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Giải bài 11 trang 92 sách bài tập toán 8 - Cánh diều

Cho tứ giác \(ABCD\) có \(\widehat C = \widehat D\) và \(AD = BC\). Chứng minh tứ giác \(ABCD\) là hình thang cân.

♔Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Cánh diều

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Đề bài

Cho tứ giác \(ABCD\) có \(\widehat C = \widehat D\) và \(AD = BC\). Chứng minh tứ giác \(ABCD\) là hình thang cân.

Phương pháp giải - Xem chi tiết

Dựa vào tính chất của hình thang cân: Trong một hình thang cân -         Hai cạnh bên bằng nhau -         Hai đường chéo bằng nhau.

Lời giải chi tiết

Gọi \(I\) là giao điểm của \(AD\) và \(BC\)Do \(\widehat C = \widehat D\) nên tam giác \(ICD\) cân tại \(I\). Suy ra \(ID = IC\)Mà \(AD = BC\), suy ra \(IA = IB\). Do đó, tam giác \(IAB\) cân tại \(I\).Vì hai tam giác \(IAB\) và \(ICD\) đều cân tại \(I\) nên\(\widehat {IAB} = \widehat D\) (cùng bằng \(\frac{{180^\circ  - \widehat I}}{2}\))Mà \(\widehat {IAB}\) và \(\widehat D\) nằm ở vị trí đồng vị, suy ra \(AB//CD\)Tứ giác \(ABCD\) có \(AB//CD\) và \(\widehat C = \widehat D\) nên \(ABCD\) là hình thang cân.

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