Squares

If you assumed rectangles to be special, you"re in because that a treat. They"re called squares, and also they"re perfect in every way. Well, not every way because no square is perfect. However squares are definitely the supermodels the the square world. (Only think healthy and balanced angles instead of sickly ones.)

A square is a rectangle with four congruent sides. Or a rhombus with 4 congruent angles. Or a parallelogram with four congruent sides and also four congruent angles. Take her pick; they"re every the same. That"s right: a square is a parallelogram, a rectangle, and a rhombus all rolled into one. It"s every that and also a bag the chips.

You are watching: A quadrilateral that is equilateral but not equiangular

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Equilateral and also equiangular average the exact same thing when you"re talking around triangles, yet quadrilaterals room a little pickier. A quadrilateral have the right to be equiangular yet not it is intended (a rectangle) or equilateral yet not equiangular (a rhombus). A square, however, is both. Every sides the a square room the same size (equilateral) and all its angles have actually the same measure (equiangular). This makes the square the just equilateral and equiangular quadrilateral. That course, we can just contact it a "regular quadrilateral."

Sample Problem

A square"s interior angle has a measure up that"s 18 time the length of that side. What is the perimeter that the square?

We recognize that each interior angle in a square has actually a measure of 90°. (Squares space equiangular and have 4 angle totaling 360°.) To find the perimeter, we have to multiply the side length of the square s through 4 due to the fact that a square has 4 congruent sides. So basically, P = 4s.

If one angle"s measure up is 18 times the of the side, we understand that 18s = 90. Us can discover the value of s by detect 90 ÷ 18. Once we"ve calculate s = 5, we can discover P = 4s = 4(5) = 20.

Sample Problem

If a quadrilateral is a rhombus, is it additionally a square?

A square is any kind of four-sided figure. A rhombus is a quadrilateral with four congruent sides. A square is a quadrilateral with four congruent sides and 4 congruent angles. So the question here becomes: must a quadrilateral with four congruent political parties have four congruent angles?

The right answer here is no. Rhombi are parallelograms, therefore opposite angles space congruent and consecutive ones room supplementary. If one is 90°, then they"re all 90°, yet where is it created that they have to be 90°? that isn"t. Not anywhere we deserve to see, anyway. So while a rhombus can it is in a square, the doesn"t need to be.

Since squares are simply rectangular rhombi (or rhomboidal rectangles), your diagonals have the very same properties together those the rhombi and rectangles. Namely, a square"s diagonals are congruent perpendicular bisectors of every other. Plus, they can tap dance, pat the cello, and do parkour. Multi-talented small rascals, aren"t they?

Sample Problem

Quadrilateral JAIM is a square. Discover the size of x.

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We have two ways to go about this, and both of "em involve best triangles. We would certainly apologize because that that, but it"s about time you acquired used come them.

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cm for JI. Because the diagonals that a square room congruent and bisect every other, we know that x is half that length, orcm ≈ 2.83 cm.

If a square drops in a forest, but nobody is over there to listen it, what is the meaning of life?

We aren"t going to resolve such terrifying existential dilemmas like this one, however we do desire to know how to tell a square is a square—regardless of the definition of life. So, if it looks choose a square and it rolls prefer a square, is the a square? (Have you ever before tried rojo a square? also downhill, it ain"t easy.) We deserve to tell a couple different ways.

If a quadrilateral is equilateral and equiangular, it"s a square.If a rhombus is equiangular, it"s a square.If a rectangle is equilateral, it"s a square.

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Basically, if you can prove the a quadrilateral is both a rhombus and also a rectangle, you"ve verified that it"s a square. Of course, we can use diagonals to prove the rhombus and also rectangle business, however we"ll leave the to you.