Question

5 multiple choice 1 point according to the markings, this triangle is ________. 6 multiple choice 1 point which of the following is not true? a right triangle can be an isosceles triangle. no scalene triangles are isosceles triangles. no equilateral triangle can have an obtuse angle. the union of any three segments is a triangle.

Explanation:

5.

• An isosceles triangle has at least two equal sides. A right - triangle has one \(90^{\circ}\) angle. An equilateral triangle has all three sides equal. A scalene triangle has all three sides of different lengths.
• Looking at the markings on the triangle (different number of tick - marks on each side), all three sides have different lengths.

• For "A right triangle can be an isosceles triangle": A right - isosceles triangle has angles \(45^{\circ},45^{\circ},90^{\circ}\) and two equal sides.
• For "No scalene triangles are isosceles triangles": By definition, a scalene triangle has all sides of different lengths and an isosceles triangle has at least two sides equal. So, they are mutually exclusive.
• For "No equilateral triangle can have an obtuse angle": The measure of each angle in an equilateral triangle is \(\frac{(3 - 2)\times180^{\circ}}{3}=60^{\circ}\) (using the formula \(\theta=\frac{(n - 2)\times180^{\circ}}{n}\) for a regular \(n\) - sided polygon, here \(n = 3\)). Since \(60^{\circ}<90^{\circ}\), there are no obtuse angles.
• For "The union of any three segments is a triangle": For three segments to form a triangle, the sum of the lengths of any two segments must be greater than the length of the third segment (triangle inequality theorem). For example, if we have segments of length \(1\), \(2\), and \(4\), \(1+2<4\), and they do not form a triangle.

Answer:

scalene

6.