Math Problem Statement

Given a parallelogram and triangle with specified angles (63° and 45°), find the unknown angles x and y.

Solution

This is a geometric problem involving angles within a figure that appears to be made up of a parallelogram and a triangle. Let's break down the situation:

  1. Parallel Lines and Transversal:

    • The parallel arrows on the sides indicate that the opposite sides of the quadrilateral are parallel, meaning this shape is likely a parallelogram.
    • In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (i.e., they add up to 180°).
  2. Given Angles:

    • The angle labeled 63° and the one labeled 45° are given.
    • The angles labeled as xx and yy need to be determined.

Let's start with solving for xx and yy:

Step 1: Solving for xx

  • Since opposite angles of a parallelogram are equal, the angle adjacent to the angle labeled xx on the same straight line will be equal to the given 63° (these are alternate interior angles because of the transversal).
  • Therefore, we use the fact that supplementary angles add up to 180°: x+63°=180°x + 63° = 180° Solving for xx: x=180°63°=117°x = 180° - 63° = 117°

Step 2: Solving for yy

  • yy is part of a triangle. The interior angles of a triangle always sum to 180°. The angles in this triangle are 45°, yy, and the angle adjacent to yy, which is equal to 63°63° (since it's also an alternate interior angle to the given 63° angle on the left side). y+45°+63°=180°y + 45° + 63° = 180° Solving for yy: y=180°45°63°=72°y = 180° - 45° - 63° = 72°

Final Results:

  • x=117°x = 117°
  • y=72°y = 72°

Would you like a detailed explanation of any specific steps?

Here are five related questions for further practice:

  1. What are the properties of angles in a parallelogram?
  2. How do alternate interior angles work with parallel lines?
  3. What is the sum of the interior angles of a triangle?
  4. If one angle of a parallelogram is 90°, what kind of parallelogram is it?
  5. Can a parallelogram have all angles equal? If so, what is it called?

Tip: In any polygon, the sum of the interior angles can be calculated by the formula (n2)×180(n-2) \times 180^\circ, where nn is the number of sides.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angles in Polygons
Parallelogram Properties

Formulas

Sum of angles in a triangle = 180°
Sum of adjacent angles in a parallelogram = 180°

Theorems

Alternate Interior Angles Theorem
Properties of Parallelograms

Suitable Grade Level

Grades 8-10