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Basic Angle Facts

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Foundational skills
Classify an angle
\[ 135^\circ \to \text{obtuse} \]
Decide if an angle is acute, right, obtuse, straight or reflex.
Angles in a right angle
\[ x = 90^\circ – 34^\circ \]
Angles that make a right angle add up to 90°.
Angles on a straight line
Two angles on a line
\[ x = 180^\circ – 65^\circ \]
Angles on a straight line add up to 180°.
Three or more angles on a line
\[ x = 180^\circ – 50^\circ – 72^\circ \]
Take all the known angles away from 180°.
Angles around a point
Angles around a point
\[ x = 360^\circ – 140^\circ – 95^\circ \]
Angles around a point add up to 360°.
Around a point with a right angle
\[ x = 360^\circ – 90^\circ – 155^\circ \]
Remember the little square counts as 90°.
Equal angles around a point
\[ 5x = 360^\circ \]
Share what is left of 360° between the equal angles.
Vertically opposite angles
Vertically opposite angle
\[ \text{opposite: } x = 48^\circ \]
Angles directly opposite each other at a crossing are equal.
Neighbouring angle at a crossing
\[ x = 180^\circ – 48^\circ \]
Angles next to each other at a crossing add up to 180°.
Angles in triangles
Missing angle in a triangle
\[ x = 180^\circ – 62^\circ – 75^\circ \]
Angles in a triangle add up to 180°.
Right-angled triangle
\[ x = 180^\circ – 90^\circ – 37^\circ \]
Include the right angle when you add up to 180°.
Isosceles: base angle from top angle
\[ x = (180^\circ – 40^\circ) \div 2 \]
Take the top angle from 180°, then halve.
Isosceles: top angle from base angle
\[ x = 180^\circ – 2 \times 70^\circ \]
The two base angles are equal, so use one twice.
Exterior angle of a triangle
\[ x = 55^\circ + 68^\circ \]
An exterior angle equals the two opposite inside angles added.
Angles in quadrilaterals
Missing angle in a quadrilateral
\[ x = 360^\circ – 85^\circ – 110^\circ – 72^\circ \]
Angles in a quadrilateral add up to 360°.
Quadrilateral with right angles
\[ x = 360^\circ – 90^\circ – 90^\circ – 118^\circ \]
Count every little square as 90°.
Two equal angles in a quadrilateral
\[ 2x = 360^\circ – 110^\circ – 70^\circ \]
Take the known angles from 360°, then halve.
Parallelogram angles
\[ x = 180^\circ – 65^\circ \]
Opposite angles are equal; neighbouring angles add to 180°.
Extensions
Triangle with an extended side
\[ 180^\circ – 130^\circ = 50^\circ \]
Use the straight line first, then the triangle.
Isosceles with an extended side
\[ 110^\circ \text{ outside} \to 70^\circ \text{ inside} \]
Use the straight line, then the equal base angles.
Crossing lines and a triangle
\[ 40^\circ \text{ opposite} \to 40^\circ \text{ inside} \]
Use vertically opposite angles, then the triangle.
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