A trapezoid , also called a trapezium in some countries, is a quadrilateral with exactly one pair of parallel sides. Show
 The parallel sides are called the bases and the non-parallel sides are the legs of the trapezoid. An isosceles trapezoid is a trapezoid in which the two non-parallel sides are congruent . The area A of a trapezoid is given by A = b 1 + b 2 2 h where b 1 and b 2 are the lengths of the two parallel sides, and h is the height, as shown in the figure below.
The perimeter of a trapezoid is the sum of the lengths of its four sides. If one or more of the lengths is not known, you can sometimes use the Pythagorean Theorem to find it.
Example: Find the area and perimeter of the trapezoid shown.
To find the area, apply the formula. A = b 1 + b 2 2 h = 3 + 11 2 ( 7 ) = 7 ( 7 ) = 49 square units To find the perimeter, add the lengths of all four sides. P = 3 + 10 + 11 + 8 = 32 units In today’s geometry lesson, we’re concluding our study of quadrilaterals, by looking at the properties of trapezoids and kites. Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher) You’ll learn all the trapezoidal properties needed to find missing sides, angles, and perimeters. In addition, we’ll explore kites and discuss their associated properties. Let’s get started! What Is A Trapezoid?A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, and the other two sides are called legs. Bases and Legs of a Trapezoid And because the bases are parallel, we know that if a transversal cuts two parallel lines, then the consecutive interior angles are supplementary. This means that the lower base angles are supplementary to upper base angles. Midsegment of a Trapezoid Additionally, the midsegment of a trapezoid is the segment joining the midpoints of the legs, and it is always parallel to the bases. But even more importantly, the midsegment measures one-half the sum of the measure of the bases.
Cool! Now, if a trapezoid is isosceles, then the legs are congruent, and each pair of base angles are congruent. In other words, the lower base angles are congruent, and the upper base angles are also congruent. Likewise, because of same-side interior angles, a lower base angle is supplementary to any upper base angle. Properties of an Isosceles Trapezoid But there’s one more distinguishing element regarding an isosceles trapezoid. A trapezoid is isosceles if and only if its diagonals are congruent. So if we can prove that the bases are parallel and the diagonals are congruent, then we know the quadrilateral is an isosceles trapezoid, as Cool Math accurately states. In the video below, we’re going to work through several examples including:
What Are The Properties Of Kites?The first thing that pops into everyone’s mind is the toy that flies in the wind at the end of a long string. But have you ever stopped to wonder why a kite flies so well? The way a toy kite is made has everything to do with mathematics! In fact, a kite is a special type of polygon. A kite is a quadrilateral that has two pairs of consecutive congruent sides. And while the opposite sides are not congruent, the opposite angles formed are congruent. Congruent Sides and Angles of a Kite Moreover, the diagonals of a kite are perpendicular, and the diagonal bisects the pair of congruent opposite angles. Perpendicular Diagonals of a Kite This means, that because the diagonals intersect at a 90-degree angle, we can use our knowledge of the Pythagorean Theorem to find the missing side lengths of a kite and then, in turn, find the perimeter of this special polygon. This framework of two pairs of consecutive congruent sides, opposite angles congruent, and perpendicular diagonals is what allows for the toy kite to fly so well. Gosh, doesn’t it make you want to get outside and play? Trapezoid Properties – Lesson & Examples (Video)41 min
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Thanks in advance to anyone who can help me out on this. I'm currently a junior in high school taking and doing well my school's honors pre-calc class, but of all of the math I've ever learned, proofs have always given me trouble. And that's what we've just begun re-learning, my kryptonite. Anyways, I just can't figure out how to prove this problem like it says.
I'm solely restricted to using things like the midpoint formula, distance formula, slope formula, etc. I can't use any theorems from geometry other than the Pythagorean Theorem. I've tried drawing a trapezoid with the points (0,0), (a, b), (a+c, b), and (d,0), and then since the given is that the diagonals are equal, finding the distances of the diagonals and setting them equal, and solving them for one variable. Then I took that variable and then plugged it into the distances of the two legs of the trapezoid, since an isosceles trapezoid has two congruent legs. Unfortunately, I just can't get the two legs of the trapezoid to end up being congruent. If anyone can explain how to solve this problem, that would be awesome! |
If the two non parallel sides of a trapezoid are congruent then the trapezoid is isosceles
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