A triangle is a three-sided, closed polygon, (many-sided figure) arguably the most important kind of polygon.
Triangles are important in science, engineering, architecture, computer graphics and other fields. A solid knowledge of all there is to know about triangles will help you to solve many problems later, even just around your home.
The basic anatomy of a triangle is shown here. The lower case letters represent the lengths of the line segments that form the sides of the triangle; upper case letters are the angles.
We say that side a is opposite angle A, side b is opposite angle B and side c is opposite angle C. Each point is called a vertex.
The square and all other closed polygons are easily deformed by changing their angles without changing the length any side. Triangles are different. A triangle cannot be distorted by changing the measure of an angle without changing the length of a side. This has profound implications for how we and nature make structures.
This animation shows two beams attached to a vertical bar, the top one reinforced with rectangular crossmembers and the bottom with the braces in a triangular arrangement. Assume that all of the joints are flexible.
As the beams are bent downward by a force (black arrows), the lengths of the crossmembers of the upper beam don't have to compress or elongate in order for the bending to occur; the joints just have to pivot.
But as the triangle-reinforced beam is bent, half of the braces must undergo a compression force and half a stretching force. Most of the materials we use for construction, like metal and wood, are very resistant to this kind of force, so triangles make structures like these very rigid.
Here is a proof that the measure of the angles of any triangle sum to 180˚. It relies on previous knowledge that alternate-interior angles of parallel lines are congruent. That means ∠d ≅ ∠a and ∠e ≅ ∠c. You can learn more about this in the section on parallel lines.
Angles e, b and d are supplementary (sum to 180˚) as drawn, so if we show that e = c and d = a, then by substitution of c for e and a for d in the upper expression, we see that the angles of the triangle must sum to 180˚.
Note that we haven't got any specific angle measures in the proof, so it's true for any triangle.
An equilateral triangle has, as the name suggests, three congruent sides. Later we'll be able to prove this, but for now take it for granted that this means all of the angles are congruent, too.
If all of the angles (we'll call their value x) are congruent, then 3x = 180˚, so x = 60˚. An equilateral triangle has 60˚ angles.
An isosceles triangle has two congruent sides. We call the third side the base, and the angles formed by the base and the two other sides the base angles, as shown.
Later, we'll be able to prove that the base angles of an isosceles triangle have to be congruent, but for now just take it for granted.
← An obtuse triangle has one angle larger than 90˚. Because the three angles must sum to 180˚, that means the other two angles must be acute (< 90˚).
A triangle can have only one obtuse angle.
All of the angles of an acute triangle are acute (smaller than 90˚).
A scalene triangle has no two congruent sides or angles.
It is worth pausing here to think a little it about the kinds of triangles we've covered and the idea of symmetry, which is a recurring theme throughout math, science, engineering, architecture, art, ...
The equilateral triangle looks the same when rotated about either of three different axes of rotation,
as shown, and those three axes are also lines of mirror symmetry: one side looks like the reflection of the other in a mirror held on that line. In addition, the equilateral triangle has three-fold rotational symmetry: When the triangle is rotated by 120˚ in the plane, it looks the same.
The isosceles triangle has a little less symmetry, but there is some. Scalene triangles have no special symmetry.
The right triangle is very important for a couple of reasons. First, it forms the basis for the whole field of trigonometry, which you will find very useful in a variety of fields for solving real problems, and second because any non-right triangle can always be made into two right triangles by drawing an altitude.
The sides of a right triangle are called sides or legs, but the side opposite the right angle is called the hypotenuse. Only a right triangle has a hypotenuse, and it's always opposite the right angle.
For right triangles only, the Pythagorean theorem is true: The square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. As it is labeled here → ,
c2 = a2 + b2
Later we'll work out the trigonometric ratios for a right triangle, but that can wait a bit.
The Pythagorean theorem is one of the most important relationships in mathematics. You will use it many times if you continue to study math. It says that
The square of the side of the hypotenuse (c) of a right triangle, is equal to the sum of the squares of the other two sides (a & b).
An altitude of a triangle is a line segment drawn from a vertex to the opposite side so that it intersects that side at a right angle. The altitude is the height of the triangle
An altitude may also intersect an extension of the base outside of the triangle:
This is just one of many proofs of the Pythagorean theorem. The figure below shows a square with sides of length (a + b). Inscribed inside is a square with sides of length c.
Now the area of the outside square is (a + b)2 and the area of the inside square is c2. If we subtract the two, we get the total area of the four pink right triangles.
The area of the inside square is c2, and we can use these two to find the sum of the areas of the four (pink) right triangles (we'll call it A):
But we can also find the area of those triangles directly using the triangle area formula A = ½bh, where b = length of base and h = height. We'll call a the base and b the height of the triangles in our figure (yeah, b stands for two different things here – sorry!):
If we equate these we get
The 2ab terms cancel:
Now we just rearrange to get the Pythagorean theorem,
How do we know that two triangles are congruent? How much information do we need about two triangles to show that they are congruent?
We define triangles to be congruent if every corresponding side and angle of each is congruent. Triangles ΔABC and ΔXYZ below are congruent because every pair of corresponding sides and corresponding angles (3 pairs each) are congruent. It doesnt' matter that these triangles appear to be mirror reflections of one-another. They are congruent if they can be superimposed by a simple rotation or "flip" of one onto the other.
In geometry we don't use the equal sign (=) to designate equivalence of two figures, like a pair of triangles, sides or angles. We say that the two figures are congruent, and use the symbol
The question we really want to answer is: How much information about corresponding parts of triangles is enough to prove they're congruent? It turns out there are several ways to prove triangle congruence using just three correspondences. We'll start with knowing the three sides. Take a look at the animation on the right.
← This animation shows the rearrangement of the sides of a scalene triangle (all sides of different length.
The black and red sides are swapped and the red side is fit in to fill the gap to re-make the third side. The triangle is then simply rotated back to its starting position, and the sides are in the same order. It's the same triangle.
This process can be repeated with either of the other two such rearrangements: swap red-black or swap red-green. The results show that if we specify the lengths of the sides of a triangle, it is specified unambiguously. Those sides simply can't form any other triangle.
For any three line segments with given lengths, one and only one triangle can be drawn. The order of joining the segments does not matter.
That leads us to a theorem about triangles we'll call the side-side-side (SSS) theorem. We've shown that if the sides are specified, we can get one and only one triangle. The angles are dependent on the lengths of the sides in ways that we're not prepared to deal with right now, but it makes sense that if two triangles are congruent, then their corresponding angles must be, too.
If there is a correspondence between the sides of two triangles such that all three pairs of corresponding sides are congruent, then the triangles are congruent.
If knowing that all pairs of corresponding sides of two triangles means that all the angles must be congruent, too, how about knowing that all pairs of corresponding angles are congruent? Does that mean that all pairs of sides are, too? Take a look at this picture on the right.
Three pairs of corresponding congruent angles (AAA) does not assure that the triangles are congruent. There are, in fact, an infinite number of triangles with any set of three angles; they are just scaled versions of one another.
Knowing that the corresponding angles of two triangles are congruent is not enough, in itself, to conclude that the triangles are congruent. We'd need a pair of congruent sides, too.
The diagram on the right shows that if the lengths of two sides of a triangle and the angle formed by those sides (called the included angle), are specified, then there is no choice about where the other side goes or the measures of the other two angles.
This gives us another way to prove that two triangles are congruent, and a new theorem we'll call the SAS theorem:
If there is a correspondence between two sides and the included angle of two triangles, such that corresponding sides are congruent and the corresponding included angles are congruent, then the triangles must be congruent.
This diagram shows that if two angles of a triangle and the side that they share (called the included side), are specified, then there is no choice about where the other sides intersect. That intersection determines both of the lengths of the other two sides and the angle between those sides, so the triangle is completely specified.
This gives us yet another way to prove that two triangles are congruent, and a new theorem we'll call the ASA theorem:
If there is a correspondence between two angles and the included side of two triangles, such that corresponding angles are congruent and the corresponding included sides are congruent, then the triangles must be congruent.
If we specify two angles of a triangle and one side, the triangle is also completely specified, meaning that no other triangle can have those measurements. The situation is illustrated on the right. If we take the upper triangle, keep the labeled angles and side BC the same, but sorten or lengthen segment AC, we spoil the triangle. We'd either have to change angle BA or the length of side BC in order to form a triangle with the altered side AC.
AAS isn't really different from ASA, though. Notice in that in the top triangle, because the angles of a triangle must sum to 180˚, if we specify two angles (∠A and ∠C), then the third is automatically specified, so we'd have an ASA situation after all.
Some books call the 180˚ sum rule for the angles of a triangle the "no-choice" theorem – if two angles are known, there is no choice about the measure of the third.
The diagram on the right shows why knowing two consecutive sides and the following angle is not enough to completely specify a triangle. If we specify the lengths of sides CB and BA, and the measure of ∠A (top triangle), you can see that there are two values of angle C that will preserve both side lengths and the angle we know (A). It's an ambiguous situation and we can't tolerate that in math ... so SSA is out as a way to prove that two triangles are congruent.
Knowing that two corresponding sides and the next angle of two triangles are congruent is not enough to conclude that the triangles are congruent.
We just showed that SSA isn't a reliable way to prove triangle congruence, but it does work in one special case — for right triangles. As you saw above (and in the top figure on the left), when consecutive sides and the next angle are known (SSA), two possible triangles could result. The third side of each forms an equal angle with the line (magenta) that would form a right triangle.
But when the triangle we begin with is already a right triangle, there is no more ambiguity. Only one right triangle can result. We name this situation HL, for "hypotenuse-leg." The leg can be either of the sides that is not the hypotenuse.
Recall that sometimes we refer to the non-hypotenuse sides of a right triangle as legs. We then have the HL theorem:
If the hypotenuse and one side (leg) of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then those triangles are congruent.
An altitude is one of two special lines that we often draw inside of triangles. An altitude is a line drawn from one vertex of a triangle to the opposite side so that it is perpendicular to the opposite side. Obviously, we can draw three different altitudes for a triangle. We'll be more specific about which one we mean when we calculate the areas of triangles later.
The figure on the right below shows that we have to do a little extra work to find some altitudes of an obtuse triangle. In those cases, simply extend one of the sides, as shown, and draw the altitude perpendicular to that extended side.
A median of a triangle is a line segment drawn from one vertex to the midpoint of the opposite side. Each triangle has three medians. The interesting thing about the medians of a triangle is that they always intersect at a common point, called the centroid of the triangle. The centroid would be the center of balance — the balance point — of the triangle if it were made of a uniform material.
The medians of some triangles are also altitudes. Sketch out for yourself the medians and altitudes of equilateral and isosceles triangles.
In the coordinate geometry section you can learn how to find the centroid of a triangle from just its (x, y) coordinates.
The centroid of a flat object of uniform thickness (and density), like a piece of metal, is the location at which it would perfectly balance on a point.
The area of any triangle is one-half of the lenth of its base multiplied by its height,
That's pretty easy to see for a right triangle inscribed in a rectangle, like the one below. The diagonal (hypotenuse of the triangle) cuts the rectangle in half.
Now let's look at a more general case – calculating the area of any old triangle. We'll still inscribe it in the same rectangle with dimensions b x h.
If we draw the altitude of the triangle (dashed line), we see that it cuts the rectangle in half, and that each part of the rectangle is then cut in half by one of the diagonal sides of the inscribed triangle. The area of each triangle is shown in the figure, and the sum of those areas is
If we factor out the 1/2 and the h, we get
Now the x's subtract away, leaving us with exactly the same formula for the area of a triangle:
Solution: In order to find the area of a triangle like this, without a known altitude, we must first determine an altitude. That will take a little geometry and algebra. We begin by drawing in an altitude, forming two new right tringles – something we can work with:
We don't know the altitude, and we don't know exactly where it intersects the base of the triangle, but we do know that the base is ten units long, so it must be divided into pieces x and 10-x long.
Now we can use the Pythagorean theorem to write two expressions for h2, one from each of our new right triangles.
Now both of the right sides of these equations are equal to h2, so (by the transitive property) they must be equal to each other. Expanding a little bit and equating the right sides gives us
Now we can distribute that negative sign on the left side, which gives us an x2 on each side. Those can be removed.
Now we have a nice expression from which we can solve for x. First put the constants on the right side:
and divide by 20 to get x.
Now we can solve for h from either of our original Pythagorean expressions (you can try the other to make sure it gives the same result – it has to).
Next we can fully label our divided triangle.
Now it's easy to find the areas of our two right triangles – we'll call them A1 and A2 – using the triangle area formula. The first is
The other is
And finally, the sum gives us the area of our triangle:
The transitive property of algebra says that if two things are equivalent to the same thing, then they are equivalent to each other.
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