Question

4. În figura alăturată patrulaterul ABCD reprezintă schița unui zmeu. Dacă AB=AC=AD,
m(BAC)= 40° și m(CAD)= 50°, atunci măsura unghiului BCD este:
a) 90°
b) 135°
c) 120°
d) 1050
​

Answer 1

Răspuns: $\bf \angle{BCD}=135^{\circ}$

Explicație pas cu pas:

Salutare!

$\bf In~\triangle{BAC}-isoscel~avem:$

$\bf \angle{BAC}=40^{\circ}$

$\bf \angle{ABC}=\angle{ACB}$

$\bf [AB]=[AC]$

$\bf \implies \angle{ABC}+\angle{BAC}+\angle{ACB}=180^{\circ}$

$\bf 2\cdot \angle{ABC}+40^{\circ}=180^{\circ}$

$\bf 2\cdot \angle{ABC}=180^{\circ}-40^{\circ}$

$\bf 2\cdot \angle{ABC}=140^{\circ}~~~\Big|:2$

$\boxed{\bf \angle{ABC}=70^{\circ}\implies \angle{ACB}=70^{\circ}}$

$\bf In~\triangle{ACD}-isoscel~avem:$

$\bf \angle{CAD}=50^{\circ}$

$\bf \angle{ADC}=\angle{ACD}$

$\bf [AD]=[AC]$

$\bf \implies \angle{ADC}+\angle{ACD}+\angle{CAD}=180^{\circ}$

$\bf 2\cdot \angle{ACD}+50^{\circ}=180^{\circ}$

$\bf 2\cdot \angle{ACD}=180^{\circ}-50^{\circ}$

$\bf 2\cdot \angle{ACD}=130^{\circ}~~~\Big|:2$

$\boxed{\bf \angle{ACD}=65^{\circ}\implies \angle{ADC}=65^{\circ}}$

$\bf \angle{BCD}=70^{\circ}+65^{\circ}$

$\boxed{\boxed{\bf \angle{BCD}=135^{\circ}}}$

Varianta b)

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