Measurement and Geometry : Module 20 Years : PDF Version of module. In contrast, there are many categories of special quadrilaterals. Apart from cyclic quadrilaterals, these special quadrilaterals and their properties have been introduced informally over several years, but without congruence, a rigorous discussion of them was not possible. Each congruence proof uses the diagonals to divide the quadrilateral into triangles, after which we can apply the methods of congruent triangles developed in the module, Congruence.
The material in this module is suitable for Year 8 as further applications of congruence and constructions. Because of its systematic development, it provides an excellent introduction to proof, converse statements, and sequences of theorems.
Considerable guidance in such ideas is normally required in Year 8, which is consolidated by further discussion in later years. Indeed, clarity about these ideas is one of the many reasons for teaching this material at school.
Most of the tests that we meet are converses of properties that have already been proven. For example, the fact that the base angles of an isosceles triangle are equal is a property of isosceles triangles. Now the corresponding test for a triangle to be isosceles is clearly the converse statement:. Remember that a statement may be true, but its converse false. We proved two important theorems about the angles of a quadrilateral:. To prove the first result, we constructed in each case a diagonal that lies completely inside the quadrilateral.
To prove the second result, we produced one side at each vertex of the convex quadrilateral. We begin with parallelograms, because we will be using the results about parallelograms when discussing the other figures. A parallelogram is a quadrilateral whose opposite sides are parallel. To construct a parallelogram using the definition, we can use the copy-an-angle construction to form parallel lines. For example, suppose that we are given the intervals AB and AD in the diagram below.
See the module, Construction. The three properties of a parallelogram developed below concern first, the interior angles, secondly, the sides, and thirdly the diagonals. The first property is most easily proven using angle-chasing, but it can also be proven using congruence. The opposite angles of a parallelogram are equal.
As an example, this proof has been set out in full, with the congruence test fully developed. Most of the remaining proofs however, are presented as exercises, with an abbreviated version given as an answer. The opposite sides of a parallelogram are equal. This is always true. Parallelograms are quadrilaterals with two sets of parallel sides. Since squares must be quadrilaterals with two sets of parallel sides, then all squares are parallelograms.
Always, Sometimes, Never. Alignments to Content Standards: 5. Student View. Irregular Quadrilaterals. Get better grades with tutoring from top-rated professional tutors. Get help fast. Want to see the math tutors near you? Parallelograms Parallelograms are special types of quadrilaterals with opposite sides parallel. Parallelograms have these identifying properties: Congruent opposite sides Congruent opposite angles Supplementary consecutive angles If the quadrilateral has one right angle, then it has four right angles Bisecting diagonals Each diagonal separates the parallelogram into two congruent triangles Parallelograms get their names from having two pairs of parallel opposite sides.
Proving A Quadrilateral is a Parallelogram Take a look at this quadrilateral: [insert drawing of quadrilateral where opposite sides are very slightly not parallel and of equal length, so it is really hard to see if it is a parallelogram] Is this quadrilateral a parallelogram? Next Lesson: Irregular Quadrilaterals.
Instructor: Malcolm M. Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher. Local and online. View Tutors. Geometry Help. A scalene quadrilateral is a four-sided polygon that has no congruent sides. Three examples are shown below. The following Venn Diagram shows the inclusions and intersections of the various types of quadrilaterals.
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