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Which three measurements fix a triangle, which allow more than one, and which allow none.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you decide whether three given measurements make a triangle, and if so whether they make only one. You will test sides with the triangle inequality, find missing angles with the 180° angle sum, find the range of lengths a missing side can have, and explain why three sides fix a triangle while three angles fix only its shape.
You can measure an angle in degrees with a protractor, and you know that a straight angle is 180°. You can also solve a two-step equation such as $4x + 40 = 180$. This lesson puts those skills together. Before you draw anything, you will be able to say whether three measurements make a triangle, and whether they make only one.
| Term | What it means |
|---|---|
| Triangle inequality | Any two sides of a triangle add to more than the third side. |
| Angle sum | The three angles of every triangle add to exactly 180°. |
| Unique triangle | Only one triangle fits the measurements. A copy that is turned or flipped over counts as the same triangle. |
| Included angle | The angle between two given sides, at the corner where they meet. |
| Included side | The side between two given angles, joining their two corners. |
| Congruent | Same shape and same size, so one fits exactly on top of the other. |
A triangle has six measurements: three sides and three angles. When you are given three of them, there are only three possible answers to the question how many triangles fit?
Two rules decide the none answers. The triangle inequality says any two sides add to more than the third. The angle sum says the three angles add to 180°.
Another way: picture
Lay a 10 inch straw on the table and hinge a 3 inch straw to its left end and a 4 inch straw to its right end. Swing the short straws up toward each other. Even lying flat along the long straw, they cover only 7 of its 10 inches, so their tips never touch. Swap the long straw for a 6 inch one and the tips meet at one point above it, and only one triangle can be made.
Another way: steps
Take any three lengths and try to build a triangle. The longest side is the hard one to close around. The two shorter sides start at its two ends and must meet above it. If they add to more than the longest side, they can bend up and meet. If they add to exactly the longest side, they meet only by lying flat on top of it, which is a line segment, not a triangle. If they add to less, a gap is left.
| Sides | Two shorter added | Longest | Triangle? |
|---|---|---|---|
| 4, 5, 6 | 9 | 6 | yes, $9 > 6$ |
| 3, 5, 8 | 8 | 8 | no, $8 = 8$ is flat |
| 2, 3, 9 | 5 | 9 | no, $5 < 9$ |
| 7, 7, 13 | 14 | 13 | yes, $14 > 13$ |
You only need to test the two shortest sides against the longest. Any other pair includes the longest side, and the longest side plus anything is already more than a shorter side.
The same rule gives a range for a missing side. If two sides are 6 and 10, the third must be less than $6 + 10 = 16$ and more than $10 - 6 = 4$. When three sides pass the test, the triangle is unique: once all three lengths are set, no corner can move.
Tear the three corners off any paper triangle and fit them together point to point. They always make a straight line, which is 180°. That is the angle sum, and it gives two quick tests. Three angles that do not add to exactly 180° cannot belong to one triangle. Two angles that already add to 180° or more leave no room for a third.
When three angles do add to 180°, a triangle exists, but it is never unique. Draw a triangle with angles of 30°, 60° and 90° with a 2 inch side. Now draw one with the same angles and a 5 inch side. Both have the same angles, and one is simply an enlargement of the other. Angles decide the shape. Only a length can decide the size. That is why three angles always give infinitely many triangles, or none.
Two known angles are enough to find the third. With angles of 47° and 68°, the third is $180 - (47 + 68) = 180 - 115 = 65$ degrees. When the angles are given as expressions such as $x$, $2x$ and $x + 40$, add them, set the total to 180 and solve the equation.
A mix of sides and angles can also fix a triangle, and where the angle sits matters.
So before you count, ask where each given angle sits. An angle between two given sides, or a side between two given angles, locks the triangle in place.
Every question in this lesson is answered with the same five moves.
Each move has a reason. Step 2 works because the two short sides must stretch around the long one. Step 3 works because the three corners of a triangle fit together into a straight angle. Step 4 works because angles can stay the same while every side grows.
How to check. After finding a missing angle, add all three angles; the total must be 180. After finding a range for a side, test a length at each edge. With sides 6 and 10, a third side of 4 gives $6 + 4 = 10$, which lies flat, and 16 gives $6 + 10 = 16$, flat again, so both edges are left out. The last check is a sketch. Draw the triangle roughly to scale. The longest side should face the largest angle. If the picture refuses to close, recheck step 2.
Look up into the attic of a house or under a bridge and you will see triangles everywhere. Builders use them because of the fact in this lesson: three fixed sides make exactly one triangle, so a triangle of beams cannot change shape. A square frame of four beams can lean over into a slanted shape with the same four sides, and it collapses unless a diagonal brace cuts it into two triangles.
Suppose a shed roof has two rafters, each 8 feet long, resting on walls 14 feet apart. First check that the truss can exist: $8 + 8 = 16$, which is more than 14, so the rafters meet above the walls. If the walls were 16 feet apart, the rafters would lie flat and the roof would have no height. At 17 feet they could not meet at all.
For these sizes each eave angle comes out at about 29°. The two eave angles are equal, because the rafters are equal, so the angle at the peak is $180 - (29 + 29) = 180 - 58 = 122$ degrees, to the nearest degree. A carpenter cuts the rafter ends to those angles, and because three sides fix the triangle, every truss built to the same sizes is the same shape.
A GPS receiver finds its position from its distances to satellites. A simpler version works on the ground. A hiker knows she is 3 miles from a fire tower and 5 miles from a ranger station, and the map shows the tower and the station are 7 miles apart. Those are three sides of a triangle: $3 + 5 = 8$, which is more than 7, so the triangle exists and is unique. She stands at its third corner. The only choice is which side of the line from tower to station she is on, which is the same triangle flipped over.
The triangle inequality also catches bad readings. If her distances had been 2 miles and 4 miles, then $2 + 4 = 6$ is less than 7, and no point on the map is that close to both. One of the readings must be wrong.
The most common mistake is to accept sides whose two shortest add to exactly the longest. Sides of 3, 5 and 8 look fine, but $3 + 5 = 8$ means the short sides lie flat along the long one. There is no inside, so there is no triangle. The sum must be greater.
The second mistake is thinking three angles fix a triangle. Angles of 60°, 60° and 60° fit an equilateral triangle 1 inch across and one 100 feet across. Angles fix the shape; a length is needed to fix the size.
The third is counting a flipped triangle as a new one. A triangle turned over is still congruent to the first, so it is the same triangle. Last, some learners test only one pair of sides that is not the two shortest. Always use the two shortest against the longest.
Find the longest side, the one the other two must reach around.
$5,\ 7,\ \mathbf{13}$
If the two shorter sides can get around the longest, every other pair can too, so one test is enough.
Add the two shorter sides.
$5 + 7 = 12$
Laid end to end, the two short sides stretch 12 cm at most.
Compare that sum with the longest side.
$12 < 13$
The triangle inequality needs the sum to be greater than the longest side, and here it is less.
Find how far short they fall.
$13 - 12 = 1$
Even lying flat along the 13 cm side, the short sides leave a 1 cm gap, so their ends can never meet.
State the answer, then change one side so it works.
$5 + 7 = 12 > 11 \;\Rightarrow\; 5,\ 7,\ 11 \text{ make a triangle}$
No triangle has sides 5, 7 and 13. A longest side under 12 cm, such as 11 cm, gives exactly one triangle, because three workable sides fix it.
Write the angle sum as an equation.
$x + 2x + (x + 40) = 180$
The three angles of a triangle add to 180°, whatever names they are given.
Combine the $x$ terms.
$4x + 40 = 180$
$x + 2x + x = 4x$, and the 40 stays as it is.
Subtract 40 from both sides.
$4x + 40 - 40 = 180 - 40 \;\Rightarrow\; 4x = 140$
Undo the addition first, keeping the equation balanced.
Divide both sides by 4.
$\dfrac{4x}{4} = \dfrac{140}{4} \;\Rightarrow\; x = 35$
Dividing undoes the multiplication by 4.
Substitute to find each angle.
$x = 35^\circ, \qquad 2x = 70^\circ, \qquad x + 40 = 75^\circ$
The question asked about the triangle, so turn $x$ back into its three angles.
Check the sum, then count the triangles.
$35 + 70 + 75 = 180$
The angles check, so a triangle exists. No side is given, so every enlargement fits too: there are infinitely many triangles with these angles.
Call the third side $c$ and write the three conditions.
$6 + 10 > c, \qquad 6 + c > 10, \qquad 10 + c > 6$
Every pair of sides must add to more than the side that is left.
Add in the first condition.
$16 > c, \text{ so } c < 16$
The third side must be shorter than the other two laid end to end.
Subtract 6 from both sides of the second condition.
$6 + c - 6 > 10 - 6 \;\Rightarrow\; c > 4$
The third side must be long enough for the 6 in side to reach around the 10 in side.
Look at the third condition.
$10 + c > 6 \text{ for every length } c$
The 10 in side is already longer than 6 in, so this one is always true and adds nothing.
Put the two bounds together.
$4 < c < 16$
The third side lies between the difference and the sum of the other two.
List and count the whole-number lengths.
$c = 5, 6, \dots, 15: \qquad 15 - 5 + 1 = 11$
Last minus first, plus one, counts the list. Each length gives one triangle.
Check a length at the edge.
$c = 4: \quad 6 + 4 = 10, \text{ not more than } 10$
At the edge the two sides lie flat along the 10 in side, so the edges are rightly left out.
Add the two angles you know.
$72 + 55 = 127$
The known angles use up 127° of the 180° total.
Subtract the total from 180.
Check that the three angles add to 180.
Could a triangle have sides of $3$ cm, $5$ cm and $6$ cm?
Two sides of a triangle are $3$ cm and $14$ cm. Complete the worked solution to find the lengths the third side $c$ can have.
Write the three conditions of the triangle inequality.
$3 + 14 > c, \qquad 3 + c > 14, \qquad 14 + c > 3$
Every pair of sides must add to more than the side that is left.
Add the two known sides: the third side must be shorter than this.
$c <$ s
The first condition says $c$ is less than the sum of the other two.
Subtract $3$ from both sides of the second condition.
$c >$ d
The third side must be longer than the difference of the other two, or the short sides cannot reach.
Look at the third condition.
$14 + c > 3 \text{ for every length } c$
The $14$ cm side is already longer than $3$ cm, so this condition tells us nothing new.
Take one away from the upper bound to get the greatest whole-number length.
$c_{\max} =$ g
The third side must be strictly less than the sum, so the largest whole number allowed is one below it.
A triangle is to be built with sides of 5 cm, 7 cm and 9 cm. How many different triangles fit?
Two angles of a triangle measure $70^\circ$ and $56^\circ$. What do the two known angles add to, and what is the third angle, in degrees?
Known angles together: t°. Third angle: c°.
The angles of a triangle measure $x^\circ$, $4x^\circ$ and $(x + 6)^\circ$. Find $x$.
Answer:
A roof truss is a triangle with two equal rafters, so the two angles at the eaves are equal. The angle at the peak is $138^\circ$. How many degrees do the two eave angles make together, and what is each one?
Both eaves together: s°. Each eave angle: e°.
A craft kit has one stick $7$ inches long and one $15$ inches long. You may cut a third stick to any whole number of inches. Between which two lengths must the third stick lie, and how many different triangles can you make?
lo in $< c <$ hi in, so n triangles.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A roof truss is a triangle with two equal rafters, so the two angles at the eaves are equal. The angle at the peak is $104^\circ$. How many degrees do the two eave angles make together, and what is each one?
Both eaves together: s°. Each eave angle: e°.
You can say whether three measurements make a triangle, and how many. Without looking: can sides 3, 4 and 9 make a triangle, and why not? Why do angles of 40°, 60° and 80° give infinitely many triangles?
15. Your turn: two angles of a triangle are 72° and 55°, step 2
$180 - 127 = 53$
The third angle is what is left of 180°.
15. Your turn: two angles of a triangle are 72° and 55°, step 3
$72 + 55 + 53 = 180$
The check confirms 53°. With no side given, the triangle can be any size.