An obtuse angle is an angle that measures more than 90 degrees and less than 180 degrees. It opens wider than a square corner but does not flatten into a straight line. Angles of 100, 135 and 179 degrees are obtuse. A triangle can have at most one obtuse angle.
An obtuse angle lives between a right angle and a straight angle. The endpoints 90° and 180° themselves have their own names.
Formal definition and notation
An angle ∠ ABC is obtuse when 90° < m∠ ABC < 180°. Both bounds are excluded: 90° is a right angle and 180° is a straight angle. Angles larger than 180° are reflex angles, not obtuse. A triangle with one obtuse angle is called an obtuse triangle.
Symbol
Read it as
Meaning
90° < θ < 180°
theta is between 90 and 180 degrees
the condition for an angle θ to be obtuse
Key facts
Obtuse comes from a Latin word meaning blunt: an obtuse angle has a blunt, wide opening.
An obtuse angle and an acute angle can sit side by side on a straight line, because they add to 180°: 130° + 50°.
A triangle has at most one obtuse angle. Two obtuse angles would already add to more than 180°.
Each interior angle of a regular hexagon is 120°, so it is obtuse. Regular polygons with 5 or more sides all have obtuse angles.
Two obtuse angles can never be supplementary or complementary, since their sum is more than 180°.
Try it: make an obtuse angle
Interactive
Drag the ray past the square corner. The angle stays obtuse until it flattens into a straight line at 180 degrees.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Obtuse Angle examples, step by step
Example 1: Classify by size
Warm-up
Name each angle as acute, right, obtuse or straight: 95°, 180°, 45°, 162°.
Compare each measure with 90° and 180°.
95° and 162° are between 90° and 180°: obtuse.
180° is straight, and 45° is acute.
Answer: Two are obtuse: 95° and 162°. 180° is straight and 45° is acute.
Example 2: Find the obtuse partner
Core
Two angles form a straight line. One measures 37°. Find the other and classify it.
Angles that form a straight line add to 180°.
Other angle = 180° − 37° = 143°.
143° is between 90° and 180°, so it is obtuse.
Answer: The other angle is 143°, an obtuse angle.
Example 3: Obtuse triangle
Test-style
In triangle PQR, ∠ P = 24° and ∠ Q = 41°. Is the triangle acute, right or obtuse?
The three angles add to 180°.
∠ R = 180° − 24° − 41° = 115°.
115° is more than 90°, so ∠ R is obtuse.
Answer:∠ R = 115°, so triangle PQR is an obtuse triangle.
Common mistakes
✗ MistakeCalling a 200° angle obtuse because it is larger than 90°.
✓ Instead200° is more than 180°, so it is a reflex angle. Obtuse angles stop below 180°.
Obtuse has an upper limit too. The opening must still be less than a straight line.
✗ MistakeReading a 120° angle as 60° on a protractor.
✓ InsteadThe angle is wider than a square corner, so the reading must be over 90: 120°.
Look first: a wide, blunt opening means use the protractor number above 90.
✗ MistakeDrawing a triangle with angles 100°, 95° and −15°.
✓ InsteadOnly one angle can be obtuse: for example 100°, 50°, 30°.
Two obtuse angles already add to more than 180°, leaving nothing for the third angle.
An acute angle is less than 90°; an obtuse angle is more than 90°. 60° is acute, 120° is obtuse.
Obtuse Anglevs.Reflex angle
A reflex angle is more than 180° and less than 360°, such as 250°. An obtuse angle is less than 180°.
Quick check: 3 questions
Question 1 of 3
Which angle is obtuse?
Show answer and explanation
Answer: B.179° is between 90° and 180°. 89° is acute, 90° is right, and 181° is past a straight line, so it is reflex.
Question 2 of 3
An obtuse angle and a second angle form a straight line. The obtuse angle is 126°. What is the second angle?
Show answer and explanation
Answer: C. Angles on a straight line add to 180°: 180° − 126° = 54°. 36° is 126° − 90°, which mixes in the right angle, and 234° subtracts from 360°.
Question 3 of 3
How many obtuse angles can one triangle have?
Show answer and explanation
Answer: A. A triangle’s angles add to 180°. One obtuse angle is possible (120° + 30° + 30°), and zero is possible (an acute triangle), but two obtuse angles already pass 180°.
Obtuse Angle: frequently asked questions
Is a 180 degree angle obtuse?
No. An angle of exactly 180° is a straight angle: its two rays point in opposite directions and form a line. Obtuse angles are strictly between 90° and 180°, so 179° is obtuse but 180° is not. Angles larger than 180° are reflex angles.
What are examples of obtuse angles in real life?
A reclining chair tipped back, a laptop opened wide, and clock hands at 4:00 (120°) all form obtuse angles. The corners of a stop sign are 135°, and the corners of a honeycomb cell are 120°. In each case the opening is wider than the corner of a book but not flat.
How do you know if a triangle is obtuse?
Find its largest angle. If that angle is more than 90°, the triangle is obtuse. With side lengths, square the longest side c: if c2 > a2 + b2, the triangle is obtuse. For sides 4, 5 and 8, 64 > 16 + 25, so the triangle is obtuse.
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