\(\Delta APB\) có:
\(\widehat{APD}=\dfrac{180^o-\widehat{PAD}-\widehat{PDA}}=\dfrac{360^o-\widehat{BAD}-\widehat{CDA}}{2}\)
\(\Delta BMC\) có:
\(\widehat{BMC}=\dfrac{180^o-\widehat{MBC}-\widehat{MCB}}=\dfrac{360^o-\widehat{ABC}-\widehat{DCB}}{2}\)
Suy ra: \(\widehat{QPN}+\widehat{QMN}=\widehat{APD}+\widehat{BMC}\\=\dfrac{360^o-\widehat{BAD}-\widehat{CDA}}{2}+\dfrac{360^o-\widehat{ABC}-\widehat{DCB}}{2}\\=\dfrac{360^o-\widehat{BAD}-\widehat{CDA}+360^o-\widehat{ABC}-\widehat{DCB}}{2}\\=\dfrac{720^o-(\widehat{BAD}+\widehat{CDA}+\widehat{ABC}+\widehat{DCB})}{2}=\\dfrac{720^o-360^o}{2}=\dfrac{360^o}{2}=180^o\)
Do đó: \(\widehat{MQP}+\widehat{MNP}=360^o-(\widehat{QPN}+\widehat{QMN})=360^o-180^o=180^o\)
Vậy tứ giác MNPQ có các góc đối bù nhau.
