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

Explanation:

Step1: Identify base angles

The lower triangle has two equal sides (marked), so it is isosceles. The exterior angle is $48^\circ$, so the base angles of this triangle are: $\frac{180^\circ - 48^\circ}{2} = 66^\circ$ The acute angle of the large right triangle is equal to this base angle, so it is $66^\circ$.

Step2: Calculate right triangle angles

In the large right triangle, the sum of angles is $180^\circ$. We know one angle is $90^\circ$, another is $66^\circ$. Let the angle adjacent to $x^\circ$ be $\alpha$: $\alpha = 180^\circ - 90^\circ - 66^\circ = 24^\circ$

Step3: Solve for $x$

The segment with equal marks splits the $24^\circ$ angle? No, correct: The lower triangle's exterior angle $48^\circ$ is equal to the sum of the two remote interior angles of the small triangle with $x^\circ$. The small triangle is isosceles (equal sides marked), so the angle opposite the equal side is equal to the acute angle of the large triangle minus $x^\circ$. Wait, corrected: The lower segment is equal, so the left angle of the large triangle is $\theta$, and the lower isosceles triangle has $\theta$ as a base angle, so $180^\circ - 2\theta = 48^\circ$, so $\theta = 66^\circ$. Then in the large right triangle, the top angle (excluding $x$) is $180-90-66=24^\circ$. The upper segment is equal, so the triangle containing $x$ is isosceles: $x = 24^\circ$? No, wait: The exterior angle $48^\circ$ is equal to $x + x$ (since the small triangle is isosceles, $x$ and the other angle are equal, and $48^\circ$ is their sum). Corrected Step1: Exterior angle theorem The $48^\circ$ angle is an exterior angle of the small isosceles triangle (with two equal sides marked). For this triangle, the exterior angle equals the sum of the two non-adjacent interior angles, which are equal ($x^\circ$ each). $48^\circ = x^\circ + x^\circ$

Step2: Solve for $x$

$2x = 48$ $x = \frac{48}{2} = 24$

Answer:

$24$