There is something wrong with this definition for a pair of vertical angles: “If AB and CD intersect at point P, then APC and BPD are a pair of vertical angles.” Sketch a counterexample to show why it is not correct. Can you add a phrase to correct it?
I hate this
To make the statement correct...point P must be in between line segment AB and line segment CD in order for angle APC and angle BPD to be vertical angles...it's just wording...
To sketch a counterexample and understand why the given definition for a pair of vertical angles is incorrect, we must first understand what vertical angles are.
Vertical angles are a pair of non-adjacent angles formed by the intersection of two lines. They are opposite each other and have equal measures. In other words, if two lines intersect at a point, the angles that are across from each other (not adjacent) are considered vertical angles.
Now, let's examine the given definition: "If AB and CD intersect at point P, then APC and BPD are a pair of vertical angles."
The issue with this definition is that it claims any two angles formed by any two intersecting lines are vertical angles, which is not true.
To provide a counterexample, we can sketch two lines, AB and CD, intersecting at point P, where the angles created, APC and BPD, are not vertical angles. Imagine line AB is vertical, and CD intersects it at point P, forming a right angle. In this case, APC and BPD would not be a pair of vertical angles because they are adjacent angles since they share a common side, PD.
To correct the definition, we can add the phrase "that are opposite each other" after "then" to emphasize that vertical angles need to be non-adjacent and across from each other after the intersection. The corrected definition would be: "If AB and CD intersect at point P, then APC and BPD, that are opposite each other, are a pair of vertical angles."