Do I have to check all three sums in the Triangle Inequality?

You can check just one, as long as you pick the right one: compare the sum of the two shortest sides to the longest side. If that sum is greater, the other two inequalities are automatically satisfied, because the longest side plus any positive length already exceeds either shorter side. Showing all three is never wrong, though, and some teachers want to see them.

What happens if two sides add up to exactly the third side?

There is no triangle. The two shorter segments meet only by lying flat along the longest one, which gives a straight-line figure with zero height and zero area. This is called a degenerate triangle and it does not count. That is why the inequality uses "greater than" and not "greater than or equal to."

Why can't three angles ever determine just one triangle?

Because angles describe shape, not size. Once the three angles add to 180^, you can scale the whole figure up or down by any factor and the angles stay identical. That produces infinitely many triangles, all similar to one another. Adding a single side length fixes the scale and reduces the answer to exactly one triangle.

Can a triangle have two obtuse angles, or two right angles?

No. Two right angles already total 180^, leaving 0^ for the third angle, which is impossible. Two obtuse angles total more than 180^ on their own. Every triangle has at most one angle that is 90^ or greater, so the other two must both be acute.

M7MATH-7.2

Conditions That Determine a Triangle

Learn to use the Triangle Inequality and the 180-degree angle sum to decide if given sides or angles make exactly one triangle, many triangles, or none at all.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Conditions That Determine a Triangle, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Conditions That Determine a Triangle free

What this lesson covers

Hand three classmates a set of straws and ask them to build a triangle. Sometimes everyone ends up with the exact same shape. Sometimes everyone gets something different. And sometimes the straws simply refuse to close up into a triangle at all. That is not luck — it follows from two rules you can check on paper before you ever pick up a ruler.

In this lesson you will use the Triangle Inequality (any two sides must together be longer than the third) and the angle sum of

180^\circ

to sort a given set of measurements into one of three buckets: exactly one triangle, more than one triangle, or no triangle. Along the way you will see why three angles never pin down a single triangle, why three sides always do, and where students most often go wrong when the numbers are close.

The Triangle Inequality: When Three Lengths Close Up

Three segment lengths form a triangle only if each pair of sides adds up to more than the third side. For sides

a

b

, and

c

you need all three of these:

a+b>c \qquad a+c>b \qquad b+c>a

Think of it physically. Lay the longest side flat on your desk. The other two sides are hinged to its endpoints and swing like a drawbridge. If the two shorter pieces together are longer than the base, they can meet somewhere above it and a triangle closes. If they are together shorter than the base, they can never reach each other — there is a gap, and no triangle exists. If they are exactly equal to the base, they meet only by lying flat on top of it, which gives a flat segment, not a triangle.

A shortcut: you only really need to check the two shortest sides against the longest side. If the sum of the two smaller lengths beats the largest length, the other two inequalities are automatically true (a bigger number plus a positive number is certainly bigger than a smaller one).

Test 4, 5, 12: is

4+5>12

? No,

9<12

. No triangle. Test 6, 6, 12: is

6+6>12

12>12

is false. No triangle — this is the flat, degenerate case. Test 7, 9, 13: is

7+9>13

16>13

, yes. A triangle exists.

Where students go wrong: checking only one of the three sums, or accepting the equal case. "Greater than" is strict.

Three Sides Give Exactly One Triangle

When three lengths pass the Triangle Inequality, they determine exactly one triangle. You can slide it, spin it, or flip it over, but every triangle built from those three lengths is congruent to every other one — same angles, same area, same shape. Triangles are rigid in a way that other polygons are not.

Compare that to a quadrilateral. Four sticks of lengths 5, 5, 5, 5 can be a square or squashed into a thin rhombus, and the sides never change length. A triangle has no such wiggle. That rigidity is why bridges, roof trusses, and bicycle frames are built out of triangles instead of rectangles.

Here is the geometric reason. Draw the longest side. From one endpoint, swing an arc with radius equal to the second length; from the other endpoint, swing an arc with radius equal to the third length. Those two arcs cross in exactly one point above the base (and one mirror-image point below it, which gives the same triangle flipped). One crossing point means one triangle.

Given	Number of triangles
Three sides passing the inequality	Exactly one
Three sides failing the inequality	None
Two shorter sides summing to exactly the longest	None (flat)

So for a side-side-side condition, the whole decision reduces to a single arithmetic check. Nothing needs to be drawn. Students sometimes answer "more than one" for three sides because the triangle can be rotated or reflected — but a rotated copy is the same triangle, just repositioned.

Three Angles Give Many Triangles — or None

The angles of every triangle add to

180^\circ

. So a set of three angle measures can only belong to a triangle if

m\angle A + m\angle B + m\angle C = 180^\circ

and each angle is greater than

0^\circ

. If the sum is anything other than

180^\circ

, no triangle exists — for example

50^\circ

60^\circ

, and

80^\circ

sum to

190^\circ

, so no triangle has those angles.

But when the sum does equal

180^\circ

, you get infinitely many triangles, not one. Angles fix the shape but say nothing about the size. A triangle with angles

60^\circ

60^\circ

60^\circ

could have sides of 2 cm each or 200 cm each. All those triangles are similar to each other (same shape, scaled differently), which connects directly to the scale-drawing ideas in this unit. Since "more than one triangle" is the answer whenever the sum works out, angle-only conditions never determine a unique triangle.

Angle set	Sum	Result
$30^\circ, 60^\circ, 90^\circ$	$180^\circ$	Infinitely many (all similar)
$45^\circ, 45^\circ, 45^\circ$	$135^\circ$	No triangle
$100^\circ, 40^\circ, 40^\circ$	$180^\circ$	Infinitely many
$90^\circ, 90^\circ, 10^\circ$	$190^\circ$	No triangle

Notice also that a triangle can have at most one angle that is

90^\circ

or larger. Two right angles alone already use up the full

180^\circ

, leaving nothing for the third.

Mixed Conditions and Finding the Range for a Missing Side

Many problems mix sides and angles, or leave one measurement unknown. Two mixed cases show up constantly.

Two sides and the angle between them (the included angle) determine exactly one triangle. Draw the angle, mark off the two given lengths along its arms, and connect the endpoints — there is only one way to finish. Two angles and any one side also determine exactly one triangle, because the third angle is forced by the

180^\circ

rule and the single length fixes the scale.

The trickier task is finding all possible values for a missing third side. If two sides are 8 and 15, let the third side be

x

. All three inequalities must hold:

8+15>x \Rightarrow x<23 \qquad 8+x>15 \Rightarrow x>7 \qquad 15+x>8 \text{ (always true)}

7<x<23

. The pattern is worth memorizing: the third side is always between the difference and the sum of the other two. Endpoints are excluded —

x=7

and

x=23

both give flat, degenerate figures.

A very common wrong answer here is

7\le x\le 23

, or listing only the upper bound and forgetting that a too-short side also fails. Check both ends every time by asking, "Could the two shorter sides still reach across the longest one?"

Key terms

Triangle Inequality.: The rule that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side.
Angle sum of a triangle.: The fact that the three interior angles of any triangle always add to exactly $180^\circ$ .
Determines a unique triangle.: A set of measurements is satisfied by exactly one triangle, up to sliding, turning, or flipping it.
Degenerate triangle.: The flat figure that results when two sides add to exactly the third side; it has zero area and is not counted as a triangle.
Included angle.: The angle formed between two named sides of a triangle; two sides plus their included angle determine one triangle.
Similar triangles.: Triangles with the same angle measures but possibly different sizes; equal angles alone produce infinitely many similar triangles.
Congruent triangles.: Triangles with exactly the same size and shape, so all corresponding sides and angles are equal.

Worked example

For each set, decide whether it determines exactly one triangle, more than one triangle, or no triangle. (a) Sides 5 cm, 9 cm, 17 cm. (b) Angles

35^\circ

65^\circ

80^\circ

. (c) Sides 10 cm, 10 cm, 4 cm. (d) Angles

70^\circ

70^\circ

50^\circ

with the side between the two

70^\circ

angles equal to 6 cm.

(a) These are side lengths, so apply the Triangle Inequality. Compare the two shortest to the longest:

5+9=14

, and

14>17

is false. The two short sides cannot reach across the 17 cm base, so there is a gap. No triangle.

(b) These are angles, so check the sum:

35+65+80=180

. The sum is exactly

180^\circ

, so triangles with these angles exist. But nothing fixes the size — one could have a shortest side of 1 cm and another a shortest side of 1 m. More than one triangle (infinitely many, all similar).

(c) Side lengths again. Two shortest against longest:

4+10=14>10

. True, so the triangle closes. Three sides always lock a triangle into one shape, so this is an isosceles triangle with a 4 cm base. Exactly one triangle.

(d) Here the angles sum to

70+70+50=180

, so the shape is possible, and now a specific length is given as well. Two angles plus a side fix both the shape and the scale: draw the 6 cm segment, build a

70^\circ

angle at each end, and the two new rays meet at exactly one point. Exactly one triangle.

The pattern: sides alone need the inequality check; angles alone can never give uniqueness; angles plus at least one length can.

Practice questions

Which set of side lengths does NOT determine a triangle?

3, 4, 5
6, 8, 15
7, 7, 7
9, 12, 20

Answer: 6, 8, 15

Check the two shortest sides against the longest in each set. For 3, 4, 5:

3+4=7>5

, fine. For 6, 8, 15:

6+8=14

, and

14>15

is false, so the two shorter sides cannot reach across the 15-unit side — no triangle. For 7, 7, 7:

7+7=14>7

, fine. For 9, 12, 20:

9+12=21>20

, just barely, but it works. Only 6, 8, 15 fails.

Two sides of a triangle measure 11 in. and 4 in. Write and solve inequalities to describe all possible lengths

x

of the third side, then explain why the endpoints are not included.

Answer: $7<x<15$

All three Triangle Inequality statements must hold. From

11+4>x

you get

x<15

. From

4+x>11

you get

x>7

. From

11+x>4

you get

x>-7

, which is automatically true for any positive length. Combining the useful ones gives

7<x<15

— the third side lies strictly between the difference and the sum of the other two. The endpoints are excluded because

x=15

would make

11+4=15

exactly, and

x=7

would make

4+7=11

exactly. In both of those cases the two shorter segments lie flat along the longest one instead of meeting above it, producing a degenerate figure with no height and no area.

Mia says the angles

40^\circ

60^\circ

, and

80^\circ

determine exactly one triangle. Is she right? Explain.

Answer: No — they determine infinitely many similar triangles, not one.

Mia is right that the angles are possible, since

40+60+80=180

. But angle measures control only the shape, never the size. You could draw one such triangle with its shortest side 2 cm long and another with its shortest side 20 cm long; both have the same three angles, and they are not congruent. To pin down exactly one triangle you would also need at least one side length. The correct classification is more than one triangle.

FAQ

Do I have to check all three sums in the Triangle Inequality?: You can check just one, as long as you pick the right one: compare the sum of the two shortest sides to the longest side. If that sum is greater, the other two inequalities are automatically satisfied, because the longest side plus any positive length already exceeds either shorter side. Showing all three is never wrong, though, and some teachers want to see them.
What happens if two sides add up to exactly the third side?: There is no triangle. The two shorter segments meet only by lying flat along the longest one, which gives a straight-line figure with zero height and zero area. This is called a degenerate triangle and it does not count. That is why the inequality uses "greater than" and not "greater than or equal to."
Why can't three angles ever determine just one triangle?: Because angles describe shape, not size. Once the three angles add to $180^\circ$ , you can scale the whole figure up or down by any factor and the angles stay identical. That produces infinitely many triangles, all similar to one another. Adding a single side length fixes the scale and reduces the answer to exactly one triangle.
Can a triangle have two obtuse angles, or two right angles?: No. Two right angles already total $180^\circ$ , leaving $0^\circ$ for the third angle, which is impossible. Two obtuse angles total more than $180^\circ$ on their own. Every triangle has at most one angle that is $90^\circ$ or greater, so the other two must both be acute.

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