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#### Define The Right Angle: List The Properties That Identify Right Angles

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# Define the right angle.

In pure mathematics and pure mathematics, a right angle is an angle of precisely 90° (degrees), adore 1 / 4 flip. If a ray is placed in order that its termination is on a line and the adjacent angles are equal, then they are right angles. The term could be a locution of Latin angelus muscle; here rectus suggests that "upright", pertaining to the vertical perpendicular to a horizontal bottom line.

Closely connected and vital geometrical ideas are perpendicular lines, which means lines that type right angles at their purpose of intersection, and orthogonality, that is that the property of forming right angles, typically applied to vectors. The presence of a right angle in a very triangle is that the shaping issue for right triangles, creating the correct angle basic to trigonometry.

Throughout history, carpenters Associate in Nursing masons have proverbial a fast thanks to make sure if an angle could be a true "right angle". It has supported the foremost wide proverbial Pythagorean triple (3, 4, 5) so known as the "rule of 3-4-5". From the angle in question, running a line on one aspect precisely three units long, and on the second aspect precisely four units long, can produce a flank (the longer line opposite the correct angle that connects the 2 measured endpoints) of precisely five units long. This measuring may be created quickly and while not technical instruments. The geometric law behind the measuring is that the Pythagorean Theorem ("The sq. of the flank of a trigon is capable the total of the squares of the 2 adjacent sides").

# Properties of Right Triangles.

A right triangle must have one interior angle of exactly 90°90°. It can be scalene or isosceles but never equilateral.

# Construct a Right Angled Triangle:

Use two uncooked spaghetti strands to make your own right triangle. Leave one alone; break the other strand into two unequal lengths. Place the two short strands a and b so they meet at two endpoints and form a 90°90° angle. Laying the third strand cc down to intersect the two endpoints of a and b creates a right triangle.

You can make a more accurate right triangle by using graph paper and a straightedge. Draw a line segment (of any desired length) along the graph paper has printed lines. Follow the lines to make a second line segment exactly 90°90° to your first line segment, of any desired length. If you connect the two endpoints of those line segments, you have a right triangle.

Geometry uses symbols as shorthand. Here are important ones to know:

means "similar"

∠∠ means "angle"

△△ means "triangle"

# The Pythagorean Theorem:

Greek mathematician Pythagoras gets the credit, but other civilizations knew about this theorem. The Pythagorean Theorem describes the relationship between the lengths of legs a and b of any right triangle to the length of hypotenuse c:

The sum of the squares of legs aa and b are equal to the square of hypotenuse c, or f x = a2 + b2 = c2.

Thousands of proofs of this theorem exist, including one by U.S. president James Garfield (before he became president). One proof is easy to make with graph paper, a straightedge, pencil, and scissors.

Construct ABCABC with legs a and b to the left and bottom and hypotenuse c at the top right. Leg a is opposite A, leg bb is opposite B, and hypotenuse c is opposite right angle C.

Let length a = 3a = 3, b = 4b = 4, and hypotenuse c = 5c = 5.

Construct a square using leg a as the right side of the square. It will be 9 square units (a2). Construct a square using leg b as the top side of its square, so it is 16 square units (b2). Cut out another 5 x 5 square and line it up with hypotenuse c, so the square is c2.

Area = a × a = a2

Area = b × b = b2

Area = c × c = c2

a2 + b2 = c2

# Right Triangle Altitude Theorem:

The right triangle altitude theorem tells us that the altitude of a right triangle drawn to the hypotenuse c forms two similar right triangles that are also similar to the original right triangle.

Construct ABC so that hypotenuse c is horizontal and opposite right angle C, meaning legs a and b are intersecting above c to form the right angle C. This puts A to the bottom left, and B to the bottom right.

Construct an altitude (or height) h from the interior right angle C to hypotenuse c (so it is perpendicular to c).

This altitude h creates two smaller triangles inside our original triangle. The altitude divided C, and created two right angles where it intersected hypotenuse c.

Call the point where the altitude h touches hypotenuse c point D. You now have two triangles, ACD and BCD. Each of these triangles is similar to the other triangle, and both are similar to the original triangle

You can prove this by seeing that new triangle's ADC = original triangle's ACB, while new triangle's CAD = original triangle's CAB.

This means two angles of ADC and ABC are similar, making the triangles themselves similar (by the Angle-Angle postulate or AA postulate):

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