Squares

If you believed rectangles to be special, you"re in for a treat. They"re dubbed squares, and they"re perfect in every way. Well, no every method because no quadrilateral is perfect. However squares are absolutely the supermodels that the square world. (Only think healthy and balanced angles instead of sickly ones.)

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

You are watching: Which of these shapes is always equiangular

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Equilateral and equiangular mean the exact same thing as soon as you"re talking around triangles, yet quadrilaterals room a little bit pickier. A quadrilateral can be equiangular yet not it is provided (a rectangle) or equilateral however not equiangular (a rhombus). A square, however, is both. All sides of a square space the same length (equilateral) and all that angles have the same measure (equiangular). This renders the square the only equilateral and equiangular quadrilateral. The course, we have the right to just call it a "regular quadrilateral."

Sample Problem

A square"s inner angle has a measure up that"s 18 times the size of the side. What is the perimeter that the square?

We know that each internal angle in a square has a measure up of 90°. (Squares space equiangular and have 4 angle totaling 360°.) To find the perimeter, we should multiply the side length of the square s by 4 due to the fact that a square has 4 congruent sides. Therefore basically, P = 4s.

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

Sample Problem

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

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

The directly answer below is no. Rhombi are parallelograms, therefore opposite angles space congruent and consecutive ones space supplementary. If one is 90°, climate they"re every 90°, however where is it written that they need to be 90°? that isn"t. Not all over we have the right to see, anyway. So while a rhombus can it is in a square, it doesn"t have to be.

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

Sample Problem

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

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We have two means to go about this, and also both that "em involve best triangles. We would apologize for that, however it"s around time you acquired used to them.

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cm for JI. Due to the fact that the diagonals of a square space congruent and bisect every other, we know that x is half that length, orcm ≈ 2.83 cm.

If a square falls in a forest, however nobody is there to hear it, what is the definition of life?

We aren"t walk to resolve such terrifying existential situations like this one, however we do want to know how to call a square is a square—regardless that the an interpretation of life. So, if it looks favor a square and also it rolls like a square, is that a square? (Have you ever tried roll a square? even downhill, the ain"t easy.) We have the right to tell a pair different ways.

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