Classifying triangles
Every triangle has three sides and three angles. Which two comparisons sort all of them into families, and what does each name tell you?
Where you are 15 in Two-dimensional shapes
When you reach for it
Triangles come in endless shapes, but two plain questions sort every one of them into families. How do its three sides compare with each other? That gives scalene, isosceles or equilateral. How does its largest angle compare with a square corner? That gives acute, right or obtuse. Every triangle gets one name from each list, as in “an isosceles right triangle” or “a scalene obtuse triangle”.
You reach for this sort whenever a triangle is in front of you, drawn or described, and you need to say what kind it is, because the kind tells you what you are allowed to use. An isosceles triangle hands you two equal sides, and an equilateral one hands you all three. A right triangle opens the Pythagorean theorem and right triangle trigonometry. A triangle with no right angle, acute or obtuse, is called oblique and needs other tools. Some theorems apply to one family only, such as the base angles theorem for isosceles triangles. In a proof, “let \triangle ABC be acute” is a real assumption, and it is worth knowing what it rules out.
You also use the names backwards. Told that a triangle is isosceles with a perimeter of 84, you can write one equation with one unknown. Told that a triangle has an angle of 110^\circ, you know at once what its other two angles must be like.
The procedure
The two sorts are separate questions, each with its own test. They are taken one at a time below, and then put together.
Classifying triangles by sides
Call the three side lengths a, b and c. The names count how many of them are equal.
- A scalene triangle has no two sides of the same length.
- An isosceles triangle has at least two sides of the same length.
- An equilateral triangle has all three sides of the same length.
The words say what they mean. Isosceles comes from the Greek isos, equal, and skelos, leg, and equilateral from the Latin aequus, equal, and latus, side.
In a drawing, the short strokes across the sides are hash marks. Sides carrying the same number of hash marks are equal in length, and a side with no partner has no mark. In Figure 1 the scalene triangle has none, because none of its sides match another.
A triangle has three pairs of sides, a with b, b with c, and a with c, and the name depends on how many of those pairs are equal. The count can be 0, 1 or 3, and never 2. Suppose a = b and b = c. Then a = c as well, because two lengths that both equal b equal each other, so a second equal pair always brings the third. That leaves three cases, and each one is a name:
| Equal pairs of sides | Name |
|---|---|
| 0 | scalene |
| 1 | isosceles, and not equilateral |
| 3 | equilateral, which is also isosceles |
A triangle that is isosceles but not equilateral has one pair of equal sides, called its legs, and one side left over, called its base. The base can be shorter than the legs or longer. Sides of 12, 12 and 13 give an isosceles triangle whose base is the longest side.
Figure 2 shows how the three names fit together. They do not cut triangles into three separate piles. Every equilateral triangle has at least two equal sides, so it is isosceles too, and the scalene triangles are everything that is left over.
To name a triangle by its sides, three things happen, each with its reason.
Put the lengths in one unit first. A length can be written in several ways, and 3\text{ cm} and 30\text{ mm} are the same length. Comparing the bare numbers 3 and 30 would call them different. Convert everything to one unit, then compare.
Compare the lengths themselves, in any order. The equal pair does not have to sit next to each other in a list. Sides of 6, 8 and 6 still have one equal pair. Compare a with b, then b with c, and compare a with c if you still need to. If the first two pairs are both equal, you can stop: the third is equal by the argument above.
From a drawing, trust the marks and the labels. Two sides are known to be equal when they carry the same hash marks or the same length. A side that only looks the same length as another, with nothing saying so, is not known to be.
The name also works backwards, from name to equation. The perimeter is the sum of the three sides, as in finding a perimeter, and a name lets you write a repeated side with one letter. If the sides are all s, or if the legs are both s and the base is b, then
That is how a problem with one unknown length is built out of a triangle that has three.
Classifying triangles by angles
The three names compare angles with a square corner, 90^\circ (see acute, right and obtuse angles). We write \triangle ABC for the triangle with corners A, B and C, and \angle A for its angle at A.
- An acute triangle has three acute angles, each less than 90^\circ.
- A right triangle has one right angle, exactly 90^\circ.
- An obtuse triangle has one obtuse angle, more than 90^\circ and less than 180^\circ.
The words “one angle” matter, and the reason is a fact about every triangle: a triangle cannot have two angles of 90^\circ or more. The angles of a triangle add to 180^\circ (Triangle angle sum), and every one of them is larger than 0^\circ. Suppose two angles were each at least 90^\circ. Together they would be at least 180^\circ, and that leaves nothing positive for the third angle. So there is no such triangle. In particular a triangle cannot have a right angle and an obtuse angle together.
Figure 4 shows the reason in a special case. Stand two sides on the ends of a base, each making the same angle with it. At 90^\circ the sides are parallel and never meet. At 110^\circ they lean apart. To close up into a triangle the sides have to lean toward each other, which takes base angles that add to less than 180^\circ.
This fact settles the whole sort. If a triangle has an angle of 90^\circ or more, that angle is the only one, so it is also the largest angle. So the largest angle decides the name:
| If the largest angle is | the triangle is | and the other two angles add to |
|---|---|---|
| less than 90^\circ | acute | more than 90^\circ |
| exactly 90^\circ | right | exactly 90^\circ |
| more than 90^\circ | obtuse | less than 90^\circ |
The last column is not a separate fact. The three angles add to 180^\circ, so the other two add to 180^\circ minus the largest. In a right triangle the two smaller angles are therefore complementary, and in a right or an obtuse triangle both of them are acute, since each is smaller than a sum that is at most 90^\circ.
Every triangle lands in exactly one row. The largest angle is less than, equal to, or greater than 90^\circ, and only one of those is true. Unlike the side names, these three leave nothing out and overlap nowhere.
To name a triangle by its angles, four things happen, each with its reason.
Get all three angles. If two are known, the third is 180^\circ minus their sum, because the three must add to 180^\circ.
Find the largest. An angle of 90^\circ or more, if there is one, must be the largest, so the largest is the only angle worth testing.
Compare it with 90^\circ. Less is acute, equal is right, more is obtuse.
Read the marks honestly. In a diagram, a small square at a corner says the angle is exactly 90^\circ, and a number says its size. A corner with neither has no known size, however it looks.
Both names at once
A full name takes one word from each list, and the two tests are run separately: the sides test never looks at an angle, and the angles test never looks at a side. Even so, the two lists are linked by two facts. Equal sides have equal angles opposite them, and the reverse (the base angles theorem and its converse), so a triangle is scalene exactly when its three angles are all different. And the longest side sits opposite the largest angle (sides and angles in order). Example 8 uses the first fact to find which pairs of names can happen at all, and Example 9 uses the second to decide acute, right or obtuse from the sides alone.
Worked examples
Comparing three lengths
Classify each triangle from its side lengths.
(a) 7\text{ cm}, 7\text{ cm}, 10\text{ cm} \qquad (b) 5\text{ cm}, 6\text{ cm}, 7\text{ cm}
(c) 9\text{ in}, 9\text{ in}, 9\text{ in} \qquad (d) 6\text{ m}, 8\text{ m}, 6\text{ m}
(e) 0.4\text{ m}, 40\text{ cm}, 400\text{ mm} \qquad (f) 2.5\text{ cm}, 25\text{ mm}, 3\text{ cm}
SolutionIn (a) to (d) all the lengths are in one unit, so the numbers can be compared directly.
(a) The first two sides are equal, 7 = 7, and 10 is different from both. That is one equal pair, so the triangle is isosceles, and not equilateral.
(b) 5 \ne 6, 6 \ne 7 and 5 \ne 7. No pair is equal, so it is scalene.
(c) All three lengths are 9, so all three pairs are equal. It is equilateral, and therefore also isosceles.
(d) The equal sides are 6 and 6, with the 8 between them in the list. The order of the list does not matter, so this is one equal pair: isosceles.
In (e) and (f) the units differ, so convert to centimeters before comparing.
(e) There are 100\text{ cm} in 1\text{ m}, so 0.4\text{ m} = 40\text{ cm}. There are 10\text{ mm} in 1\text{ cm}, so 400\text{ mm} = 40\text{ cm}. The sides are 40, 40 and 40 centimeters, so the triangle is equilateral.
(f) 25\text{ mm} = 2.5\text{ cm}. The sides are 2.5, 2.5 and 3 centimeters. One pair is equal and the third side is different, so the triangle is isosceles.
Naming from the angles
Name the triangle with each set of angles. First check that the three can be the angles of a triangle.
(a) 35^\circ, 65^\circ, 80^\circ \qquad (b) 28^\circ, 62^\circ, 90^\circ
(c) 20^\circ, 35^\circ, 125^\circ \qquad (d) 89^\circ, 46^\circ, 45^\circ
Then name (e) the triangle in Figure 5, and (f) the triangle with two angles of 112^\circ and 41^\circ.
A set of angles can belong to a triangle only if each is positive and they add to 180^\circ. Then the largest angle is compared with 90^\circ.
(a) 35 + 65 + 80 = 180. The largest angle is 80^\circ, which is less than 90^\circ, so the triangle is acute.
(b) 28 + 62 + 90 = 180. The largest angle is exactly 90^\circ, so the triangle is right. As the table says, the other two angles add to 90^\circ: 28 + 62 = 90.
(c) 20 + 35 + 125 = 180. The largest angle is 125^\circ, more than 90^\circ, so the triangle is obtuse.
(d) 89 + 46 + 45 = 180. The largest angle is 89^\circ. It is very close to a square corner but it is less than 90^\circ, so the triangle is acute. Only the comparison with 90^\circ matters, not how near it is.
In (e) and (f) only two angles are known. The third is 180^\circ minus their sum, and after that the largest of all three decides.
(e) 47 + 38 = 85, so x = 180 - 85 = 95. The angles are 47^\circ, 38^\circ and 95^\circ. The largest is 95^\circ, so the triangle is obtuse.
(f) 112 + 41 = 153, so the third angle is 180 - 153 = 27. The angles are 112^\circ, 41^\circ and 27^\circ. This time the largest angle was one of the two we were given. It is 112^\circ, so the triangle is obtuse. The third angle by itself, 27^\circ, does not name the triangle. All three have to be in hand before the largest can be picked.
Angles from algebra
(a) The angles of a triangle are in the ratio 1 : 2 : 6. Name the triangle. Do the same for the ratios (b) 1 : 2 : 3 and (c) 2 : 3 : 4. Then (d): the angles of a triangle are (7x)^\circ, (x + 10)^\circ and (2x + 10)^\circ; find x, the three angles, and name the triangle.
Solution(a) A ratio 1 : 2 : 6 means the angles are x^\circ, 2x^\circ and 6x^\circ for some x. They add to 180^\circ:
The angles are 20^\circ, 40^\circ and 120^\circ. The largest is 120^\circ, so the triangle is obtuse.
(b) x + 2x + 3x = 180 gives 6x = 180 and x = 30. The angles are 30^\circ, 60^\circ and 90^\circ, so the triangle is right.
(c) 2x + 3x + 4x = 180 gives 9x = 180 and x = 20. The angles are 40^\circ, 60^\circ and 80^\circ, so the triangle is acute.
(d) The three expressions add to 180, which gives an equation in x:
Substitute into each expression: 7(16) = 112, 16 + 10 = 26 and 2(16) + 10 = 42. The angles are 112^\circ, 26^\circ and 42^\circ. All three are positive, so this is a real triangle. The largest angle is 112^\circ, which is more than 90^\circ, so the triangle is obtuse.
Check. Each set adds to 180^\circ: 20 + 40 + 120, 30 + 60 + 90, 40 + 60 + 80 and 112 + 26 + 42 all give 180.
Finding sides from the perimeter
(a) The perimeter of an equilateral triangle is 57 inches. How long is each side?
(b) A triangular pennant, shown in Figure 6, is isosceles. Its third side is 30\text{ cm} and its perimeter is 84\text{ cm}. How long is each of the two equal sides?
(a) Let s be the length of one side. All three sides are equal, so the perimeter is the sum of three copies of s:
Dividing both sides by 3 undoes the multiplication, so s = 19. Each side is 19 inches. Check. 19 + 19 + 19 = 57.
(b) Let s be the length of each equal side. The perimeter is the two equal sides plus the third:
Subtract 30 from both sides, then divide by 2:
Each equal side is 27\text{ cm}.
Check. 27 + 27 + 30 = 84. The sides must also make a triangle, which needs every side to be shorter than the other two together, as in the triangle inequality. The longest side is 30, and the other two together make 27 + 27 = 54, which is more than 30. They do make a triangle.
Sides written as expressions
The triangle in Figure 7 is equilateral. Its sides, in centimeters, are (2x + 5), (4x - 7) and (3x - 1). Find x, and find the length of a side.
All three sides are equal. Take two of them and set them equal:
Put x = 6 into the first expression to get the side: 2(6) + 5 = 17.
There are three expressions and only one unknown, so one equation was enough to find x and the third side is a check on it. The other two expressions must give the same side:
They do, so each side is 17\text{ cm}. Had the third expression given anything else, no value of x could have made the triangle equilateral.
One part of an isosceles triangle, two cases
(a) An isosceles triangle has perimeter 20\text{ cm}, and one of its sides is 4\text{ cm} long. Find the other two sides.
(b) An isosceles triangle has an angle of 40^\circ. Find the other two angles and name the triangle. Then do the same for an isosceles triangle with an angle of 100^\circ.
SolutionBoth problems hand you one part of an isosceles triangle without saying whether it belongs to the equal pair or is the odd one out, so there are two cases to try each time, and each case must be tested to see that it gives a real triangle.
(a) Case 1. The 4\text{ cm} side is one of the two equal sides. Then two sides are 4 and 4, and the third is 20 - 4 - 4 = 12. The sides would be 4, 4 and 12. But two sides of 4 laid end to end stretch only 8, which cannot reach across a side of 12. The right-hand panel of Figure 8 shows the gap. These lengths do not make a triangle, so this case is impossible.
Case 2. The 4\text{ cm} side is the base. Let each equal side be s:
The sides would be 8, 8 and 4. Here 8 + 8 = 16 is more than 4, and 8 + 4 = 12 is more than 8, so the legs do meet, as in the left-hand panel. The triangle has sides 8\text{ cm}, 8\text{ cm} and 4\text{ cm}.
(b) In an isosceles triangle the angles opposite the equal sides are equal (base angles theorem), so two of its three angles are equal.
The 40^\circ angle is one of the equal pair. Then the other equal angle is also 40^\circ, and the third is 180 - 40 - 40 = 100. The angles are 40^\circ, 40^\circ, 100^\circ, and the triangle is obtuse.
The 40^\circ angle is the odd one. Then the other two are equal and add to 180 - 40 = 140, so each is 70^\circ. The angles are 40^\circ, 70^\circ, 70^\circ, and the triangle is acute.
Both cases give a genuine triangle, so the angle of 40^\circ does not decide the family. There are two different isosceles triangles with an angle of 40^\circ.
For 100^\circ, try the same two cases. If 100^\circ were one of an equal pair, the pair would add to 200^\circ, which is more than 180^\circ. That is impossible. So 100^\circ is the odd one, the other two are equal and add to 80^\circ, and each is 40^\circ. The angles are 100^\circ, 40^\circ, 40^\circ, and the triangle is obtuse. Here only one case survives, just as in (a).
Every triangle with whole-number sides and a perimeter of 12
Find every triangle whose sides are whole numbers of centimeters and whose perimeter is 12\text{ cm}, and name each one by its sides.
SolutionList the sides from smallest to largest, a \le b \le c, so that each triangle is listed once.
Two facts limit the largest side c. The other two sides together must be longer than c, and together they are 12 - c, so
And c is the largest of three numbers adding to 12, so it is at least a third of 12: c \ge 4. That leaves c = 4 and c = 5.
The case c = 4. Then a + b = 8 with a \le b \le 4, which only 4 + 4 does. The sides are 4, 4, 4.
The case c = 5. Then a + b = 7 with a \le b \le 5. That gives 2 + 5 and 3 + 4, since 1 + 6 has a side bigger than c. The sides are 2, 5, 5 and 3, 4, 5.
Each of these passes the test that the two shorter sides together beat the longest: 4 + 4 > 4, 2 + 5 > 5 and 3 + 4 > 5.
Now name them: 4, 4, 4 is equilateral, 2, 5, 5 is isosceles, and 3, 4, 5 is scalene. A perimeter of 12 gives exactly one triangle of each kind.
Angles and sides together
A triangle can be sorted by its angles (acute, right, obtuse) and by its sides (scalene, isosceles but not equilateral, equilateral). Every equilateral triangle is also isosceles, so the middle side family is taken without the equilateral ones, and then every triangle has exactly one name from each list. That makes nine boxes. Which of them hold a triangle?
SolutionEqual sides have equal angles opposite them, and equal angles have equal sides opposite them (base angles theorem and its converse). So a triangle is scalene exactly when its three angles are all different, and isosceles but not equilateral exactly when exactly two of its angles are equal.
Equilateral. All three sides are equal, so all three angles are equal (equilateral and equiangular). They add to 180^\circ, so each is 180 \div 3 = 60. An equilateral triangle is always acute. The right and obtuse equilateral boxes are empty.
Right and isosceles. Two angles are equal, and the third is different. They cannot both be the right angle, because a triangle has at most one angle of 90^\circ or more. So the right angle is the odd one, and the other two are equal and add to 90^\circ: each is 45^\circ. A right isosceles triangle always has angles 45^\circ, 45^\circ, 90^\circ.
Obtuse and isosceles. By the same argument the obtuse angle is the odd one, and the two equal angles are acute. If each equal angle is x^\circ, the obtuse angle is 180 - 2x, which is more than 90 exactly when x is less than 45. So any x below 45 works, for example x = 30.
The other examples are angles that add to 180^\circ and are either all different (scalene) or exactly two equal (isosceles but not equilateral).
| The triangle is | An example, or why there is none |
|---|---|
| acute and scalene | 50^\circ, 60^\circ, 70^\circ |
| acute, isosceles, not equilateral | 80^\circ, 50^\circ, 50^\circ |
| acute and equilateral | 60^\circ, 60^\circ, 60^\circ |
| right and scalene | 30^\circ, 60^\circ, 90^\circ |
| right and isosceles | 45^\circ, 45^\circ, 90^\circ |
| right and equilateral | none: every angle is 60^\circ |
| obtuse and scalene | 20^\circ, 40^\circ, 120^\circ |
| obtuse and isosceles | 30^\circ, 30^\circ, 120^\circ |
| obtuse and equilateral | none: every angle is 60^\circ |
Seven of the nine boxes hold a triangle, and the two that do not are empty for a reason, not by accident.
Acute, right or obtuse from the sides
A triangle has sides 6, 7 and 8. Is it acute, right or obtuse? Then decide the same for (b) 6, 8, 10 and (c) 4, 5, 8.
SolutionNo angle is given, but three side lengths fix the triangle, and so its angles. Call the longest side c and the other two a and b, and let C be the angle opposite c. The longest side sits opposite the largest angle (sides and angles in order), so C is the largest angle and C alone decides the name.
The converse of the Pythagorean theorem says C is exactly 90^\circ when c^2 = a^2 + b^2. Now picture the sides a and b hinged at the corner between them. Closing the hinge brings their free ends nearer, so the side across them gets shorter; opening it pushes the ends apart, so that side gets longer (Hinge theorem). A side c shorter than the one that makes a right angle, with c^2 < a^2 + b^2, therefore sits across an angle less than 90^\circ, and a longer one, with c^2 > a^2 + b^2, across an angle more than 90^\circ. The Pythagorean converse lesson proves this test in full:
| Compare c^2 with a^2 + b^2 | the triangle is |
|---|---|
| c^2 < a^2 + b^2 | acute |
| c^2 = a^2 + b^2 | right |
| c^2 > a^2 + b^2 | obtuse |
(a) The sides 6, 7, 8: here a^2 + b^2 = 36 + 49 = 85 and c^2 = 64. Since 64 < 85, the side across the largest angle is shorter than a right angle would need, and the triangle is acute. Its sides are all different, so its full name is a scalene acute triangle.
(b) The sides 6, 8, 10: 36 + 64 = 100 = 10^2, so the triangle is right.
(c) The sides 4, 5, 8: first check that they make a triangle, since 4 + 5 = 9 is more than 8. Then a^2 + b^2 = 16 + 25 = 41 and c^2 = 64. Since 64 > 41, the triangle is obtuse.
The same test names the scalene triangle of Example 7: 3^2 + 4^2 = 25 = 5^2, so the triangle with sides 3, 4, 5 is a scalene right triangle.
Boundaries
The parts must make a triangle. Naming only applies once the triangle exists. Sides 5, 5 and 11 have an equal pair, yet 5 + 5 = 10 is less than 11, so no triangle has them. Sides 2, 3 and 5 fail in a different way: the two short sides together measure exactly 5, so they lie flat along the long one. A triangle needs each side strictly shorter than the other two together. In the same way, angles of 90^\circ, 60^\circ and 40^\circ add to 190^\circ, so there is no triangle to name, and an angle of exactly 180^\circ is not allowed either: the three corners would lie on one line. That is why an obtuse angle is strictly between 90^\circ and 180^\circ.
“Isosceles” has two conventions. This lesson says at least two equal sides, so equilateral triangles are isosceles. Euclid’s own wording, in Book I, Definition 20, was stricter: an isosceles triangle “has two of its sides alone equal”, which is exactly two. Modern treatments mostly prefer the wider meaning, but when an answer turns on it, check which one a book is using. Neither choice changes a single triangle; it only decides whether an equilateral triangle may also be called isosceles. The wider one is used here because then whatever is true of every isosceles triangle is automatically true of every equilateral one.
A picture is not a measurement, and equal means exactly equal. The eye cannot tell 89^\circ from 90^\circ or from 91^\circ, yet as the largest angle of a triangle those three sizes give an acute, a right and an obtuse triangle. Unless a figure carries a square mark or a number, or the text states it, an angle is not known to be right. Lengths are the same: a length read off a ruler marked in millimeters is only known to about the nearest millimeter, so two sides that both read 5.0\text{ cm} might differ by a fraction of a millimeter. A measurement can show that sides are equal to within the ruler’s precision. It cannot prove that they are equal.
Each name answers only its own question. Acute does not mean small, and a right triangle may be tall, flat, or turned on its side. Knowing that a triangle is isosceles says nothing yet about whether it is acute, right or obtuse: a triangle with legs of equal length and a right angle between them is isosceles and right, and one with sides 3, 4 and 5 is scalene and right. The two lists are linked only through the facts in Example 8 and Example 9.
Flat surfaces only. The angle sort rests on the angle sum being 180^\circ, and that is a fact about triangles drawn on a flat surface. On the surface of a sphere it fails: take the north pole and two points on the equator a quarter of the way round from each other, join them by the shortest paths along the surface, and the triangle they make has three right angles, and is equilateral too. The sorts in this lesson are for triangles in the plane.
Where this goes next
A name about sides turns into a statement about angles. In an isosceles triangle, the two angles opposite the equal sides are equal, and in an equilateral triangle all three angles are 60^\circ; both are in the base angles theorem, with their converses.
Each angle family opens a different set of tools. A right triangle has the Pythagorean theorem, its converse, and the trigonometric ratios, with the special right triangles as the ones with exact values; the right isosceles triangle of Example 8 is the first of them. Acute and obtuse triangles are the oblique triangles, handled by the law of sines and the law of cosines; the second turns the side test of Example 9 into the exact size of the largest angle.
The fact that drove the angle sort, that the angles add to 180^\circ, has its own lesson and its own proof: Triangle angle sum and Proving the triangle angle sum. It also leads to the exterior angle theorem.
The checks in Examples 6 and 7, that the shorter sides must beat the longest, are one theorem stated in general: the triangle inequality. When three lengths do satisfy it, they make a triangle, and every triangle with those lengths is the same shape and size up to sliding, turning and flipping. That is where congruent triangles begins.
An equilateral triangle can be built with a compass and no measuring at all, by drawing two circles through each other’s centers, in constructing an equilateral triangle. It is the very first proposition of Euclid’s Elements, Book I, Proposition 1. Finding a missing side from the perimeter, as in Examples 4 and 6, is part of Perimeter, with the name of the triangle telling you which sides match. To decide whether a triangle drawn on a coordinate grid is right or isosceles, see showing a plotted triangle is right or isosceles.
Exercises
Naming by sides
Classify each triangle from its side lengths. Convert units first where they differ.
- 6\text{ cm}, 9\text{ cm}, 6\text{ cm}
- 0.5\text{ m}, 50\text{ cm}, 600\text{ mm}
- \displaystyle \frac{3}{4}\text{ cm}, \displaystyle \frac{6}{8}\text{ cm}, \displaystyle \frac{9}{12}\text{ cm}
- Use Figure 1. For each of the three triangles, say how many of its three pairs of sides are equal.
Naming by angles
Name each triangle from its three angles.
- 42^\circ, 71^\circ, 67^\circ
- 91^\circ, 45^\circ, 44^\circ
- 88^\circ, 47^\circ, 45^\circ
Finding the third angle
Use Figure 9 for Exercise 11.
- A triangle has angles of 64^\circ and 38^\circ. Find the third angle and name the triangle.
- A triangle has angles of 29^\circ and 61^\circ. Find the third angle and name the triangle.
- A triangle has angles of 126^\circ and 22^\circ. Find the third angle and name the triangle.
- In each triangle of Figure 9, find x and name the triangle.
From a perimeter
- The perimeter of an equilateral triangle is 135\text{ cm}. Find the length of each side.
- An isosceles triangle has a base of 18\text{ m} and a perimeter of 62\text{ m}. Find the length of each of the two equal sides.
- An isosceles triangle has two equal sides of 12\text{ cm} each and a perimeter of 37\text{ cm}. Find the third side. Is it shorter or longer than the equal sides?
- An isosceles triangle has a perimeter of 25\text{ cm} and one side of 7\text{ cm}. Find every possible set of three side lengths.
Using algebra
- A triangle has sides (x + 7)\text{ cm}, (2x - 5)\text{ cm} and 13\text{ cm}, and its first two sides are equal. Find x and all three sides.
- An equilateral triangle has sides (5x - 4)\text{ cm}, (3x + 10)\text{ cm} and (4x + 3)\text{ cm}. Find x and the length of a side.
- A triangle has sides (x + 2)\text{ cm}, (2x - 1)\text{ cm} and (3x - 7)\text{ cm}, and its perimeter is 42\text{ cm}. Find the three sides, and name the triangle both by its sides and by its angles.
- The angles of a triangle are in the ratio 2 : 3 : 5. Find them and name the triangle.
- The angles of a triangle are {(3x + 5)^\circ}, {(4x - 15)^\circ} and {(2x + 10)^\circ}. Find x, the three angles, and name the triangle by its angles and by its sides.
- The smallest angle of a triangle is 20^\circ, and the largest angle is three times the middle angle. Find all three angles and name the triangle.
Isosceles triangles
- An isosceles triangle has an angle of 50^\circ. Find every possible set of three angles, and name each triangle.
- An isosceles triangle has an angle of 110^\circ. Find the other two angles and name the triangle.
Whole-number sides
- List every triangle with whole-number sides and a perimeter of 11, and name each one by its sides.
Acute, right or obtuse from the sides
- A triangle has sides 9, 12 and 15. Is it acute, right or obtuse?
- A triangle has sides 7, 8 and 9. Is it acute, right or obtuse?
- A triangle has sides 5, 6 and 10. Is it acute, right or obtuse?
Reasoning
In these exercises “isosceles” means at least two equal sides, as in this lesson.
- True or false? Give a reason or an example for each. (a) Every equilateral triangle is isosceles. (b) Every isosceles triangle is equilateral. (c) Every scalene triangle is isosceles. (d) Some triangle has exactly two pairs of equal sides.
- True or false? Give a reason or an example for each. (a) Every equilateral triangle is acute. (b) Some right triangle is isosceles. (c) Some obtuse triangle is equilateral. (d) Every isosceles triangle is acute. (e) Some triangle has both a right angle and an obtuse angle.
- A triangle has sides 7, 7 and x, where x is a whole number. (a) List every value of x that makes a triangle, and say which side names each triangle can have. (b) For which of these values is the triangle acute, right or obtuse?
- The two equal angles of an isosceles triangle are each x^\circ. For which values of x is the triangle acute, for which right, and for which obtuse?
- You measure a triangle with a ruler marked in millimeters and read 5.0\text{ cm}, 5.0\text{ cm} and 5.0\text{ cm}. Can you be certain that it is equilateral? Explain.
Answers
- Isosceles. The first and third sides are both 6, and 9 is different, so one pair is equal.
- Isosceles. In centimeters the sides are 50, 50 and 60, since 0.5\text{ m} = 50\text{ cm} and 600\text{ mm} = 60\text{ cm}. One pair is equal.
- Equilateral. \displaystyle \frac{6}{8} = \frac{3}{4} and \displaystyle \frac{9}{12} = \frac{3}{4}, so all three sides are \displaystyle \frac{3}{4}\text{ cm}.
- Scalene: 0 pairs. Isosceles: 1 pair, the two marked sides. Equilateral: 3 pairs, since each of its three sides matches the other two.
- Acute. The angles add to 42 + 71 + 67 = 180, and the largest, 71^\circ, is less than 90^\circ.
- Obtuse. 91 + 45 + 44 = 180, and the largest angle, 91^\circ, is more than 90^\circ.
- Acute. 88 + 47 + 45 = 180, and the largest angle, 88^\circ, is less than 90^\circ.
- The third angle is 78^\circ, and the triangle is acute. 64 + 38 = 102 and 180 - 102 = 78. The angles are 64^\circ, 38^\circ, 78^\circ, and the largest is less than 90^\circ.
- The third angle is 90^\circ, and the triangle is right. 29 + 61 = 90 and 180 - 90 = 90.
- The third angle is 32^\circ, and the triangle is obtuse. 126 + 22 = 148 and 180 - 148 = 32. The largest angle is the given 126^\circ, which is more than 90^\circ.
- P: x = 70^\circ, acute. Q: x = 90^\circ, right. R: x = 98^\circ, obtuse. S: x = 85^\circ, acute. In each, x is 180 minus the two marked angles: 48 + 62 = 110, 34 + 56 = 90, 38 + 44 = 82, 60 + 35 = 95. The largest angle of S is 85^\circ, less than 90^\circ, even though it looks nearly square.
- 45\text{ cm}. From 3s = 135, s = 45. Check: 45 + 45 + 45 = 135.
- 22\text{ m}. From 2s + 18 = 62, 2s = 44 and s = 22. Check: 22 + 22 + 18 = 62.
- 13\text{ cm}, which is longer than the equal sides. The third side is 37 - 12 - 12 = 13. Check: 12 + 12 = 24 > 13, so the triangle exists. The base of an isosceles triangle can be the longest side.
- Two possibilities: 7, 7, 11 and 9, 9, 7. If the 7\text{ cm} side is one of the equal pair, the third side is 25 - 7 - 7 = 11, and 7 + 7 = 14 > 11 so it is a triangle. If the 7\text{ cm} side is the base, then 2s + 7 = 25, so s = 9, and 9 + 7 > 9 and 9 + 9 > 7 so it is a triangle too. Both cases work.
- x = 12, and the sides are 19\text{ cm}, 19\text{ cm} and 13\text{ cm}. Setting the first two equal, x + 7 = 2x - 5 gives x = 12, so x + 7 = 19. Check: 19 + 13 > 19, so it is a triangle. The triangle is isosceles.
- x = 7, and each side is 31\text{ cm}. Setting two sides equal, 5x - 4 = 3x + 10 gives 2x = 14 and x = 7, so 5(7) - 4 = 31. The third side must agree, and it does: 4(7) + 3 = 31.
- x = 8, the sides are 10\text{ cm}, 15\text{ cm} and 17\text{ cm}, and the triangle is scalene and acute. The perimeter gives (x + 2) + (2x - 1) + (3x - 7) = 42, so 6x - 6 = 42 and x = 8. The sides are 8 + 2, 2(8) - 1 and 3(8) - 7. All three differ, and 10 + 15 = 25 > 17, so it is a triangle. By the square test, 10^2 + 15^2 = 325 and 17^2 = 289, and 289 < 325, so it is acute.
- The angles are 36^\circ, 54^\circ, 90^\circ, and the triangle is right. From 2x + 3x + 5x = 180, 10x = 180 and x = 18, so the angles are 2(18), 3(18) and 5(18).
- x = 20, the angles are 65^\circ, 65^\circ, 50^\circ, and the triangle is acute and isosceles. From (3x + 5) + (4x - 15) + (2x + 10) = 180, 9x = 180. Then 3(20) + 5 = 65, 4(20) - 15 = 65 and 2(20) + 10 = 50. Two equal angles have equal sides opposite them, and the third angle differs, so exactly two sides are equal.
- The angles are 20^\circ, 40^\circ, 120^\circ, and the triangle is obtuse. Call the middle angle m, so the largest is 3m and 20 + m + 3m = 180. Then 4m = 160, m = 40, and 3m = 120. The order 20 < 40 < 120 agrees with the description.
- 50^\circ, 50^\circ, 80^\circ and 50^\circ, 65^\circ, 65^\circ, and both triangles are acute. If 50^\circ is one of the equal pair, the third angle is 180 - 100 = 80. If it is the odd one, the other two share 130^\circ and are 65^\circ each.
- The other two angles are 35^\circ and 35^\circ, and the triangle is obtuse. Two angles of 110^\circ would add to 220^\circ, so 110^\circ is the odd angle and the other two are equal and add to 70^\circ.
- 1, 5, 5 and 3, 3, 5 and 3, 4, 4 are isosceles, and 2, 4, 5 is scalene. The largest side c satisfies 3c \ge 11 and c < 5.5, so c = 4 or 5. With c = 4 the other two add to 7 and neither is above 4, which only 3 + 4 does. With c = 5 they add to 6, giving 1 + 5, 2 + 4 and 3 + 3. All four pass the test: 1 + 5 > 5, 2 + 4 > 5, 3 + 3 > 5 and 3 + 4 > 4. There is no equilateral one, since 11 is not a multiple of 3.
- Right. 9^2 + 12^2 = 81 + 144 = 225 = 15^2.
- Acute. 7^2 + 8^2 = 49 + 64 = 113, and 113 > 9^2 = 81, so the square on the longest side is smaller than the sum.
- Obtuse. 5 + 6 = 11 > 10, so it is a triangle. 5^2 + 6^2 = 25 + 36 = 61, and 10^2 = 100 > 61.
- (a) True. An equilateral triangle has three equal sides, so it has at least two. (b) False. Sides of 5, 5 and 7 make a triangle, since 5 + 5 = 10 > 7, and it is isosceles but not equilateral. (c) False. A scalene triangle has no two sides equal, and isosceles needs at least two. (d) False. Two pairs among three sides must share a side, so they read a = b and b = c. Then a = c as well, a third equal pair. The number of equal pairs is 0, 1 or 3.
- (a) True. All three angles are equal and add to 180^\circ, so each is 60^\circ, less than 90^\circ. (b) True. The triangle with angles 45^\circ, 45^\circ, 90^\circ is right, and two of its angles are equal, so two of its sides are equal and it is isosceles. (c) False. Every equilateral triangle has three angles of 60^\circ, so none is obtuse. (d) False. The triangle with angles 100^\circ, 40^\circ, 40^\circ is isosceles and obtuse. (e) False. The angles of a triangle add to 180^\circ. A right angle and an obtuse angle would already add to more than 90^\circ + 90^\circ = 180^\circ, leaving nothing for the third angle. The same angle-sum argument rules out two obtuse angles, and two right angles, which would leave 0^\circ.
- (a) x = 1, 2, 3, \dots, 13. The third side must be shorter than 7 + 7 = 14, and each 7 must be shorter than 7 + x, which holds whenever x \ge 1. Every one of these triangles has the two equal sides of 7, so each is isosceles. The case x = 7 is also equilateral. None is scalene, since two sides are always equal. (b) Acute for x = 1 to 9, obtuse for x = 10 to 13, and never right. When x \le 7 a longest side is a 7, and 7^2 = 49 is less than 49 + x^2, so the triangle is acute. When x > 7 the longest side is x, compared with 7^2 + 7^2 = 98: 8^2 = 64 and 9^2 = 81 are less than 98, while 10^2 = 100 is more. A right triangle would need x^2 = 98, and no whole number squares to 98.
- Acute for 45 < x < 90, right for x = 45, obtuse for 0 < x < 45. The third angle is 180 - 2x, which must be positive, so x < 90. The two equal angles are x^\circ, less than 90^\circ. The third is less than 90^\circ when 180 - 2x < 90, that is x > 45, equal to 90^\circ when x = 45, and more than 90^\circ when x < 45. For x < 45 the third angle is the largest, so it decides the name.
- No. A ruler marked in millimeters gives a length only to about the nearest millimeter, so each true side could differ from its reading by a fraction of a millimeter. You can say the sides are equal to within the ruler’s precision. To be certain they are exactly equal you need their lengths given exactly, marked equal, or proved equal.