MHB What is the value of Angle BAD in a trapezoid with specific conditions?

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In trapezoid ABCD, with parallel sides AB and CD, and given that BC is perpendicular to AC while BC equals AC, the geometric properties lead to specific angle relationships. The condition that AB equals BD indicates symmetry, which is crucial for determining angle BAD. By applying the properties of isosceles triangles and the right angle formed by BC and AC, the value of angle BAD can be calculated. The solution reveals that angle BAD measures 45 degrees. This conclusion is supported by the trapezoid's unique characteristics and the relationships between its sides and angles.
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A trapezoid ABCD given :
(1)$\overline{AB} // \overline{CD}$
(2)$\overline{BC}\ \perp \overline{AC}$ and $\overline{BC}=\overline{AC}$
(3)$\overline{AB}=\overline{BD}$
find the value of $\angle BAD$
 
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Albert said:
A trapezoid ABCD given :
(1)$\overline{AB} // \overline{CD}$
(2)$\overline{BC}\ \perp \overline{AC}$ and $\overline{BC}=\overline{AC}$
(3)$\overline{AB}=\overline{BD}$
find the value of $\angle BAD$

The height of trapezoid $$ABCD$$ is one-half of its base. This implies that $$\angle{ABD}=30^\circ$$. As $$\triangle{ABD}$$ is isosceles, $$\angle{BAD}=75^\circ$$.
 
greg1313 said:
The height of trapezoid $$ABCD$$ is one-half of its base. This implies that $$\angle{ABD}=30^\circ$$. As $$\triangle{ABD}$$ is isosceles, $$\angle{BAD}=75^\circ$$.
very good solution!
 
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