Angles & Triangles (Geometry)
Angle Definitions
Acute Angle
An angle less than $90^\circ$.
Right Angle
An angle equal to $90^\circ$.
Obtuse Angle
An angle greater than $90^\circ$ but less than $180^\circ$.
Straight Angle
An angle equal to $180^\circ$.
Reflex Angle
An angle greater than $180^\circ$ but less than $360^\circ$.
Full Angle
An angle equal to $360^\circ$.
Complementary Angles
Two angles whose sum is $90^\circ$.
Supplementary Angles
Two angles whose sum is $180^\circ$.
Parallel Lines and Transversals
Angle Relationships
When a transversal line intersects two parallel lines, several pairs of angles are formed with specific relationships:
- Vertically Opposite Angles: Equal. (e.g., $a=d$, $b=c$, $e=h$, $f=g$)
- Corresponding Angles: Equal. (e.g., $a=e$, $b=f$, $c=g$, $d=h$)
- Alternate Interior Angles: Equal. (e.g., $c=f$, $d=e$)
- Alternate Exterior Angles: Equal. (e.g., $a=h$, $b=g$)
- Consecutive Interior Angles (Same-Side Interior): Supplementary (sum to $180^\circ$). (e.g., $c+e=180^\circ$, $d+f=180^\circ$)
Angle Units & Conversion
Degrees ($^\circ$): A common unit where a full circle is $360^\circ$.
Minutes ($'$): $1^\circ = 60'$.
Seconds ($''$): $1' = 60''$. So, $1^\circ = 3600''$.
Radians (rad): The SI unit. $2\pi \text{ rad} = 360^\circ$. $1 \text{ rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ$.
Degrees to DMS / Radians
DMS to Degrees / Radians
Radians to Degrees / DMS
Relevant Tools
To further enhance your learning and problem-solving skills, explore these additional resources
Loading relevant links...
Angle Arithmetic (DMS)
Enter two angles in Degrees, Minutes, Seconds format.
Types of Triangles
Equilateral
All three sides are equal, and all three angles are $60^\circ$.
Isosceles
Two sides are equal, and the angles opposite those sides are equal.
Scalene
All three sides and all three angles are different.
Acute Triangle
All three angles are acute (less than $90^\circ$).
Right Triangle
One angle is a right angle ($90^\circ$). The side opposite the right angle is the hypotenuse.
Obtuse Triangle
One angle is obtuse (greater than $90^\circ$).
Key Triangle Properties
- Sum of Angles: The sum of the interior angles of any triangle is always $180^\circ$. $A + B + C = 180^\circ$.
- Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. (e.g., $a+b > c$, $a+c > b$, $b+c > a$).
- Law of Sines: Relates the sides of a triangle to the sines of its opposite angles. $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$ (where R is the circumradius).
- Law of Cosines: Relates the lengths of the sides of a triangle to the cosine of one of its angles.
- $a^2 = b^2 + c^2 - 2bc \cos A$
- $b^2 = a^2 + c^2 - 2ac \cos B$
- $c^2 = a^2 + b^2 - 2ab \cos C$
- Area of a Triangle:
- Given base $b_s$ and height $h_s$: Area $= \frac{1}{2} b_s h_s$.
- Given two sides $a, b$ and the included angle $C$: Area $= \frac{1}{2} ab \sin C$.
- Heron's Formula (given sides $a,b,c$ and semi-perimeter $s = \frac{a+b+c}{2}$): Area $= \sqrt{s(s-a)(s-b)(s-c)}$.