A pair of
angleIn geometry and trigonometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle...
s is said to be
vertical (US English) or
vertically opposite (British English) if the angles share the same
vertexIn geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3D models, where each such point is given as a vector.-Of an angle:The vertex of an angle is the...
and are bounded by the same pair of
linesIn Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely...
but are opposite to each other. Such angles are
congruentIn geometry, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections...
and thus have equal measure.
If two
line segmentIn geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
s, EF and GH,
intersectIn mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
at the point P, they form four angles, EPG, GPF, FPH, and HPE. These angles can be grouped into two pairs of vertical angles: one vertical pair contains EPG and FPH, and the other pair contains GPF and HPE.
A pair of
angleIn geometry and trigonometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle...
s is said to be
vertical (US English) or
vertically opposite (British English) if the angles share the same
vertexIn geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3D models, where each such point is given as a vector.-Of an angle:The vertex of an angle is the...
and are bounded by the same pair of
linesIn Euclidean geometry, a line is a straight curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height. Lines are an idealisation of such objects and have no width or height at all and are usually considered to be infinitely...
but are opposite to each other. Such angles are
congruentIn geometry, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections...
and thus have equal measure.
Vertical Angle Theorem
If two
line segmentIn geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
s, EF and GH,
intersectIn mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
at the point P, they form four angles, EPG, GPF, FPH, and HPE. These angles can be grouped into two pairs of vertical angles: one vertical pair contains EPG and FPH, and the other pair contains GPF and HPE. Any angle in the first pair is supplementary to any angle in the second pair. The most obvious way to tell if two angles are vertical angles are if they form an "X".
A vertical angle is an angle that intersects with another one.
Proof of the Vertical Angle Theorem
This theorem can be proved in an algebraic way. Assume that angle A in the picture is equal to the measure x, and then subtract x from 180 (the amount of degrees in a line) to get angle C, or 180-x. Then, to find angle B, subtract the quantity "180-x" from 180. This means that angle B is equal to 180-(180-x), or 180-180+x, which equals x. Finally, angle D can be found algebraically by subtracting angle B from 180, which equals 180-x. Angles A and B are both equal to x, so angles A and B are
equal. As well, angles C and D are both equal to 180-x, so angles C and D are equal.
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