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To solve for \( w \) in this geometric problem, we can use the fact that the sum of the angles in any triangle is always \( 180^\circ \).

The three angles in the triangle are labeled as:
1. \( w + 46^\circ \)
2. \( w + 7^\circ \)
3. \( 2w - 19^\circ \)

So, we can set up the equation:
\[
(w + 46^\circ) + (w + 7^\circ) + (2w - 19^\circ) = 180^\circ
\]

Now, let's simplify and solve this equation.

1. Combine like terms:
\[
w + w + 2w + 46^\circ + 7^\circ - 19^\circ = 180^\circ
\]
\[
4w + 34^\circ = 180^\circ
\]

2. Subtract 34 from both sides:
\[
4w = 146^\circ
\]

3. Divide by 4:
\[
w = 36.5^\circ
\]

Thus, the value of \( w \) is \( 36.5^\circ \).

Do you want more details on this, or do you have any questions?

Here are 5 related questions to further explore this topic:
1. What is the sum of the angles in a quadrilateral?
2. How do you calculate the third angle in a triangle if two angles are known?
3. What is the exterior angle theorem in triangles?
4. Can a triangle have more than one obtuse angle? Why or why not?
5. How do we apply the properties of parallel lines to solve for unknown angles?

**Tip:** In a triangle, knowing the sum of angles equals \(180^\circ\) is often key to finding unknown angles.

Learn how to solve for w in a triangle where the angles are expressed as w + 46°, w + 7°, and 2w - 19°. This problem involves basic algebra and the triangle angle sum theorem. Suitable for students in Grades 8-10.

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