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Geometry Definitions, Postulates and Theorems
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Gravity
Terms in this set (67)
Complementary angles
two angles whose measures have a sum of 90°
Supplementary angles
two angles who measures have a sum of 180°
Theorem
a statement that can be proven
Vertical Angles
two angles formed by intersecting lines and facing in the opposite direction
Transversal
a line that intersects two lines in the same plane at different points
Corresponding angles
pairs of angles formed by two parallel lines and a transversal that make an F pattern
Same-side interior angles
pairs of angles formed by two parallel lines and a transversal that make a C pattern
Alternate interior angles
pairs of angles formed by two parallel lines and a transversal that make a Z pattern
Congruent triangles
triangles in which corresponding parts (sides and angles) are equal in measure
Similar triangles
triangles in which corresponding angles are equal in measure and corresponding sides are in proportion (ratios equal)
Angle bisector
a ray that begins at the vertex of angle and divides the angle into two angles or equal measure
Segment bisector
a ray, line or segment that divides a segment into two parts of equal measure
Legs of an isosceles triangle
the sides of equal measure in an isosceles triangle
Base of an isosceles triangle
the third side of an isosceles triangle
Equiangular
having angles that are all equal in measure
Perpendicular bisector
a line that bisects a segment and is perpendicular to it
Altitude
a segment from a vertex of a triangle perpendicular to the line containing the opposite side
Geometric mean
the value of x in proportion a/x = x/b where a,b, and x are positive numbers (x is the geometric mean between a and b)
Sine, sin
for an acute angle of a right triangle, the ratio of the side opposite the angle to the measure of the hypotenuse (opp/hyp)
Cosine, cos
for an acute angle of a right triangle the ratio of the side adjacent to the angle to the measure of the hypotenuse (adj/hyp)
Tangent, tan
for an acute angle of a right triangle, the ratio of the side opposite to the angle to the measure of the side adjacent (opp/adj)
Addition Property of Equality
If the same number is added to equal numbers, then the sums are equal.
Subtraction Property of Equality
If the same number is subtracted from equal numbers, then the difference are equal.
Multiplication Property of Equality
If equal numbers are multiplied by the same number, then the products are equal.
Division Property of Equality
If equal numbers are divided by the same number, then the quotients are equal.
Reflexive Property of Equality
A number is equal to itself.
Symmetric Property of Equality
If a = b then b = a.
Substitution Property of Equality
If values are equal, then one value may be substituted for the other.
Transitive Property of Equality
If a = b and b = c then a = c.
Distributive Property
a(b+c) = ab + ac
Reflexive Property of Congruence
A ≅ A
Symmetric Property of Congruence
If A ≅ B, then B ≅ C then A ≅ C
Angle Addition postulate
For any angle, the measure of the whole is equal to the sum of the measures of its non-overlapping parts
Linear Pair Theorem
If two angles form a linear pair, then they are supplementary.
Congruent supplements theorem
If two angles are supplements of the same angle, then they are congruent.
Congruent complements theorem
If two angles are complements of the same angle, then they are congruent.
Right Angle Congruence Theorem
All right angles are congruent.
Vertical Angles Theorem
Vertical angles are equal in measure.
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.
Segment Addition postulate
For any segment, the measure of the whole is equal to the sum of the measures of its non-overlapping parts.
Common Segments Theorem
Given collinear points A,B,C and D arranged as shown, if segment AB ≅ segment CD then segment AC ≅ segment BC
Corresponding Angles Postulate
If two parallel lines are intersected by a transversal, then the corresponding angles are equal in measure.
Converse of Corresponding Angles Postulate
If two lines are intersected by a tranversal and corresponding angles are equal in measure, then the lines are parallel.
Alternate Interior Angles Theorem
If two parallel lines are intersected by a transversal, then alternate interior angles are equal in measure
Alternate Exterior Angles Theorem
If two parallel lines are intersected by a transversal, then alternate exterior angles are equal in measure.
Same-side Interior Angles Theorem
If two parallel lines are intersected by a transversal, then same-side interior angles are supplementary.
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 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 Same-side Interior Angles Theorem
If two lines are intersected by a transversal and same-side interior angles are supplementary, then the lines are parallel.
Perpendicular Transversal Theorem
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one.
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Converse of the Perpendicular Bisector 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.
Parallel Lines Theorem
In a coordinate plane, two nonvertical lines are parallel IFF they have the same slope.
Perpendicular Lines Theorem
In a coordinate plane, two nonvertical lines are perpendicular IFF the product of their slopes is -1.
Two-Transversals Proportionality Corollary
If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.
Angle-Angle (AA) Similarity Postulate
If two angles of one triangle are equal in measure to two angles of another triangle, then the two triangles are similar.
Side-side-side (SSS) Similarity Theorem
If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar.
Side-angle-side (SAS) Similarity Theorem
If two sides of one triangle are proportional to two sides of another triangle, then the third pair of angles are congruent.
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.
Side-Angle-Side Congruence Postulate SAS
If two sides and the included angle of one triangle are equal in measure to the corresponding sides and angle of another triangle, then the triangles are congruent.
Side-side-side Congruence Postulate SSS
If three sides of one triangle are equal in measure to the corresponding sides of another triangle, then the triangles are congruent.
Angle-side-angle Congruence Postulate ASA
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
Triangle Sum Theorem
The sum of the measure of the angles of a triangle is 180°
Exterior angle theorem
An exterior angle of a triangle is equal in measure to the sum of the measures of its two remote interior angles.
Triangle Proportionality Theorem
If a line parallel to a side of triangle intersects the other two sides, then it divides those sides proportionally.
Converse of Triangle Proportionality Theorem
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
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