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?
To sketch a counterexample and show why the given definition for vertical angles is not correct, we need to understand what vertical angles are.
Vertical angles are a pair of non-adjacent angles formed by the intersection of two lines. They have equal measures and are formed in opposite directions. In other words, if two lines intersect with each other, the pairs of angles opposite to each other are called vertical angles.
Now, let's consider the given definition: "If AB and CD intersect at point P, then APC and BPD are a pair of vertical angles."
This definition is incorrect because it states that any two angles formed by the intersection of lines AB and CD are vertical angles, which is not true. To correct this, we can add a phrase to clarify the condition for vertical angles. An improved version of the definition could be:
"If AB and CD intersect at point P, and no other lines are present in the vicinity that intersect AB and CD at point P, then APC and BPD are a pair of vertical angles."
By adding the condition that no other lines should be present that intersect AB and CD at point P, we ensure that the angles APC and BPD are the only pair of angles formed by the intersecting lines and hence are vertical angles.