Understanding Theorems about Parallelograms
Understanding the theorems about parallelograms involves exploring the properties and relationships inherent within this specific type of quadrilateral. A parallelogram is defined by its two sets of opposite sides that are parallel, meaning they will never intersect, even when extended. This geometric figure adheres to specific theorems that help establish rules regarding the angles and sides of the shape, providing a foundation for problem-solving in mathematics. Key among these theorems is that when a transversal intersects two parallel lines, it creates congruent angles, leading to various relationships between angles inside the parallelogram.
In practical terms, the congruency of opposing angles and the presence of both obtuse and acute angles are essential for understanding the balance and stability of structures in the real world. These geometric principles are not only crucial for academic study but also play a vital role in engineering and architecture, where ensuring safety and stability is paramount. Overall, the study of parallelogram theorems offers valuable insights into the consistent rules governing geometric shapes and their applications in everyday life.
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Subject Terms
Understanding Theorems about Parallelograms
A parallelogram is a quadrilateral—a shape or closed figure that has four sides—but not all quadrilaterals are parallelograms. The parallelogram gets its name from being a shape that has two sets of parallel lines that are opposite to one another and will never touch or intersect when extended. Parallelograms are also known as polygons.
Overview
Mathematics requires a set of rules that stay true to a geometric shape regardless of how big, small, long, or wide it is. Having a consistent set of rules allows the novice mathematician an efficient way to problem solve while reinforcing basic geometric concepts. To make sure the geometric rules of parallelograms are consistent with a diagram, theorems are used to prove statements about the relationship between the parallelogram’s lines and angles.
Theorem 1. If a transversal intersects two parallel lines, then congruent angles are created.
Proof 1. Angles 3 and 6 are congruent and angles 4 and 5 are alternate interior angles.
Proof 2. Angles 1 and 8 and angles 2 and 7 are angles exterior angles.
Proof 3. Angles 1 and 5, angles 2 and 6, angles 3 and 7, and angles 4 and 8 are corresponding angles that are in the same position.
Congruency between opposing angles do occur when a transverse line intersects through a parallelogram. If a shape is truly a parallelogram, it will have four sides with a set of two obtuse angles and two obtuse angles. The opposing angles will be congruent, or exactly the same measurement in degrees. If the shape is traced with extending lines, one should be able to create two transverse lines that run through the horizontal set of parallel lines.
Real-World Perspective
When considering how proofs and theorems in geometry pertain to everyday life, one could simply think about how many things have angles that require congruency for balance and stability. All structures must be built with angles that have a level of standards to provide covering for individuals. But more importantly is structurally sound for safety. Simply looking at a problem involving measurement requires one to be able to hypothesize, test, and deduce the validity of all proposed solutions.
Bibliography
McKellar, Danica. Girls Get Curves. New York: Hudson Street, 2012.
Stankowski, James F., ed. Geometry and Trigonometry. New York: Rosen, 2015.
Wyss, Walter. "Perfect Parallelograms." The American Mathematical Monthly (2012): 1.