Final answer:
To prove that the interior angles of a triangle add to 180° using the given figure, we can use the properties of parallel lines and the definition of a triangle. By labeling the angles and using algebraic manipulation, we can show that the sum of the angles is equal to 180°.
Step-by-step explanation:
To prove that the interior angles of a triangle add to 180°, consider the given figure with angle APQ with AB || PQ. First, let's label the angles in the figure. Angle PAQ is denoted as angle 1, angle AQB is denoted as angle 2, and angle PAB is denoted as angle 3.
To show that the interior angles of a triangle add to 180°, we need to prove that angle 1 + angle 2 + angle 3 = 180°.
In the figure, angle 1 is equal to angle APQ because they are alternate interior angles formed by the parallel lines AB and PQ, which are cut by the transversal AP. So we can replace angle 1 with angle APQ in our equation: angle APQ + angle 2 + angle 3 = 180°.
Now, let's look at angle APQ. We can see that it is a straight line, which is a straight angle, and it measures 180°. Therefore, angle APQ = 180°. We can substitute this value in our equation: 180° + angle 2 + angle 3 = 180°.
Since we have 180° on both sides of the equation, we can subtract 180° from both sides to isolate angle 2 and angle 3: angle 2 + angle 3 = 0°.
Now, consider the triangle APQ. It is a triangle on a plane, and by definition, the angles in a triangle add up to 180°. So we can replace angle 2 + angle 3 with 180° in our equation: 180° = 180°.
This equation is true, which means our original statement that the interior angles of a triangle add to 180° is proven.