Math Problem Statement

Find the sizes of the three unknown angles in the image above

Solution

To find the sizes of the unknown angles aa, bb, and cc in the image, we will use the properties of vertically opposite angles and the fact that angles on a straight line sum to 180180^\circ.

  1. Identifying vertically opposite angles:

    • The angle labeled 3737^\circ is vertically opposite to angle bb.
    • Vertically opposite angles are equal, so: b=37b = 37^\circ
  2. Using the fact that angles on a straight line sum to 180180^\circ:

    • Angle aa and the 3737^\circ angle form a linear pair. Therefore: a+37=180a + 37^\circ = 180^\circ
    • Solving for aa: a=18037=143a = 180^\circ - 37^\circ = 143^\circ
  3. Using the fact that angles around a point sum to 360360^\circ:

    • The angles aa, bb, and cc are formed around a point. Therefore: a+b+c=360a + b + c = 360^\circ
    • We already know a=143a = 143^\circ and b=37b = 37^\circ. Substituting these values: 143+37+c=360143^\circ + 37^\circ + c = 360^\circ
    • Solving for cc: 180+c=360180^\circ + c = 360^\circ c=360180=180c = 360^\circ - 180^\circ = 180^\circ

So, the sizes of the unknown angles are: a=143,b=37,c=180a = 143^\circ, \quad b = 37^\circ, \quad c = 180^\circ

Would you like more details or have any questions? Here are 5 additional questions you might find interesting:

  1. How do you find the value of an unknown angle in a right triangle?
  2. What is the relationship between corresponding angles and parallel lines?
  3. How can you use the exterior angle theorem to solve problems?
  4. What are the properties of supplementary and complementary angles?
  5. How do you determine the angles in an equilateral triangle?

Tip: Always look for known angle relationships, such as vertically opposite angles and angles on a straight line, to simplify the process of finding unknown angles.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angles
Properties of Angles

Formulas

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Theorems

Vertically Opposite Angles Theorem
Angles on a Straight Line Theorem
Angles around a Point Theorem

Suitable Grade Level

Grades 7-9