https://file.aiquickdraw.com/mathbot/file/1729530325840wsd284v4.jpg

Let's go through each problem one by one. The instruction is to find the measure of the exterior angle shown.

### Problem 17:
The diagram shows a triangle with one exterior angle of \( (2x - 2)^\circ \), and two interior angles: \( x^\circ \) and \( 45^\circ \). The exterior angle theorem tells us that an exterior angle is equal to the sum of the two opposite interior angles.

Using the exterior angle theorem:
\[
(2x - 2)^\circ = x^\circ + 45^\circ
\]
Now solve for \( x \):
\[
2x - 2 = x + 45
\]
Subtract \( x \) from both sides:
\[
x - 2 = 45
\]
Add 2 to both sides:
\[
x = 47
\]

Now, substitute \( x = 47 \) back into the expression for the exterior angle:
\[
(2x - 2) = 2(47) - 2 = 94 - 2 = 92^\circ
\]

Thus, the exterior angle is **92°**.

---

### Problem 18:
We have an exterior angle of \( 24^\circ \), and the two adjacent interior angles are given as \( (3x + 6)^\circ \) and \( (2x + 18)^\circ \).

Again, using the exterior angle theorem:
\[
24^\circ = (3x + 6)^\circ + (2x + 18)^\circ
\]
Now solve for \( x \):
\[
24 = 3x + 6 + 2x + 18
\]
Combine like terms:
\[
24 = 5x + 24
\]
Subtract 24 from both sides:
\[
0 = 5x
\]
Thus:
\[
x = 0
\]

Substitute \( x = 0 \) back into the expressions for the interior angles:
\[
(3x + 6) = 3(0) + 6 = 6^\circ
\]
\[
(2x + 18) = 2(0) + 18 = 18^\circ
\]
Check: \( 6^\circ + 18^\circ = 24^\circ \), which matches the given exterior angle.

Thus, the exterior angle is **24°**, as given.

---

### Problem 19:
This is a right triangle, where one angle is \( 90^\circ \), and the exterior angle shown is \( (3x + 2)^\circ \), with an interior angle of \( x^\circ \).

The sum of the interior angle and its exterior angle is always \( 180^\circ \) because they form a linear pair:
\[
x^\circ + (3x + 2)^\circ = 180^\circ
\]
Now solve for \( x \):
\[
x + 3x + 2 = 180
\]
Combine like terms:
\[
4x + 2 = 180
\]
Subtract 2 from both sides:
\[
4x = 178
\]
Divide by 4:
\[
x = 44.5
\]

Now, substitute \( x = 44.5 \) back into the expression for the exterior angle:
\[
(3x + 2) = 3(44.5) + 2 = 133.5 + 2 = 135.5^\circ
\]

Thus, the exterior angle is **135.5°**.

---

Would you like further details or clarifications on any of these problems?

### Related Questions:
1. How does the exterior angle theorem apply to different types of triangles?
2. What is the difference between an exterior angle and an interior angle?
3. Can you explain how to solve for unknown angles when given supplementary angles?
4. How would the process change if one of the angles was labeled differently?
5. Can the exterior angle theorem be applied in polygons other than triangles?

### Tip:
The exterior angle theorem is a powerful tool that allows you to solve for unknown angles in triangles quickly. Always remember that the exterior angle is equal to the sum of the two opposite interior angles.

Find the measure of the exterior angles in Problems 17, 18, and 19 as shown in the provided image.

Explore how to find the measure of exterior angles in geometric problems using the Exterior Angle Theorem and Linear Pair of Angles. This detailed step-by-step solution covers Problems 17, 18, and 19, ideal for students in Grades 8-10.

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