Start with grouping the statements in all possible combinations and try to find an example for each group.
Statements I and III.
Practice makes perfect
We have been given the following three statements.
I. In right &△ ABC, m∠ B = 90.
II. In right &△ ABC, m∠ A = 80.
III. In right &△ ABC, m∠ C = 90.
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. In right △ ABC, m∠ B = 90.
I. In right △ ABC, m∠ B = 90.
II. In right △ ABC, m∠ A = 80.
II. In right △ ABC, m∠ A = 80.
III. In right △ ABC, m∠ C = 90.
III. In right △ ABC, m∠ C = 90.
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 right triangle with m∠ B = 90 and m∠ A = 80. By the Triangle Angle-Sum Theorem, the interior angles of a triangle must add up to 180^(∘).
m∠ A + m∠ B + m∠ C = 180
To find m∠ C, let's substitute m∠ A = 80 and m∠ B = 90 in the above equation.
We can see that it is possible to have a right triangle with an interior angle m∠ 80. Therefore, statements I and II do not contradict each other.
Group B
Let's continue with Group B. We are trying to find a right triangle with two right-angles. We will apply the Triangle Angle-Sum Theorem.
m∠ A + m∠ B + m∠ C = 180
To find m∠ A, let's substitute m∠ B = 90 and m∠ C = 90 in the above equation.
Therefore, it is not possible to give an example of a triangle and that has two right angles. Statements I and III contradict each other.
Group C
For the last group, like the Group A, we will examine the angles m∠ A =80 and m∠ C = 90. By the Triangle Angle-Sum Theorem, the interior angles of a triangle must add to 180^(∘).
m∠ A + m∠ B + m∠ C = 180
To find m∠ B, let's substitute m∠ A = 80 and m∠ C = 90 in the above equation.
