Math Problem Statement

Determine the values of the angles labeled 1 to 5 in the provided diagram, given the angles 50°, 65°, 68°, and 140°.

Solution

To solve this diagram, we need to determine the values of the numbered angles labeled 1 through 5. Here's the step-by-step analysis:

Step 1: Identify Relationships and Given Angles

  • The diagram shows two intersecting lines with angles formed around them.
  • Angles given in the triangles are 50°, 65°, 68°, and 140°.
  • The numbered angles (1 to 5) are labeled, and we need to find their values.

Step 2: Use Triangle Angle Sum Property

  1. Triangle with 50° and 65°:

    • The sum of the angles in any triangle is 180°.
    • Let's find the third angle (angle 1): 1805065=65180^\circ - 50^\circ - 65^\circ = 65^\circ So, angle 1 = 65°.
  2. Triangle with 68°:

    • The other angle needed is angle 5: 1806850=62180^\circ - 68^\circ - 50^\circ = 62^\circ So, angle 5 = 62°.

Step 3: Use Linear Pairs and Vertical Angles

  1. Angle 2:

    • Since angle 1 is 65° and angles on a straight line sum to 180°, angle 2=18065=115.\text{angle 2} = 180^\circ - 65^\circ = 115^\circ.
  2. Angle 3 and angle 4:

    • Both are vertical angles opposite 50° and 140°, respectively, and follow vertical angle rules, thus the values remain:

      • Angle 3 = 50°
      • Angle 4 = 140°

Conclusion:

  • Angle 1: 65°
  • Angle 2: 115°
  • Angle 3: 50°
  • Angle 4: 140°
  • Angle 5: 62°

Would you like further clarifications or additional information?

Related Questions:

  1. How do the linear pair relationships affect angle measurements?
  2. Why are the vertical angles always equal?
  3. How can you determine an unknown angle in a triangle?
  4. What role does symmetry play in these angle relationships?
  5. What are alternate angles, and how do they differ from vertical?

Tip:

Always look for straight lines or intersecting lines first to leverage basic geometric rules like vertical angles, linear pairs, and triangle sum rules.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angle Relationships
Triangle Angle Sum Property

Formulas

Triangle angle sum: 180°
Linear pair: Angles on a straight line sum to 180°
Vertical angles: Opposite angles are equal

Theorems

Triangle Angle Sum Theorem
Vertical Angle Theorem
Linear Pair Postulate

Suitable Grade Level

Grades 7-10