Math Problem Statement

What is the measure of each interior angle of a regular pentagon? In a parallelogram, adjacent angles are equal. Find the measure of all angles of this parallelogram. What is the name of this special parallelogram? A line graph which is a whole unbroken line is called a ________ graph. The standard form for 0.000064 is _______. Is it possible to have a quadrilateral in which all the angles are obtuse? Explain your answer. The area of a rhombus is 120 cm². If its one diagonal is 20 cm, find the other diagonal. What will be the side of a cube whose surface area is 2400 cm²?

Solution

Let me extract the questions from the image and solve them for you:

Questions Extracted:

  1. What is the measure of each interior angle of a regular pentagon?
  2. In a parallelogram, adjacent angles are equal. Find the measure of all angles of this parallelogram. What is the name of this special parallelogram?
  3. A line graph which is a whole unbroken line is called a ________ graph.
  4. The standard form for 0.0000640.000064 is _______.
  5. Is it possible to have a quadrilateral in which all the angles are obtuse? Explain your answer.
  6. The area of a rhombus is 120cm2120 \, \text{cm}^2. If its one diagonal is 20cm20 \, \text{cm}, find the other diagonal.
  7. What will be the side of a cube whose surface area is 2400cm22400 \, \text{cm}^2?

Answers:

  1. Measure of each interior angle of a regular pentagon: The formula for the measure of an interior angle of a regular polygon is: Interior Angle=(n2)×180n\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n} For a pentagon (n=5n = 5): Interior Angle=(52)×1805=3×1805=108\text{Interior Angle} = \frac{(5-2) \times 180^\circ}{5} = \frac{3 \times 180^\circ}{5} = 108^\circ Answer: 108°

  2. Angles of a parallelogram with equal adjacent angles: In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. If adjacent angles are equal, both angles must be 9090^\circ, making the parallelogram a rectangle. Answer: All angles are 9090^\circ, and the parallelogram is a rectangle.

  3. A line graph which is a whole unbroken line: Answer: A continuous graph.

  4. Standard form of 0.0000640.000064: Standard form is also known as scientific notation. We move the decimal point 5 places to the right: 0.000064=6.4×1050.000064 = 6.4 \times 10^{-5} Answer: 6.4×1056.4 \times 10^{-5}.

  5. Is it possible to have a quadrilateral with all obtuse angles? No, it is not possible. The sum of the interior angles of any quadrilateral is 360360^\circ. An obtuse angle is greater than 9090^\circ. If all four angles were obtuse, their sum would exceed 360360^\circ, which is not possible. Answer: No, it is not possible.

  6. Find the other diagonal of a rhombus: The area of a rhombus is given by: Area=12×d1×d2\text{Area} = \frac{1}{2} \times d_1 \times d_2 where d1d_1 and d2d_2 are the diagonals. Given: Area=120cm2,d1=20cm\text{Area} = 120 \, \text{cm}^2, \quad d_1 = 20 \, \text{cm} Substituting into the formula: 120=12×20×d2120 = \frac{1}{2} \times 20 \times d_2 Solving for d2d_2: d2=120×220=12cmd_2 = \frac{120 \times 2}{20} = 12 \, \text{cm} Answer: The other diagonal is 12cm12 \, \text{cm}.

  7. Find the side of a cube with a surface area of 2400cm22400 \, \text{cm}^2: The surface area SS of a cube is given by: S=6s2S = 6s^2 where ss is the side length of the cube. Given: 2400=6s22400 = 6s^2 Solving for s2s^2: s2=24006=400s^2 = \frac{2400}{6} = 400 Taking the square root: s=400=20cms = \sqrt{400} = 20 \, \text{cm} Answer: The side length is 20cm20 \, \text{cm}.


Let me know if you want more details or explanations on any of these answers. Here are 5 related questions to consider:

  1. What is the sum of interior angles of a regular hexagon?
  2. How do you calculate the exterior angle of any regular polygon?
  3. What are the properties of a rhombus in relation to its diagonals?
  4. Can a quadrilateral have exactly two obtuse angles? Why or why not?
  5. How do you convert a number like 8.5×1038.5 \times 10^3 into standard form?

Tip: Always remember that the sum of interior angles in any polygon is given by (n2)×180(n-2) \times 180^\circ, where nn is the number of sides.

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Math Problem Analysis

Mathematical Concepts

Geometry
Polygons
Angles
Graphs
Scientific Notation
Properties of Quadrilaterals
Surface Area
Diagonals

Formulas

Interior angle of a regular polygon: (n-2) × 180° / n
Area of a rhombus: (1/2) × d1 × d2
Surface area of a cube: 6s²

Theorems

Properties of Parallelograms
Angle Sum Property of Quadrilaterals
Definition of Scientific Notation

Suitable Grade Level

Grades 6-8