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:
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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°).
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Given Angles:
- The angle labeled 63° and the one labeled 45° are given.
- The angles labeled as and need to be determined.
Let's start with solving for and :
Step 1: Solving for
- Since opposite angles of a parallelogram are equal, the angle adjacent to the angle labeled 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°: Solving for :
Step 2: Solving for
- is part of a triangle. The interior angles of a triangle always sum to 180°. The angles in this triangle are 45°, , and the angle adjacent to , which is equal to (since it's also an alternate interior angle to the given 63° angle on the left side). Solving for :
Final Results:
Would you like a detailed explanation of any specific steps?
Here are five related questions for further practice:
- What are the properties of angles in a parallelogram?
- How do alternate interior angles work with parallel lines?
- What is the sum of the interior angles of a triangle?
- If one angle of a parallelogram is 90°, what kind of parallelogram is it?
- 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 , where 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