A geometry theorem is a statement that has been proven to be true using logical reasoning within the field of geometry. Geometry theorems provide a way to make precise statements about the relationships between various geometric objects, such as points, lines, angles, and shapes. Some examples of well known geometry theorems include the Pythagorean theorem, which relates to the sides of a right triangle, and the theorem of Thales, which describes the relationship between angles in a circle and chords intersecting those angles. These theorems and others like them are used extensively in many branches of mathematics and engineering, and they have practical applications in fields such as architecture, surveying, and computer graphics.
Properties
- Reflexive property of angle congruence - For any angle \(\large{ \angle A }\) , \(\large{ m\angle A= m\angle A }\)
- Symmetric property of angle congruence - If \(\large{ m\angle A \cong m\angle B }\) , then \(\large{ m\angle B\cong m\angle A }\)
- Transitive property of angle congruence - If \(\large{ m\angle A\cong m\angle B }\) and \(\large{ m\angle B\cong m\angle C }\) , then \(\large{ m\angle A\cong m\angle C }\)
- Reflexive property of segment congruence - For any segment \(\large{ AB }\) , \(\large{ \overline{AB} \cong \overline{AB} }\)
- Symmetric property of segment congruence - If \(\large{ \overline{AB} \cong \overline{CD} }\) , then \(\large{ \overline{CD} \cong \overline{AB} }\)
- Transitive property of segment congruence - If \(\large{ \overline{AB} \cong \overline{CD} }\) and \(\large{ \overline{CD} \cong \overline{EF} }\) , then \(\large{ \overline{AB} \cong \overline{EF} }\)
- Reflexive property of triangle congruence - Every triangle is congruent to itself.
- Symmetric property of triangle congruence - If \(\large{ \triangle ABC \cong \triangle DEF }\) , then \(\large{ \triangle DEF \cong \triangle ABC }\)
- Transitive property of triangle congruence - If \(\large{ \triangle ABC \cong \triangle DEF }\) and \(\large{ \triangle DEF \cong \triangle JKL }\) , then \(\large{ \triangle ABC \cong \triangle JKL }\)
Angle
- Alternate exterior angles theorem - If a transversal intersects two parallel lines, then the alternate exterior angles are congruent.
- Alternate interior angles theorem - If a transversal intersects two parallel lines, then the alternate interior angles are congruent.
- Angle bisector theorem - If a point is on the biscetor of an angle, then it is equidistant from the side of the angle.
- Congruent complements theorem - If two angles are complements of the same angle (or of congruent angles), then they are congruent.
- Congruent supplements theorem - If two angles are supplements of the same angle (or of congruent angles), then they are congruent.
- Converse of the angle bisector theorem - If a point in the interior of an angle is equidistant from the side of the angle, then it is on the bisector of the angle.
- Linear pair theorem - If two angles are form a linear pair, then they are supplementary.
- Right angle congruence theorem - All right angles are congruent.
- Same-side interior angles theorem - If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.
- Theorem - If two congruent angles are supplementary, then each is a right angle.
- Vertical angles theorem - All right angles are equal in measurement.
Circle
- Chord-chord product theorem - If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal.
- Inscribed angle theorem - The measure of an inscribed angle is half the measure of its intercepted arc.
- Secant-secant product theorem - If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.
- Secant-tangent product theorem - If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared.
- Theorem - If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
- Theorem - If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle.
- Theorem - If two segments are tangent to a circle from the same external point, then the segments are congruent.
- Theorem - In a circle or congruent circles: congruent central angles have congruent chords, congruent chords have congruent arcs and congruent arcs have congruent central angles.
- Theorem - In a circle, if a radius (or diameter) is perpendicilar to a chord, then it bisects the chord and its arc.
- Theorem - In a circle, the perpendicilar bisector of a chord is a radius (or diameter).
- Theorem - An inscribed angle subtends a semicircle IFF the angle is a right angle.
- Theorem - If a quadrilateral is inscribed in a circle, then its opposite angles are supplementart.
- Theorem - If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc.
- Theorem - If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of the intercepted arcs.
- Theorem - If a tangent and a secant, two tangents or two secants intersect in the exterior of a circle, then the measure of the angle formed is half the difference of the measure of its intercepted arc.
Lines
- Alternate exterior angles theorem - If two parrallel lines are intersected by a transversal, then alternating exterior angles are equal in measurement.
- Alternate interior angles theorem - If two parrallel lines are intersected by a transversal, then alternating interior angles are equal in measurement.
- Comment segments theorem - Given collinear points \(\large{ A, B, C, D }\) , if \(\large{ \bar{A} \bar{B} \cong \bar{C} \bar{D} }\) then \(\large{ \bar{A} \bar{C} \cong \bar{B} \bar{C} }\)
- Converse of alternate exterior angles theorem - If two lines are intersected by a transversal and alternate exterior angles are equal in measure, then the lines are parallel.
- Converse of alternate interior angles theorem - If two lines are intersected by a transversal and alternate interior angles are equal in measure, then the lines are parallel.
- Converse of same side interior angles theorem - If two lines are intersected by a transversal and same side interior angles are supplememtary, then the lines are parallel.
- Converse of the perpendicular biscetor theorem - If a point is the same distance from both the endpoints of a segment, then it lies on the perpendicular bisector of the segment.
- Theorem - If two intersecting lines from a linear pair of congruent angles, then the lines are perpendicular.
- Theorem - If two lines are perpendicular to the same transversal, then they are parallel.
- Parallel lines theorem - In a coordinate plane, two nonvertical lines are parallel if they have the same slope.
- Perpendicular biscetor theorem - If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of the segment.
- Perpendicular lines theorem - In a coordinate plane, two nonvertical lines are perpendicular if the product of their slopes is -1.
- Perpendicular transversal theorem - If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other side.
- Same side interior angles theorem - If two parrallel lines are intersected by a transversal, then same side interior angles are supplementary.
- Two transversals porportionality corollary - If three or more parallel lines intersect two transversals, then they divide the transversals porportionally.
Quadrilateral
- Porportional perimeters and area theorem - If the similarity ratio of two similar figures is \(\large{ \frac{a}{b} }\) , then the ratio of their perimeter is \(\large{ \frac{a}{b} }\) and the ratio of their areas is \(\large{ \frac{a^2}{b^2} }\) or \(\large{ \left( \frac{a}{b} \right)^2 }\)
- Theorem - If a quadrilateral is a parallelogram, then its opposite sides are congruent.
- Theorem - If a quadrilateral is a parallelogram, then its opposite angles are congruent.
- Theorem - If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
- Theorem - If a quadrilateral is a parallelogram, then its diagonals bisect each other.
- Theorem - If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.
- Theorem - If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
- Theorem - If both pairs of opposite angles are congruent, then the quadrilateral is a parallelogram.
- Theorem - If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.
- Theorem - If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
- Theorem - If a quadrilateral is a rectangle, then it is a parallelogram.
- Theorem - If a parallelogram is a rectangle, then its diagonals are congruent.
- Theorem - If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.
- Theorem - If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
- Theorem - If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
- Theorem - If a quadrilateral is a rhombus, then it is a parallelogram.
- Theorem - If a parallelogram is a rhombus, then its diagonals are perpendicular.
- Theorem - If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.
- Theorem - If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
- Theorem - If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.
- Theorem - If a quadrilateral is a kite, then its diagonals are perpendicular.
- Theorem - If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
- Theorem - If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.
- Theorem - If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles.
- Theorem - If a trapezoid is isosceles, if and only if its diagonals are congruent.
- Trapezoid midsegment theorem - The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases.
Triangle
- 30° - 60° - 90° triangle theorem - In a \(\large{ 30°-60°-90° }\) triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times the square root of 3.
- 45° - 45° - 90° triangle theorem - In a \(\large{ 45°-45°-90° }\) triangle, both legs are congruent, and the length of the hypotenuse is the length of a length times the square root of 2.
- Angle-angle-side congruence theorem (AAS) - If two angles and a non-included side of one triangle are equal in measure to the corresponding angles and side of another triangle, then the triangles are congruent. If \(\large{ \triangle{ABC} \cong \triangle{DEF} }\) , \(\large{ \angle{A} \cong \angle{D} }\) , \(\large{ \angle{C} \cong \angle{F} }\) , then \(\large{ \overline{AB} \cong \overline{DE} }\)
- Centriod theorem - The centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side.
- Circumcenter theorem - The circumcenter of a triangle is equidistant from the vertices of the triangle.
- Converse of hinge theorem - If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side.
- Converse of isosceles triangle theorem - If two angles of a triangle are equal in measure, then the sides opposite those angles are equal in measure.
- Converse of triangle proportionality theorem - If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
- Converse of the pythagorean theorem - If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. If \(\large{ c^2 = a^2+b^2 }\) , then \(\large{ ABC }\) is an acute triangle or right triangle or obtuse triangle
- Corollary - The acute angles of a right triangle are complementary.
- Corollary - If a triangle is equilateral, then it is equiangular.
- Corollary - The measure of each angle of an equiangular triangle is 60°.
- Corollary - If a triangle is equiangular, then it is also equilateral.
- Exterior angle theorem - An exterior angle of a triangle is equal in measure to the sum of the measures of its two remote internal angles.
- Geometric means corollary a - The length of the altitude to the hypotenuse of a tighr triangle is the geometric mean of the lengths of the two segments of the hypotenuse.
- Geometric means corollary b - The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
- Hinge theorem - If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the longer third side is across from the larger included angle.
- Hypotenuse-leg congruence (HL) - If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. If \(\large{ \overline{AB} = \overline{DE} }\) , \(\large{ \overline{BC} = \overline{EF} }\) , \(\large{ \angle{ACB} = \angle{DFE} }\) , then \(\large{ \triangle{ABC} \cong \triangle{DEF} }\)
- Incenter theorem - the incenter of a triangle is equidistant from the sides of the triangle.
- Isosceles triangle theorem - If two sides of a triangle are equal in measure, then the angles opposite those sides are equal in measure.
- Law of cosines - For any triangle \(\large{ ABD }\) with sides \(\large{ a, b }\) and \(\large{ c }\) , \(\large{ a^2 = b^2+c^2-2bc\;cos\;A }\) , \(\large{ b^2 = a^2+c^2-2ac\;cos\;B }\) , \(\large{ c^2 = a^2+b^2-2ac\;cos\;C }\)
- Law of sines - For any triangle \(\large{ ABD }\) with side lengths \(\large{ a, b }\) and \(\large{ c }\) , \(\large{ \frac{sin\;A}{a} = \frac{sin\;B}{b} = \frac{sin\;C}{c} }\)
- Pythagorean theorem - In any right triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs. \(\large{ c^2 = a^2+b^2 }\)
- Pythagoream Inequalities theorem - In \(\large{ \triangle ABD }\) , \(\large{ c }\) is the length is the length of the longest side. If \(\large{ c^2 > a^2+b^2 }\) , then \(\large{ \triangle ABD }\) is an obtuce triangle. If \(\large{ c^2 < a^2+b^2 }\) , then \(\large{ \triangle ABD }\) is acute.
- Side-side-side similarity theorem (SSS) - If the three sides of one triangle are porportional to the three corresponding sides of another triangle, then the triangles are similar. \(\large{ \triangle ABC \sim \triangle DEF }\)
- Side-angle-side similarity theorem (SAS) - If two sides of one triangle are porportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. \(\large{ \triangle ABC \sim \triangle DEF }\)
- Theorem - If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the origional triangle and to each other.
- Theorem - If two sides of a triangle are not congruent, then the large angle is opposite the longer side.
- Theorem - If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.
- Third angles theorem - If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. If \(\large{ \triangle{ABC} \cong \triangle{DEF} }\) , \(\large{ \angle{B} \cong \angle{E} }\) , \(\large{ \angle{C} \cong \angle{F} }\) , then \(\large{ \angle{A} \cong \angle{D} }\)
- Triangle angle bisector theorem - An angle bisector of a triangle divides the opposite sides into two segments whose lengths are porportional to the lengths of the other two sides.
- Triangle inequality theorem - The sum of any two side lengths of a triangle is greater than the third side length.
- Triangle midsegment theorem - The midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.
- Triangle porportionality theorem - If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides porportionally.
- Triangle sum theorem - The sum of the measurement of the angle of a triangle is 180°. \(\large{ \angle{A} + \angle{B} + \angle{C} = 180° }\)
