Final answer:
To prove that ∠ 1+∠ 2+∠ 3=180°, we can use the fact that the sum of the angles in a triangle is always 180 degrees. By showing that the alternate interior angles formed by the parallel line and the triangle angles are congruent, we can equate them and solve for the value of one angle. Substituting this value into the equation confirms that the sum of the three angles is indeed 180°.
Step-by-step explanation:
To prove that ∠ 1+∠ 2+∠ 3=180°, we can use the fact that the sum of the angles in a triangle is always 180 degrees. Since the line on top is parallel to the base, it creates alternate interior angles with the three angles of the triangle.
- ∠ 1 is an alternate interior angle with ∠ 2, so they are congruent.
- ∠ 2 is an alternate interior angle with ∠ 3, so they are congruent.
Therefore, ∠ 1 = ∠ 2 and ∠ 2 = ∠ 3. By substituting these equalities into the equation, we get ∠ 1+∠ 2+∠ 3 = ∠ 1+∠ 1+∠ 1 = 3∠ 1.
Since the sum of the angles in a triangle is 180 degrees, we have 3∠ 1 = 180. By dividing both sides by 3, we find that ∠ 1 = 60. Therefore, ∠ 1+∠ 2+∠ 3 = 60+60+60 = 180°.