Math Problem Statement

Solve for the unknown angles in the pentagon given that one internal angle is 100°, another is 50°, an exterior angle is 62°, and two angles are labeled x and 2x.

Solution

The image shows a pentagon with various angles marked, and the goal is likely to solve for the unknown angles based on the given information.

Information from the image:

  • One of the internal angles is 100100^\circ.
  • Another internal angle is 5050^\circ.
  • The angle adjacent to the exterior angle 6262^\circ is labeled as 2x2x.
  • There is an internal angle labeled xx.
  • The angle adjacent to 5050^\circ and forming a straight line is 2x2x.

We can solve for the unknown angle xx by using the following facts:

  1. Sum of interior angles of a pentagon: The sum of the interior angles of any pentagon is given by the formula: Sum of interior angles=(n2)×180\text{Sum of interior angles} = (n - 2) \times 180^\circ where nn is the number of sides. For a pentagon, n=5n = 5, so: Sum of interior angles=(52)×180=3×180=540.\text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.

  2. Straight-line angles: The angle adjacent to 2x2x and 5050^\circ is part of a straight line, so they should sum to 180180^\circ: 50+2x=180    2x=130    x=65.50^\circ + 2x = 180^\circ \implies 2x = 130^\circ \implies x = 65^\circ.

Now that we have x=65x = 65^\circ, we can calculate the remaining angles and verify the sum of the interior angles.

  • The angle marked 2x2x is 2×65=1302 \times 65^\circ = 130^\circ.
  • The remaining angle is already given as 100100^\circ.

Now let's check the sum of all the angles:

65+130+100+50+62=540.65^\circ + 130^\circ + 100^\circ + 50^\circ + 62^\circ = 540^\circ.

Thus, everything checks out.

Final Answer:

  • x=65x = 65^\circ.

Would you like further details or clarification on any part of this solution?

Here are five related questions you might explore:

  1. How do we calculate the sum of the interior angles of any polygon?
  2. What is the relationship between exterior and interior angles in polygons?
  3. How can we determine angles when they form a straight line?
  4. How do we use algebraic expressions in geometry to find unknown angles?
  5. How can the knowledge of basic angle properties be applied to more complex polygons?

Tip: Always remember that adjacent angles on a straight line sum up to 180180^\circ.

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Math Problem Analysis

Mathematical Concepts

Geometry
Polygons
Interior Angles
Exterior Angles
Linear Pair of Angles

Formulas

Sum of Interior Angles of a Polygon: (n - 2) * 180°
Linear Pair of Angles: Sum of two adjacent angles on a straight line equals 180°

Theorems

Sum of Interior Angles Theorem
Linear Pair Theorem

Suitable Grade Level

Grades 8-10