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 I and II.
We have been given the following three statements. I. &△ PQR is a right triangle. II. &△ PQR is an obtuse triangle. III. &△ PQR is scalene. 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. △ PQR is a right triangle | I. △ PQR is a right triangle | II. △ PQR is an obtuse triangle. |
| II. △ PQR is an obtuse triangle. | III. △ PQR is scalene. | III. △ PQR is scalene. |
Next, we will find an example for each group. If we cannot, this means that the two statements contradict each other. Let's start with Group A.
An example for Group A must be a triangle which is both a right triangle and an obtuse triangle at the same time.
Any triangle which meets both of these requirements would need to have one 90^(∘) angle and one angle with a measure x which is greater than 90^(∘) but less than 180^(∘). The sum of these measures would be greater than 180^(∘). 90 < x < 180 ⇒ 90+x > 180 However, by the Triangle Angle-Sum Theorem, the interior angles of a triangle must add to 180^(∘). In this case, the sum of the two angle measures is already greater than 180. Therefore, it is not possible to give an example of a triangle that meets these requirements. Statements I and II contradict each other.
Let's continue with Group B. The triangle must be both right and scalene.
Let's graph a right triangle with legs that are 2 and 4 units long. The last side of the triangle, the hypotenuse, will have to be more than 4 units long and all of the sides of the triangle will have different lengths.
This triangle is a scalene right triangle. Since we are able to give an example for Group B, Statements I and III do not contradict each other.
For the last group, we will try to draw an scalene obtuse triangle.
Looking at our diagram, we can see that all of the sides have different lengths and ∠ Q is obtuse. Therefore, △ PQR is a scalene obtuse triangle. Statements II and III do not contradict each other.