Geometry Conjectures and Theorems

Flashcards for Geometry Review

Converse of Exterior Supplements Conjecture

If two consecutive (same side) exterior angles are supplements, then the lines intersected by a transversal are parallel.

Congruent and Supplementary Conjecture

If two angles are both congruent and supplementary, then each is a right angle.

Supplements of Congruent Angles Conjecture

Supplements of congruent angles are congruent.

Right Angles Are Congruent Conjecture

All right angles are congruent.

Perpendicular to Parallel Conjecture

If two lines in the same plane are perpendicular to a third line, then the two lines are parallel to each other.

Parallel Transitivity Conjecture

If two lines in the same plane are parallel to a third, then they are parallel to each other.

Perpendicular Bisector Conjecture

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints.

Converse of the Perpendicular Bisector Conjecture

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Shortest Distance Conjecture

The shortest distance from a point to a line is measured along the perpendicular segment from the point to the line.

Angle Bisector Conjecture

If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

Angle Bisector Concurrency Conjecture

The three angle bisectors of a triangle are concurrent at the incenter.

Perpendicular Bisector Concurrency Conjecture

The three perpendicular bisectors of a triangle are concurrent at the circumcenter.

Altitude Concurrency Conjecture

The three altitudes of a triangle are concurrent at the orthocenter.

Circumcenter Conjecture

The circumcenter of a triangle is equidistant from the vertices.

Incenter Conjecture

The incenter of a triangle is equidistant from the sides.

Median Concurrency Conjecture

The three medians of a triangle are concurrent at the centroid.

Centroid Conjecture

The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the opposite side.

Triangle Sum Conjecture

The sum of the measures of the angles in every triangle is 180°.

Third Angle Conjecture

If two angles of one triangle are equal in measure to two angles of another triangle, then the third angle in each triangle are congruent to each other.

Isosceles Triangle Conjecture

If a triangle is isosceles, then the base angles are congruent.

Converse of the Isosceles Triangle Conjecture

If a triangle has two congruent angles, then it is an isosceles triangle.

Triangle Inequality Conjecture

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Side-Angle Inequality Conjecture

In a triangle, if one side is longer than another side, then the angle opposite the longer side is larger in measure than the angle opposite the shorter side.

Triangle Exterior Angle Conjecture

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

SSS Congruence Conjecture

If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.

SAS Congruence Conjecture

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

ASA Congruence Conjecture

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

SAA Congruence Conjecture

If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent.

Vertex Angle Bisector Conjecture

In an isosceles triangle, the bisector of the vertex angle is also the altitude and median to the bases (it's also the perpendicular bisector of the base).

Equilateral/Equiangular Triangle

Every equilateral triangle is equiangular, and conversely, every equiangular triangle is equilateral.

Quadrilateral Sum Conjecture

The sum of the measures of the four angles in any quadrilateral is 360°.

Pentagon Sum Conjecture

The sum of the measures of the five angles in any pentagon is 540°.

Polygon Sum Conjecture

The sum of the measures of the n interior angles of an n-gon is 180° (n-2).

Exterior Angle Sum Conjecture

For any polygon, the sum of the measures of a set of exterior angles is 360°.

Equiangular Polygon Conjecture

You can find the measure of each interior angle of an equiangular n-gon by using 180(n-2) / n

Kite Angles Conjecture

The non-vertex angles of a kite are congruent.

Kite Diagonals Conjecture

The diagonals of a kite are perpendicular.

Kite Diagonal Bisector Conjecture

The diagonal connecting the vertex angles of a kite is the perpendicular bisector of the other diagonal.

Kite Angle Bisector Conjecture

The vertex angles of a kite are bisected by a diagonal.

Trapezoid Consecutive Angles Conjecture

The consecutive angles between the bases of a trapezoid are supplementary.

Isosceles Trapezoid Conjecture

The base angles of an isosceles trapezoid are congruent.

Isosceles Trapezoid Diagonals Conjecture

The diagonals of an isosceles trapezoid are congruent.

Three Midsegments Conjecture

The three midsegments of a triangle divide it into four congruent triangles.

Triangle Midsegment Conjecture

A midsegment of a triangle is parallel to the third side and half the length of the third side.

Trapezoid Midsegment Conjecture

The midsegment of a trapezoid is parallel to the bases and is equal in length to the average of the lengths of the bases.

Parallelogram Opposite Angles Conjecture

The opposite angles of a parallelogram are congruent.

Parallelogram Consecutive Angles Conjecture

The consecutive angles of a parallelogram are supplementary.

Parallelogram Opposite Sides Conjecture

The opposite sides of a parallelogram are congruent.

Parallelogram Diagonals Conjecture

The diagonals of a parallelogram bisect each other.

Double-Edged Straightedge Conjecture

If two parallel lines are intersected by a second pair of parallel lines that are the same distance apart as the first pair, then the parallelogram formed is a rhombus.

Rhombus Diagonals Conjecture

The diagonals of a rhombus are perpendicular, and they bisect each other.

Rhombus Angles Conjecture

The diagonals of a rhombus bisect the angles of the rhombus.

Rectangle Diagonals Conjecture

The diagonals of a rectangle are congruent and bisect each other.

AA Similarity Conjecture

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

SSS Similarity Conjecture

If the three sides of one triangle are proportional to the three sides of another triangle, then the two triangles are similar.

SAS Similarity Conjecture

If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.

Proportional Parts Conjecture

If two triangles are similar, then the corresponding altitudes, medians, and angle bisectors are proportional to the corresponding sides.

Angle Bisector/Opposite Side Conjecture

A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the two sides forming the angle.

Proportional Areas Conjecture

If corresponding sides of two similar polygons, or the radii of two circles, compare in the ratio of m/n, then their areas compare in the ratio of (m/n)².

Proportional Volumes Conjecture

If corresponding sides of two similar polygons, or the radii of two circles, compare in the ratio of m/n, then their volumes compare in the ratio of (m/n)³.

Parallel/Proportionality Conjecture

If a line parallel to one side of a triangle passes through the other two sides, then it divides the other two sides proportionally. Conversely, if a line cuts two sides of a triangle proportionally, then it is parallel to the third side.

Extended Parallel/Proportionality Conjecture

If two or more lines pass through two sides of a triangle parallel to the third side, then they divide the two sides proportionally.

SAS Triangle Area Conjecture

The area of a triangle is given by the formula A = 1/2 * a * b * sinC where a and b are the lengths of two sides and C is the angle between them.

Law of Sines

For a triangle with angles A, B, and C with side lengths of a, b, and c respectively, SinA/a = SinB/b = SinC/c.

Pythagorean Identity

For any angle A, (sinA)² + (cosA)² = 1.

Law of Cosines

For any triangle with sides of length a, b, and c with the angle C being opposite side c, then c² = a² + b² -2ab*cosC.

Reflections Rule Conjecture (x-axis)

(x,y) → (x, -y)

Reflections Rule Conjecture (y-axis)

(x,y) → (-x, y)

Reflection over line y = x

(x,y) → (y, x)

Translations Rule Conjecture

When translating a pre-image a given distance a left/right and b up/down, it can be denoted as (x,y) → (x +/- a, y +/- b).

Reflections over Parallel Lines Conjecture

A composition of reflections over parallel lines results in a translation. Furthermore the distance of the translation is double the distance between the parallel lines.

Rotations Rules Conjecture (90° counterclockwise)

(x,y) → (-y, x)

Rotations Rules Conjecture (180°)

(x,y) → (-x,-y)

Rotations Rules Conjecture (270° counterclockwise)

(x,y) → (y, -x)

Reflections over Intersecting Lines Conjecture

A composition of reflections over intersecting lines results in a rotation. Furthermore the angle of rotation is double the angle of intersection between the two lines.

Obtuse Angle Bisector Conjecture

If an obtuse angle is bisected, then the result is two congruent acute angles.

Overlapping Segments Conjecture

If AD has points A, B, C, D in that order with AB = CD, then the overlapping segments AC and BD are congruent.

Linear Pair Conjecture

If two angles form a linear pair, then the measures of the angles add up to 180°.

Vertical Angles Conjecture

If two angles are vertical angles, then they are congruent.

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then corresponding angles are congruent.

Converse of Corresponding Angles Postulate

If two corresponding angles are congruent, then the lines intersected by a transversal are parallel.

Alternate Interior Angles Conjecture

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

Converse of Alternate Interior Angles Conjecture

If two alternate interior angles are congruent, then the lines intersected by a transversal are parallel.

Alternate Exterior Angles Conjecture

If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.

Converse of Alternate Exterior Angles Conjecture

If two alternate exterior angles are congruent, then the lines intersected by a transversal are parallel.

Interior Supplements Conjecture

If two parallel lines are cut by a transversal, then consecutive (same side) interior angles are supplements.

Converse of Interior Supplements Conjecture

If two consecutive (same side) interior angles are supplements, then the lines intersected by a transversal are parallel.

Exterior Supplements Conjecture

If two parallel lines are cut by a transversal, then consecutive (same side) exterior angles are supplements.

Circumference Conjecture

If C is the circumference and d is the diameter of a circle, then there is a number π such that C = π *d. If d = 2r where r is the radius, then C = 2πr.

Intersecting Chords Conjecture

The measure of the angle formed by two intersecting chords is equal to the average of the two intercepted arcs.

Intersecting Secants Conjecture

The measure of an angle formed by two secants through a circle is equal to half the difference of the intercepted larger arc and smaller arc measures.

Arc Length Conjecture

The length of an arc equals the circumference times the measure of the central angle divided by 360°.

Rectangle Area Conjecture

The area of a rectangle is given by the formula A = bh, where A is the area, b is the length of the base, and h is the height of the rectangle

Parallelogram Area Conjecture

The area of a parallelogram is given by the formula A = bh, where A is the area, b is the length of the base, and h is the height of the parallelogram.

Triangle Area Conjecture

The area of a triangle is given by the formula A = ½bh, where A is the area, b is the length of the base, and h is the height of the triangle.

Trapezoid Area Conjecture

The area of a trapezoid is given by the formula A = ½ (b₁ + b₂)h, where A is the area, b₁ and b₂ are the lengths of the two bases, and h is the height of the trapezoid.

Kite Area Conjecture

The area of a kite is given by the formula A = ½ d₁d₂, where A is the area, d₁ and d₂ are the lengths of the two diagonals.

Regular Polygon Area Conjecture

The area of a regular polygon is given by the formula A = asn, where A is the area, a is the apothem, s is the length of each side, and n is the number of sides. Or since perimeter is P = sn you can also use A = ½aP.

Circle Area Conjecture

The area of a circle is given by the formula A = πr², where A is the area and r is the radius of the circle.

Sector Area Conjecture

The area (A) in terms of the circle's radius (r) of a sector is given by the formula A = (a/360) * πr².