Start by grouping the statements in all possible combinations and try to find an example for each group.
Statements I and II.
Practice makes perfect
We have been given the following three statements.
I. &△ ABC is a right triangle.
II. &△ ABC is equilateral.
III. &△ ABC is isosceles.
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. △ ABC is a right triangle.
II. △ ABC is equilateral.
I. △ ABC is a right triangle.
II. △ ABC is equilateral.
III. △ ABC is isosceles.
III. △ ABC is isosceles.
Next, we will find an example for each group. If we cannot, we will try to verify whether the statements contradict each other or not. Let's start with Group A.
Group A
An example for Group A must be a triangle which is both an equilateral triangle and a right triangle at the same time. Let's recall the definitions of these two types of triangle.
Equilateral Triangle: All sides have the same length and all the angles have the same measure.
Right Triangle: There is a 90^(∘) angle.
Any triangle that meets both of these requirements will have three 90^(∘) angles. However, by the Triangle Angle-Sum Theorem, the interior angles of a triangle must add up to 180^(∘).
90+90+90≠ 180
Therefore, it is not possible for a triangle to satisfy these two statements. Statements I and II contradict each other.
Group B
Let's continue with Group B. The triangle must be both equilateral and isosceles.
Equilateral Triangle: All sides have the same length and all the angles have the same measure.
Isosceles Triangle: At least two of the sides have the same length.
Since an equilateral triangle has three equal side lengths, and an isosceles triangle has at least two equal side lengths, any equilateral triangle is also isosceles.
Statements I and III do not contradict each other.
Group C
For the last group, we will draw an isosceles right triangle by making sure that both of the legs of the right triangle have the same length.
Since we are able to give an example for Group C, Statement II and III do not contradict each other.
