https://file.aiquickdraw.com/mathbot/file/1727361367053mmlppghj.jpg

The image contains two geometry problems. Let me walk through each one:

### Problem 1 (Upper Problem):
In this problem, triangle \( \triangle ABC \) is depicted with angles at points \( A \), \( B \), and \( C \). The provided angles are:
- \( \angle A = 110^\circ \)
- \( \angle B = 120^\circ \)

The task is likely to find \( \angle C \), based on the fact that the sum of angles in a triangle always equals \( 180^\circ \).

#### Solution for Problem 1:
Since we know two angles of the triangle, the third angle can be calculated as follows:
\[
\angle C = 180^\circ - (\angle A + \angle B)
\]
Substituting the given values:
\[
\angle C = 180^\circ - (110^\circ + 120^\circ) = 180^\circ - 230^\circ = -50^\circ
\]
This suggests that the problem might have additional geometric context or external angles that need to be considered. The illustration may represent exterior angles as well, or the values might have been for external angles.

### Problem 2 (Lower Problem):
This problem involves intersecting lines, with one angle labeled \( 120^\circ \) and another angle involving a bisector. Specifically:
- Line \( AB \) is the bisector of \( \angle X \), and we are asked to find the value of \( \angle X \).

Given that \( AB \) bisects \( \angle X \), and the adjacent angle is \( 120^\circ \), we use the fact that angles on a straight line sum to \( 180^\circ \). Hence:
\[
\text{Total angle } = 180^\circ
\]
\[
\angle X = 180^\circ - 120^\circ = 60^\circ
\]
Since \( AB \) bisects \( \angle X \), each half of \( \angle X \) is:
\[
\frac{\angle X}{2} = \frac{60^\circ}{2} = 30^\circ
\]

---

Would you like more details on either problem or clarification? Here are five related questions you can explore:

1. How do you calculate exterior angles in a triangle?
2. How can bisectors help solve geometric problems involving angles?
3. What are vertical angles, and how do they relate to intersecting lines?
4. Can a triangle have negative angle measures? If not, how would external angles play a role?
5. What is the sum of all interior angles in any polygon?

**Tip:** Always verify if the angles given are interior or exterior when solving problems involving polygons.

Two geometry problems: one involving the angles in a triangle (110°, 120°), and another involving a bisector and a 120° angle.

Solve two geometry problems involving a triangle with angles 110° and 120°, and a bisector problem with a 120° angle. This problem explores the triangle angle sum theorem and straight angle theorem, suitable for Grades 7-9.

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