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The image shows a geometry problem involving a triangle \( \triangle ABC \) with angles labeled \( m \angle 1 \), \( m \angle 2 \), and \( m \angle 3 \). The question asks which statement must be true to prove that the sum of these angles equals 180°.

Let's break down the problem:

The goal is to show that \( m \angle 1 + m \angle 2 + m \angle 3 = 180^\circ \), which is a known fact from the Triangle Angle Sum Theorem (the sum of the interior angles of a triangle is always 180°).

To prove this, let's look at the answer choices:
- **Option A: \( m \parallel n \)** — If lines \( m \) and \( n \) are parallel, then by the Alternate Interior Angle Theorem, angles \( m \angle 1 \) and \( m \angle 2 \) will relate to angles on the same transversal. This would help establish that their sum along with \( m \angle 3 \) is 180°.
- **Option B: \( m \angle 1 + m \angle 2 = 180^\circ - m \angle 3 \)** — This would seem to reflect the Triangle Angle Sum Theorem directly, but it is not a rule or property typically used in the proof.
- **Option C: \( m \angle 1 + m \angle 2 = 90^\circ \)** — This suggests a right triangle, but there is no indication that this is a right triangle.
- **Option D: \( m \angle 1 = m \angle 2 = m \angle 3 \)** — This would describe an equilateral triangle, but again, there's no information to support that.

**Correct answer: Option A** — Parallel lines \( m \parallel n \) must be true to use properties like the Alternate Interior Angle Theorem to prove the angle sum.

Would you like further clarification or details on this?

Here are 5 related questions:
1. How does the Triangle Angle Sum Theorem work?
2. What are alternate interior angles, and how are they used in proofs?
3. How can you prove that two lines are parallel in geometry?
4. What is the significance of transversal lines in geometry?
5. Can the Triangle Angle Sum Theorem be applied to non-Euclidean triangles?

**Tip:** Always check if parallel lines are involved in geometry problems as they often imply angle relationships through transversal lines!

Which statement must be true to prove m∠1 + m∠2 + m∠3 = 180°?

In this geometry problem, we explore the proof that the sum of the angles in a triangle equals 180° using the Triangle Angle Sum Theorem and properties of parallel lines.

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