1. Though there are two ways in which we classify triangles, we often name a triangle using both its side length and angle measure classifications (e.g., an obtuse isosceles triangle). Given this, answer true or false to the following statements. If the statement is true, draw an example of the triangle. If the statement is false, explain why. Statement True False Why? A triangle can have exactly 2 acute angles. A triangle can have an obtuse angle and a right angle. A triangle can be scalene and right. A triangle can be equilateral and obtuse. A triangle can be isosceles and right. None
The Correct Answer and Explanation is:
Let’s analyze each statement and determine whether it is true or false, with explanations.
1. A triangle can have exactly 2 acute angles.
✅ True
Explanation:
A triangle must have exactly three interior angles that sum to 180°. An acute angle is an angle less than 90°.
If two angles are acute (e.g., 60° and 50°), the third angle would be 70°, which is also acute. But we want exactly two acute angles.
So, if we have two acute angles (e.g., 60° and 25°), the third angle would be 95°, which is obtuse.
Hence, a triangle can have exactly two acute angles when the third is obtuse or right.
2. A triangle can have an obtuse angle and a right angle.
❌ False
Explanation:
A triangle can have only one angle greater than or equal to 90°, because the sum of all three angles must be 180°.
A right angle is 90°, and an obtuse angle is greater than 90°.
So, 90° + any obtuse angle (>90°) would exceed 180°, which is not possible.
Therefore, a triangle cannot have both an obtuse and a right angle.
3. A triangle can be scalene and right.
✅ True
Explanation:
A scalene triangle has no equal sides or angles.
A right triangle has one 90° angle.
A triangle can have one right angle and two other unequal acute angles with all three sides of different lengths.
Example: A triangle with angles 90°, 60°, and 30° and side lengths 3, 4, 5 is a classic example.
4. A triangle can be equilateral and obtuse.
❌ False
Explanation:
An equilateral triangle has three equal sides and three equal angles, each measuring 60°.
Since all the angles are 60°, they are all acute, not obtuse.
Therefore, an equilateral triangle cannot be obtuse.
5. A triangle can be isosceles and right.
✅ True
Explanation:
An isosceles triangle has two equal sides and two equal angles.
A right triangle has one angle of 90°.
If a triangle has one right angle and the other two angles are 45° each, it’s an isosceles right triangle.
So, yes — such a triangle is possible (e.g., a triangle with angles 45°, 45°, 90°).
Summary Table:
| Statement | True/False | Explanation Summary |
|---|---|---|
| A triangle can have exactly 2 acute angles. | ✅ True | Third angle can be right or obtuse. |
| A triangle can have an obtuse angle and a right angle. | ❌ False | Sum would exceed 180°. |
| A triangle can be scalene and right. | ✅ True | Unequal sides with a 90° angle is possible. |
| A triangle can be equilateral and obtuse. | ❌ False | Equilateral triangles have only 60° angles. |
| A triangle can be isosceles and right. | ✅ True | 45°-45°-90° triangle is valid. |
