A rhombus is a special kind of parallelogram, in which all the sides are equal. We've seen that one of the properties of a rhombus is that its diagonals are perpendicular to each other.Here we … [Read more...] about A parallelogram with Perpendicular Diagonals is a Rhombus
Quadrangles (also called quadrilaterals) are polygons with 4 edges, and 4 angles, or vertices, at the corners.
There are several types of special polygons, with unique properties that depend on factors such as:
1. If their sides are parallel to each other or not - and if they are parallel, are both pairs of opposite edges parallel, or just one set
2. If their sides are all equal or if they just have two pairs of equal sides.
3. If the angles are right angles.
We have a section for each special type of polygon, describing and proving their properties, which are very common in high school geometry problems.
The sum of the angles in a quadrangle
One property that is common to all quadrangles, other than 4 sides and 4 vertices, is that the sum of the angles in a quadrangle is always 360°. The proof of this is simple.
Proof: Show that the sum of interior angles in a simple convex quadrangle is always 360°
Let’s think about the strategy to do this. 360 is a familiar number - it is the measure of angles in a circle, but there are no circles here, so let’s rule that out.
360 is also 2x180, and that is also a familiar number – it is the sum of the angles in a triangle – so, if we can show that this quadrangle is composed of two triangles we will have the proof- and that’s all we need.
Proof: In any simple convex polygon, a line connecting 2 points on the perimeter of the polygon is entirely within the polygon, per the definition of convex polygons. So, let’s draw a line connecting two opposite corners of the quadrangle (such a line is a called a ‘diagonal’) – say from C to A:
(1) m∠A1 + m∠D+ m∠C1 = 180° //sum of the interior angles in triangle ΔADC
(2) m∠A2 + m∠B+ m∠C2 = 180° //sum of the interior angles in triangle ΔABC
(3) m∠A1 + m∠D+ m∠C1+m∠A2 + m∠B+ m∠C2 = 360° //add both equations
(4) m∠A1 + m∠A2 +m∠D+ m∠C1+ m∠C2 + m∠B= 360° //re-arrange terms
(5) m∠A1 + m∠A2 = m∠A // angle addition postulate
(6) m∠C1 + m∠C2 = m∠C // angle addition postulate
(7) m∠A+m∠D+ m∠C + m∠B= 360° ∎ //substitution property of equality
Now that we've explained the basic concept of quadrangles in geometry, let's scroll down to work on specific geometry problems relating to this topic.
We are used to seeing the term 'radius' in the context of circles - the radius of the circle is the distance from its center to any point on its circumference.But in geometry, we also use the … [Read more...] about Radius of a Regular Polygon
Each of the diagonals of a parallelogram divides it into two congruent triangles, as we saw when we proved properties like that the opposite sides are equal to each other or that the two pairs of … [Read more...] about The Diagonals Divide a Parallelogram Into 4 Equal Areas
In a square, the diagonals bisect each other. This is a general property of any parallelogram. And as a square is a special parallelogram, which has all the parallelogram's basic properties, this is … [Read more...] about The Diagonals of a Square Bisect Each Other
We have shown that in any parallelogram, the opposite angles are congruent. Since a rhombus is a special kind of parallelogram, it follows that one of its properties is that both pairs of opposite … [Read more...] about Properties of Rhombus: The Opposite Angles are Congruent
A square is a geometric shape which is fully determined by the lengths of its side, a. If we know the length of the side of a square, we know its perimeter, its area, the length of its diagonals, … [Read more...] about How to Find the Diagonal of a Square
We have looked at a couple of special lines within a trapezoid that are parallel to the bases - the area bisector which partitions the trapezoid into two smaller trapezoids with equal areas, and the … [Read more...] about Similar Trapezoids
There are a number of interesting features of lines drawn parallel to the bases of a trapezoid. We saw one of these in this problem- a line parallel to the bases and connecting the midpoints of the … [Read more...] about Line Parallel to the Base of a Trapezoid
The triangle midsegment theorem states that the line connecting the midpoints of two sides of a triangle, called the midsegment, is parallel to the third side, and its length is equal to half the … [Read more...] about Trapezoid Midsegment Theorem
One of the proofs done by Euclid in his "Elements" book, where he laid the foundation for geometry, is that if you draw lines parallel to the sides of a parallelogram and through any point on the … [Read more...] about Parallel Lines Drawn Through a Point on the Diagonal of a Parallelogram