Question - Calculation of Angles in Triangles
Solution:
To find the value of $$ x $$, we need to look at the triangle $$ ABC $$. Since we know two angles in the triangle, $$ 15^\circ $$ (angle $$ A $$) and $$ 45^\circ $$ (angle $$ B $$), we can find angle $$ C $$ because the sum of the angles in a triangle is $$ 180^\circ $$.Let's calculate angle $$ C $$:$$ C = 180^\circ - A - B = 180^\circ - 15^\circ - 45^\circ = 120^\circ $$Now we can focus on triangle $$ BCD $$, it appears to be an isosceles triangle since $$ BD = DC $$. Therefore, angle $$ B $$ and angle $$ C $$ in this triangle should be equal. However, were also given that $$ DE $$ is congruent to $$ BD $$, which means triangle $$ BDE $$ is also isosceles, and angles $$ B $$ and $$ E $$ must be equal.Since the large angle $$ C $$ of the big triangle $$ ABC $$ is split into two angles $$ C $$ and $$ E $$ within the triangle $$ BCD $$ and $$ BDE $$ respectively, we must have:$$ C $$ (angle within triangle $$ BCD $$) $$ + E = 120^\circ $$ (angle C of $$ ABC $$)Since $$ C $$ (within $$ BCD $$) is equal to $$ B = 45^\circ $$, we get:$$ 45^\circ + E = 120^\circ $$Now we can solve for $$ E $$:$$ E = 120^\circ - 45^\circ = 75^\circ $$This is the value of angle $$ E $$, and since $$ DE $$ is congruent to $$ BD $$, angle $$ D $$ in triangle $$ BDE $$ is also $$ 75^\circ $$. Our goal is to find the angle $$ x $$ which is the angle $$ D $$ in triangle $$ ADE $$. Since $$ ADE $$ is an isosceles triangle with $$ AE = DE $$, the angles at $$ A $$ and $$ D $$ are equal.Therefore, angle $$ x $$ (which is angle $$ D $$ in $$ ADE $$) equals $$ 75^\circ $$.
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