments
AC
,
CB
,
BD
and
AD
are congruent, then ray
AB
is the bisector of ∠CAD and ray
CD
is the bisector of ∠ACB. Prove that segments
AB
and
CD
are perpendicular.

Question

Sasha06 Sasha06

11-10-2018
Mathematics

contestada

Segments
AB
and
CD
intersect (see picture to the right). Prove that if segments
AC
,
CB
,
BD
and
AD
are congruent, then ray
AB
is the bisector of ∠CAD and ray
CD
is the bisector of ∠ACB. Prove that segments
AB
and
CD
are perpendicular.

Respuesta :

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JackelineCasarez JackelineCasarez · Answer 1 · 2018-10-22T10:35:24+02:00

Given

AB and CD intersect

AC, CB, BD and AD are congruent.

Prove that AB is the bisector of ∠CAD and ray CD is the bisector of ∠ACB.

and AB and CD are perpendicular.

To proof

Bisector

A bisector is that which cut an angle in two equal parts.

In ΔACB and ΔADB

AD = AC ( Given )

AB = AB ( common )

BC = DB ( Given )

by SSS congurence property

we have

ΔACB ≅ΔADB

∠CAB =∠ DAB

∠CBA = ∠DBA

( By corresponding sides of the congurent triangle )

Thus AB is the bisector of the ∠CAD.

InΔ DAC and ΔDBC

AD = DB (Given)

AC = CB ( Given )

CD = CD (common)

By SSS congurence property

ΔDAC≅ Δ DBC

∠ ACD =∠ BCD

∠ADC =∠BDC

( By corresponding sides of the congurent triangle )

Therefore CD is the bisector of the CAD.

In ΔBOC andΔ BOD

BO = BO ( Common )

∠BCO = ∠BDO

( As prove above ΔACB ≅ΔADB

Thus ∠ACB = ∠ADB by corresponding sides of the congurent triangle , CD is a bisector

∠BCO = ∠BDO )

CB = DB ( given )

by SAS congurence property

ΔBOC ≅ ΔBOD

∠BOC =∠ BOD

∠BOC +∠ BOD = 180 °( Linear pair )

2∠ BOC = 180°

∠BOC = 90°

∠BOC =∠ BOD = 90°

also

In ΔCOA and ΔAOD

AO = AO ( Common )

∠ACO =∠ ADO

( As prove above ΔACB ≅ΔADB Thus ACB = ADB by corresponding sides of congurent triangle ,CD is a bisector

thus ∠ACO = ∠ADO )

AC =AD ( given )

by SAS congurence property

Δ COA ≅ ΔAOD

∠AOC = ∠AOD

( By corresponding angle of corresponding sides )

∠AOC + ∠AOD = 180°

2∠ AOC = 180° ( Linear pair )

∠AOC = 90°

∠AOC = ∠AOD = 90 °

Thus AB and CD are perpendicular.

Hence proved

Segments AB and CD intersect (see picture to the right). Prove that if segments AC , CB , BD and AD are congruent, then ray AB is the bisector of ∠CAD and ray CD is the bisector of ∠ACB. Prove that segments AB and CD are perpendicular.

Respuesta :

Otras preguntas

Segments
AB
and
CD
intersect (see picture to the right). Prove that if segments
AC
,
CB
,
BD
and
AD
are congruent, then ray
AB
is the bisector of ∠CAD and ray
CD
is the bisector of ∠ACB. Prove that segments
AB
and
CD
are perpendicular.