What Is an Obtuse Angle?
An obtuse angle is an angle whose measure is greater than $90°$ but less than $180°$. In symbols, an angle $\angle ABC$ is obtuse when $90° < \angle ABC < 180°$. It is wider than a right angle (the square $90°$ corner) and narrower than a straight angle (the flat $180°$ line).
The word comes from the Latin obtusus, meaning "blunt" or "dull" — the opposite of the sharp, narrow acute angle. An obtuse angle looks open and blunt: think of the angle at the wide base of an isosceles triangle that has been squashed flat, not the needle-sharp point at its tip.
Because the boundaries are strict, $90°$ itself is not obtuse (that is a right angle) and $180°$ is not obtuse (that is a straight angle). An obtuse angle lives in the open interval — anything from $91°$ up to $179°$ in whole degrees, and every value between.
Properties of an Obtuse Angle
The defining range gives the obtuse angle a small set of properties worth holding on to.
Range. It always satisfies $90° < \theta < 180°$. Outside that range, the angle is not obtuse.
One per triangle, at most. A triangle's three angles sum to $180°$, so it can hold at most one obtuse angle. Two obtuse angles would already exceed $180°$ on their own.
Its supplement is acute. Two angles that add to $180°$ are supplementary. The supplement of an obtuse angle is always acute, because if $\theta > 90°$ then $180° - \theta < 90°$.
Its reflex partner is between $180°$ and $360°$. Every angle has a reflex partner that completes the full turn. For an obtuse angle of $\theta$, that partner is $360° - \theta$, which lands between $180°$ and $270°$.
It cannot be a right angle's complement. Complementary angles add to $90°$, so no obtuse angle has a complement — it is already larger than $90°$ by itself.
A quick reader question worth answering here: is $90°$ an obtuse angle? No. $90°$ is exactly a right angle. The obtuse band starts after $90°$, the moment the angle swings past the square corner.
How Do You Identify an Obtuse Angle?
The fastest check is to compare the angle against a right angle by eye, then confirm with measurement.
By sight: picture a square corner ($90°$) at the vertex. If the angle's arms open wider than that corner but have not flattened into a straight line, it is obtuse.
By measure: read the angle with a protractor. If the reading is between $91°$ and $179°$, it is obtuse. The key habit is deciding first whether the angle looks bigger or smaller than a right angle, then choosing the protractor scale that matches.
Obtuse Angles Inside a Triangle
A triangle with one obtuse angle is called an obtuse triangle (or obtuse-angled triangle). Its other two angles must both be acute, because the three together can only reach $180°$. The side opposite the obtuse angle is always the longest side of the triangle — the wider the angle opens, the longer the side it faces.
This connects to a wider idea your study of triangles will reach: a triangle is classified by its largest angle. If the biggest angle is below $90°$ it is acute; exactly $90°$, right; above $90°$, obtuse. You will meet the same three-way split again with the law of cosines, where the sign of one term tells you directly whether a triangle is obtuse.
Examples of Obtuse Angles
With the definition and properties in hand, here is the obtuse angle doing real work. The problems build from a one-step classification up to a triangle solve.
Example 1
Is an angle measuring $115°$ obtuse?
Check the range. Since $90° < 115° < 180°$, the angle is obtuse.
Example 2
An angle measures $90°$. A student says it is the smallest obtuse angle. Is that right?
Wrong attempt. The reasoning goes: "obtuse means at least $90°$, and $90°$ is the smallest value that qualifies, so $90°$ is the smallest obtuse angle." That reads the boundary as included.
Why it breaks. An angle of exactly $90°$ is a right angle, by definition — it has its own name precisely because it is the dividing line. If $90°$ counted as obtuse, the right angle would have no distinct category. The obtuse range is strictly greater than $90°$.
Correct. $90°$ is a right angle, not obtuse. There is no single "smallest obtuse angle" — the values get arbitrarily close to $90°$ (such as $90.0001°$) without ever reaching it. The smallest whole-degree obtuse angle is $91°$.
Example 3
Two angles are supplementary. One of them is $130°$. Classify both.
Supplementary angles add to $180°$, so the other is $180° - 130° = 50°$. The $130°$ angle is obtuse; the $50°$ angle is acute. (The supplement of any obtuse angle is acute — a property in action.)
Example 4
An obtuse angle measures $145°$. Find the measure of its reflex partner.
The reflex partner completes the full turn: $360° - 145° = 215°$. So a $145°$ obtuse angle pairs with a $215°$ reflex angle.
Example 5
In a triangle, two angles measure $35°$ and $40°$. Find the third angle and classify the triangle.
The angles sum to $180°$, so the third is $180° - (35° + 40°) = 180° - 75° = 105°$. Since $105°$ is between $90°$ and $180°$, it is obtuse, so the triangle is an obtuse triangle.
Example 6
The angle $\angle XYZ$ is given by the expression $(4k + 30)°$, and it is known to be obtuse with $k = 25$. Confirm it is obtuse.
Substitute: $(4 \times 25 + 30)° = (100 + 30)° = 130°$. Since $90° < 130° < 180°$, the angle $\angle XYZ$ is indeed obtuse.
Why the Obtuse Angle Matters Beyond the Classroom
Naming an angle obtuse is not a label for its own sake — the moment an angle crosses $90°$, the physics and the engineering change with it.
Structural stability. Roof trusses and bridge supports are designed around specific angles; an obtuse joint spreads a load differently from a right-angled one, and getting the angle wrong changes how force travels through the frame.
Ergonomics and design. A laptop hinge, a reclining seat, a desk lamp — each is set to an obtuse "rest" angle because that open position is comfortable and stable. Designers specify these angles to the degree.
Navigation and bearings. A course change can turn through an obtuse angle; whether a turn is acute, obtuse, or reflex completely changes a vessel's heading and the distance travelled.
Sport and motion. The angle a ball leaves a bat or a cue, the spread of a gymnast's limbs mid-air — many of these are obtuse, and the exact opening governs the resulting path.
For a Grade 6 student, the obtuse angle is one of the first "in-between" categories in geometry: not the sharp acute point, not the tidy right-angle corner, but the wide middle — and learning to spot that middle by eye is the skill every later angle topic builds on.
Where Students Trip Up on Obtuse Angles
Mistake 1: Treating $90°$ or $180°$ as obtuse
Where it slips in: A student remembers "obtuse is the big one" and lumps the right angle ($90°$) or the straight angle ($180°$) in with it.
Don't do this: Call a $90°$ angle the smallest obtuse angle, or a $180°$ straight angle a "very obtuse" angle.
The correct way: The range is strict: $90° < \theta < 180°$. Exactly $90°$ is a right angle; exactly $180°$ is a straight angle. Obtuse is everything strictly between them.
Mistake 2: Calling a reflex angle obtuse
Where it slips in: An angle is clearly bigger than a right angle, so the student stops checking and labels it obtuse — even when it has swung past $180°$.
Don't do this: Read a $230°$ angle as "obtuse" just because it is large.
The correct way: Obtuse stops at $180°$. Once an angle passes the straight-angle line it is a reflex angle. Always ask: has this angle crossed $180°$? The second-guesser who measures with a protractor often reads the small arc when the angle is actually reflex — decide the category by eye first.
Mistake 3: Reading the wrong protractor scale
Where it slips in: A protractor shows two rows of numbers — one running $0°$ to $180°$ left-to-right, the other right-to-left. The reader follows the wrong row.
Don't do this: Read a clearly-obtuse angle as $60°$ by following the inner scale when the outer one applies.
The correct way: Judge by eye first — does this angle look bigger or smaller than a square corner? An obtuse-looking angle cannot read as $60°$. Pick the scale whose $0°$ sits on one arm, then read along it.
Key Takeaways
An obtuse angle measures strictly more than $90°$ and strictly less than $180°$.
The named boundaries are not obtuse: $90°$ is a right angle and $180°$ is a straight angle.
A triangle can hold at most one obtuse angle, and the side opposite it is the longest side.
The supplement of an obtuse angle is always acute; its reflex partner is $360°$ minus the angle.
The most common mistake is including the $90°$ boundary or confusing an obtuse angle with a reflex angle — always check whether the angle has crossed $90°$ and whether it has crossed $180°$.
Practice These Problems to Solidify Your Understanding
Classify each angle as acute, right, obtuse, straight, or reflex: $48°$, $90°$, $134°$, $180°$, $300°$.
Two angles are supplementary; one is $158°$. Find the other and classify both.
A triangle has angles $25°$ and $43°$. Find the third angle and state whether the triangle is obtuse.
Answer to Question 1: acute, right, obtuse, straight, reflex. Answer to Question 2: the other is $22°$; the $158°$ angle is obtuse and the $22°$ angle is acute. Answer to Question 3: the third angle is $112°$, so the triangle is obtuse. If you called $300°$ obtuse, revisit Mistake 2.
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