find the value of $x$.\n\nif necessary, you may learn what the markings on a figure indicate.\n\n$x =…

Explanation:

Step1: Identify the properties of the right triangle

The large triangle is a right triangle with a $90^{\circ}$ angle and a $27^{\circ}$ angle.

Step2: Calculate the third angle of the large triangle

The sum of angles in a triangle is $180^{\circ}$. $$180^{\circ} - 90^{\circ} - 27^{\circ} = 63^{\circ}$$

Step3: Identify the properties of the inner triangle

The inner triangle is isosceles because two sides are marked equal.

Step4: Determine the base angles of the isosceles triangle

The angle opposite one equal side is $27^{\circ}$, so the other base angle is also $27^{\circ}$.

Step5: Calculate the vertex angle of the isosceles triangle

The vertex angle at the top of the isosceles triangle is: $$180^{\circ} - (27^{\circ} + 27^{\circ}) = 126^{\circ}$$

Step6: Relate the angles at the top vertex

The total angle at the top vertex of the large triangle is $63^{\circ}$. The angle $x$ is part of this angle. However, looking at the geometry, the angle $x$ is the complement of the base angle of the small right triangle.

Step7: Calculate the exterior angle of the isosceles triangle

The exterior angle to the isosceles triangle is the base angle of the small right triangle: $$27^{\circ} + 27^{\circ} = 54^{\circ}$$

Step8: Solve for $x$ in the small right triangle

In the small right triangle, the sum of $x$ and the exterior angle is $90^{\circ}$. $$x + 54^{\circ} = 90^{\circ}$$ $$x = 90^{\circ} - 54^{\circ}$$

Answer:

$x = 36$