Below are 42 absolutely indispensable Geometry Theorems you ought to master!
Segment Addition Postulate: Q is the midpoint of PR. Show that PQ and QR are each equal to ½ PR.
Congruent Supplements Theorem: If 2 angles are supplementary to the same angle (or to congruent angles) then they are congruent.
Vertical angles Theorem: Vertical angles are congruent
Congruent Complements Theorem: If 2 angles are complementary to the same angle (or to congruent angles) then they are congruent.
If two lines are perpendicular, then they intersect to form four right angles.
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of alternate interior angles are congruent.
Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of alternate exterior angles are congruent.
Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then both pair of consecutive interior angles are supplementary.
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Alternate Interior Angles Converse Theorem: If two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.
Alternate Exterior Angles Converse Theorem: If two lines are cut by a transversal such that the alternate exterior angles are congruent, then the lines are parallel.
Consecutive Interior Angles Converse Theorem: If two lines are cut by a transversal such that the consecutive interior angles are supplementary, then the lines are parallel.
Triangle Sum Theorem: The sum of the measures of the interior angles is 180 degrees.
Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two nonadjacent interior angles.
3rd Angles Theorem: If two angles of 1 triangle are congruent to two angles of another, then the 3rd angle is also congruent.
Angle-Angle-Side (AAS) Congruence Theorem: If two angles and an non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.
Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite of them are congruent.
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Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the end points of the segment.
Perpendicular Bisector Theorem Converse: If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of the segment.
Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.
Angle Bisector Converse: If a point is in the interior of an angle, equidistant from its sides, then it lies on the bisector of the angle.
Mid Segment Theorems:
* The segments connecting the midpoints of two sides of a triangle is parallel to the third side and half the third side.
* Given: ABC with mid-segments DE, EF, FD. Prove: ADE is congruent to DBF
Triangle Inequality: If one side of a triangle is longer than another side, then the angle opposite the longest side is larger than the angle opposite of the shorter side.
Parallelogram Theorems:
* If a quadrilateral is a parallelogram, then its opposite sides are congruent.
* If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
* If a quadrilateral is a parallelogram, then its opposite angles are congruent.
* If a quadrilateral is a parallelogram, then its diagonals bisect each other.
* If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
* If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
* If an angle of a quadrilateral is supplementary to both its consecutive angles, then the quadrilateral is a parallelogram.
* If a parallelogram is a rhombus, then its diagonals are perpendicular.
* If the diagonals of a parallelogram are perpendicular, then its a rhombus.
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SSS Similarity Theorem: If the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.
SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the length of the sides including these angles are proportional, then the triangles are similar.
Triangle Proportionality Theorem: If a line parallel to one side of a triangle, intersects the other two sides, then it divides 2 sides proportionally.
Pythagorean Theorem
Tangents to a Circle Theorem: If two segments from the same exterior point are tangents to a circle, then they are congruent.
Inscribed Angle Theorems:
* If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.
* If two inscribed angles of a circle intersect the same arc, then the angles are congruent.
* The angle in a semicircle is a right angle (if one side of an inscribed triangle is the diameter of a circle, then the triangle is a right triangle).
* If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle.
Polygon Angle Theorems:
* The sum of the interior angles of an n-sided polygon is (n-2)180 degrees.
* The sum of the exterior angles of an n-sided polygon is 360 degrees.
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