Math Problem Statement

For the figure shown on the right, find the value of the variable and the measures of the angles.

Solution

To solve this problem, we need to determine the value of xx and find the measures of the angles in the triangle PQR\triangle PQR.

Since PQRPQR is a triangle, the sum of its interior angles must equal 180180^\circ.

The angles are given as follows:

  • P=(2x13)\angle P = (2x - 13)^\circ
  • Q=(x+37)\angle Q = (x + 37)^\circ
  • R=x\angle R = x^\circ

Step 1: Set up the equation

Using the triangle angle sum property: (2x13)+(x+37)+x=180(2x - 13) + (x + 37) + x = 180

Step 2: Simplify the equation

Combine like terms: 4x+24=1804x + 24 = 180

Step 3: Solve for xx

Subtract 24 from both sides: 4x=1564x = 156

Now, divide by 4: x=39x = 39

Step 4: Substitute xx back to find the angles

Now that we know x=39x = 39, we can substitute this value into each angle expression to find their measures.

  1. P=2x13=2(39)13=7813=65\angle P = 2x - 13 = 2(39) - 13 = 78 - 13 = 65^\circ
  2. Q=x+37=39+37=76\angle Q = x + 37 = 39 + 37 = 76^\circ
  3. R=x=39\angle R = x = 39^\circ

Final Answer:

  • x=39x = 39
  • P=65\angle P = 65^\circ
  • Q=76\angle Q = 76^\circ
  • R=39\angle R = 39^\circ

Would you like further explanation or have any other questions?

Here are five related questions to deepen understanding:

  1. What is the triangle angle sum property, and why does it hold?
  2. How can we verify the values of xx and the angles we found?
  3. What would change if the triangle were not a standard triangle but an isosceles or equilateral triangle?
  4. How do supplementary angles relate to solving for angles in triangles?
  5. What other geometric shapes have specific angle sum properties?

Tip: When solving for unknowns in geometric figures, always start by applying known properties like the angle sum for triangles or polygons. This ensures that your setup is grounded in a fundamental rule.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Properties
Algebra

Formulas

Sum of interior angles of a triangle = 180°

Theorems

Triangle Angle Sum Theorem

Suitable Grade Level

Grades 8-10