Math Problem Statement

In the figure given, QT ⊥ PR, ∠TQR = 40° and ∠SPR = 30°. Find x and y.

Solution

Let's analyze the problem step by step.

Given Information:

  • QTPRQT \perp PR (QT is perpendicular to PR, meaning QTR=90\angle QTR = 90^\circ).
  • TQR=40\angle TQR = 40^\circ.
  • SPR=30\angle SPR = 30^\circ.

You are tasked with finding the angles xx and yy marked in the diagram.

Step-by-Step Solution:

  1. Angle in a Triangle (TQR): In triangle TQRTQR, the sum of interior angles must equal 180180^\circ. TQR+QTR+QRT=180\angle TQR + \angle QTR + \angle QRT = 180^\circ Substitute the given values: 40+90+QRT=18040^\circ + 90^\circ + \angle QRT = 180^\circ Solving for QRT\angle QRT: QRT=180130=50\angle QRT = 180^\circ - 130^\circ = 50^\circ So, QRT=50\angle QRT = 50^\circ.

  2. Angle SPR: From the diagram, SPR=30\angle SPR = 30^\circ is given, which corresponds to the marked angle yy. Hence: y=30y = 30^\circ

  3. Angle x (Exterior Angle Theorem): Using the exterior angle theorem, the exterior angle xx at point PP is equal to the sum of the two opposite interior angles in triangle TQRTQR: x=TQR+QRT=40+50=90x = \angle TQR + \angle QRT = 40^\circ + 50^\circ = 90^\circ

Thus, the values of xx and yy are: x=90,y=30x = 90^\circ, \quad y = 30^\circ

Final Answer:

  • x=90x = 90^\circ
  • y=30y = 30^\circ

Would you like further clarification on any part of the solution?

Follow-up Questions:

  1. How can the exterior angle theorem be applied to different triangle problems?
  2. What are the different properties of perpendicular lines in triangles?
  3. How do we calculate angles when one is given as a perpendicular?
  4. Why is the sum of the interior angles of a triangle always 180180^\circ?
  5. How does knowing one angle help in determining other angles in a geometric figure?

Tip:

Always look for right angles in geometric figures as they simplify calculations by providing clear angle relationships.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangles
Angles

Formulas

Sum of angles in a triangle = 180°
Exterior Angle Theorem: exterior angle = sum of opposite interior angles

Theorems

Exterior Angle Theorem
Triangle Angle Sum Theorem

Suitable Grade Level

Grades 8-10