Start with 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. & In right△ ABC, m∠ A = 60.
II. &In right△ ABC, ∠ A ≅ ∠ C.
III. &In right△ ABC, m∠ B = 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∠ A = 60.
I. In right △ ABC, m∠ A = 60.
II. In right △ ABC, ∠ A ≅ ∠ C.
II. In right △ ABC, ∠ A ≅ ∠ C.
III. In right △ ABC, m∠ B = 90.
III. In right △ ABC, m∠ B = 90.
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.
Group A
An example for Group A must be a right triangle △ ABC in which m∠ A = 60 and ∠ A ≅ ∠ C at the same time. This means that m∠ A = m ∠ C, so m∠ C = 60.
l m∠ A = 60 m∠ A = m∠ C ⇒ m∠ C = 60
Since △ ABC is a right triangle, the only angle left, ∠ B, has to measure 90^(∘). However, by the Triangle Angle-Sum Theorem, the interior angles of a triangle must add to 180^(∘).
60+90+60≠ 180
Therefore, it is not possible to give an example of a triangle and that meets these requirements. Statements I and II contradict each other.
Group B
Let's continue with Group B. In right △ ABC, we must have m∠ A = 60 and m∠ B = 90 at the same time. Let's consider the following triangle.
As we can see, △ ABC from the diagram is a right triangle. Furthermore, m∠ A= 60 and m∠ B = 90. Since we are able to give an example for Group B, statements I and III do not contradict each other.
Group C
For the last group, we will draw a right △ ABC by making sure that ∠ A ≅ ∠ C and m∠ B = 90. Let's consider the following triangle.
As we can see, △ ABC is a right triangle with m∠ B = 90. Also, both the m∠ A and the m∠ C is 45. This means that angles ∠ A and ∠ C are congruent. Therefore, the triangle is an example for Group C. As such, statements II and III do not contradict each other.
