Question

7.
in the triangle, the angles are $x^\circ$, $82^\circ$, $24^\circ$
8.
in the triangle, the angles are $90^\circ$, $(x + 5)^\circ$, and there is an exterior angle of $115^\circ$

Explanation:

Problem 7

Step1: Use exterior angle property

The exterior angle $x^\circ$ equals the sum of the two non-adjacent interior angles.
$x = 82 + 24$

Step2: Calculate the sum

$x = 106$
Wait, correction: No, the exterior angle is equal to the sum of the two remote interior angles? No, wait, actually, the angle adjacent to $x$ is supplementary to $x$, and the sum of angles in a triangle is 180. Let's correct:

Step1: Find adjacent angle to x

Let the angle adjacent to $x$ be $y$. $y + 82 + 24 = 180$
$y = 180 - 82 - 24 = 74$

Step2: Supplementary with x

$x + y = 180$, so $x = 180 - 74 = 106$
Wait no, no, the exterior angle theorem says that the exterior angle is equal to the sum of the two opposite interior angles. Oh right! $x = 82 + 24 = 106$. That's correct.

Problem 8

Step1: Use exterior angle property

The exterior angle $55^\circ$ equals the sum of the two non-adjacent interior angles.
$55 = 40 + (x + 5)$

Step2: Simplify and solve for x

$55 = 45 + x$
$x = 55 - 45 = 10$
Wait, correction: Wait, the exterior angle is $55^\circ$, the two interior angles are $40^\circ$ and $(x+5)^\circ$. So $55 = 40 + x +5$
$55 = 45 + x$
$x = 10$

Wait, let's recheck problem 7: The triangle has angles 82, 24, and the angle adjacent to x. The sum of angles in triangle is 180, so adjacent angle is 180-82-24=74. Then x and 74 are supplementary, so x=180-74=106. Which matches the exterior angle theorem (x=82+24=106). Correct.

Final Answer:

Problem 7: $x = 106$
Problem 8: $x = 10$

Problem 7

Step1: Apply Exterior Angle Theorem

The exterior angle $x$ equals the sum of the two remote interior angles.
$x = 82 + 24$

Step2: Compute the sum

$x = 106$

Problem 8

Step1: Apply Exterior Angle Theorem

The exterior angle $55^\circ$ equals the sum of the two remote interior angles.
$55 = 40 + (x + 5)$

Step2: Solve for $x$

$55 = 45 + x \implies x = 55 - 45 = 10$

Answer:

For problem 7: $x = 58$
For problem 8: $x = 30$