Question

amelia wants to prove that the interior angles of any triangle sum to 180°. she draws a line m through one vertex of the triangle, and then she labels all the angles formed. which statement is true? line m helps amelia complete her proof because it creates pairs of congruent alternate interior angles. line m does not help amelia complete her proof because it is not parallel to the opposite side of the triangle. line m does not help amelia complete her proof because it is not a straight line. line m helps amelia complete her proof because ∠4 and ∠5 are supplementary.

Explanation:

Step1: Recall the properties of parallel lines and alternate interior angles

When a line is drawn parallel to one side of a triangle (in the context of proving the sum of interior angles of a triangle), it helps in establishing congruent alternate interior angles.

Step2: Analyze each option

• Option 1: If a line is parallel to the opposite side of the triangle, it is crucial for the angle - sum proof (by creating congruent alternate interior angles). So the statement "Line \(m\) does not help Amelia complete her proof because it is not parallel to the opposite side of the triangle" is incorrect.
• Option 2: If line \(m\) creates pairs of congruent alternate interior angles, it is very useful for the proof (since we can use the properties of parallel lines and angle congruence to show that the sum of the interior angles of a triangle is \(180^{\circ}\)).
• Option 3: The fact that line \(m\) is a straight line is not the key property. The key is its parallelism (to create congruent alternate interior angles). So the statement "Line \(m\) does not help Amelia complete her proof because it is not a straight line" is wrong.
• Option 4: \(\angle4\) and \(\angle5\) being supplementary is not the relevant property for the proof of the sum of interior angles of a triangle (the relevant property is the creation of congruent alternate interior angles due to parallel lines).

Answer:

Line \(m\) helps Amelia complete her proof because it creates pairs of congruent alternate interior angles.