Chapter Test
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Start with grouping the statements in all possible combinations and try to find an example for each group.
Statements II and III.
We have been given the following three statements. I. & ∠ DOS ≅ ∠ CAT II. & ∠ DOS and ∠ CAT are vertical. III. & ∠ DOS and ∠ CAT are adjacent. In order to identify the two statements that contradict each other, we will first group the statements in all possible combinations.
| Group A | Group B | Group C |
|---|---|---|
| I. ∠ DOS ≅ ∠ CAT. | I. ∠ DOS ≅ ∠ CAT. | II. ∠ DOS and ∠ CAT are vertical. |
| II. ∠ DOS and ∠ CAT are vertical. | III. ∠ DOS and ∠ CAT are adjacent. | III. ∠ DOS and ∠ CAT are adjacent. |
An example for Group A must be a pair of angles ∠ DOS and ∠ CAT that are congruent and vertical at the same time.
By the Vertical Angles Theorem, vertical angles are always congruent. Therefore, any pair of vertical angles will be an example satisfying both Statements from Group (I).
Since we are able to give an example for Group A, Statement I and II do not contradict each other.
Let's continue with Group B. The angles ∠ DOS and ∠ CAT must be both congruent and adjacent.
In order to give an example for Group B, we will draw any angle and divide it in half by creating an angle bisector. The resulting angles will have the same measure and they will be adjacent.
On the diagram, ∠ DOS and ∠ CAT are congruent. They are also adjacent — they do not overlap and they share side AT and vertex A. Therefore, statements I and III do not contradict each other.
For the last group, we would have to draw angles ∠ DOS and ∠ CAT such that they are vertical and adjacent at the same time.
Any two angles that would meet both of these conditions would have to lie on the opposite sides of an intersection of lines and they would have to have a common side. However, this is impossible. To see why, let's consider an arbitrary intersection of two lines.
We have marked both pairs of vertical angles created by the two lines. As we can see, neither pair of angles has a common side, because the vertical angles are opposite from each other. This will always be the case so it is impossible to give an example for Group C. Therefore, statements II and III contradict each other.