Geometry Terms & Theorems through Congruent Triangles

Circumcenter Theorem

Circumcenter Theorem. The vertices of a triangle are equidistant from the circumcenter.

Side-Side-Side Similarity Theorem (SSS)

1. Angle-Angle (AA) Similarity Postulate - If two angles of one triangle are congruent to two angles of another, then the triangles must be similar. 2. Side-Side-Side (SSS) Similarity Theorem - If the lengths of the corresponding sides of two triangles are proportional, then the triangles must be similar.

Pythagorean Theorem

relating to or characteristic of the Greek philosopher Pythagoras or his ideas.

Altitude

the line containing the opposite side is called the extended base of the

Transversal

(of a line) intersecting a system of lines.

Angle Bisector

A line that splits an angle into two equal angles. ("Bisect" means to divide into two equal parts.

Linear Pair Theorem

A linear pair is a pair of adjacent, supplementary angles. Adjacent means next to each other, and supplementary means that the measures of the two angles add up to equal 180 degrees.

Linear Pair

A linear pair of angles is formed when two lines intersect. Two angles are said to be linear if they are adjacent angles formed by two intersecting lines. The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees.

Scalene Triangle

A scalene triangle is a triangle that has three unequal sides, such as those illustrated above. SEE ALSO: Acute Triangle, Equilateral Triangle, Isosceles Triangle, Obtuse Triangle, Triangle.

Acute Triangle

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted.

Obtuse Triangle

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted.

Addition Property of Equality

Additive Property of Equality. The additive property of equality states that if the same amount is added to both sides of an equation, then the equality is still true.

Alternate Interior Angles

Alternate Interior Angles. When two lines are crossed by another line (called the Transversal): Alternate Interior Angles are a pair of angles on the inner side of each of those two lines but on opposite sides of the transversal.

Isosceles Triangle

An isosceles triangle is a triangle with (at least) two equal sides. In the figure above, the two equal sides have length and the remaining side has length . ... An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides and angles equal.

Interior Angle of a Triangle

Because of this, only one of the angles can be 90° or more. In a right triangle, since one angle is always 90°, the other two must always add up to 90° A triangle is simply a polygon that has 3 sides. See interior angles of a polygon for the properties of the interior angles of a polygon with any number of sides.

Centroid Theorem

Centroid of a Triangle. The centroid of a triangle is the point where the three medians coincide. The centroid theorem states that the centroid is of the distance from each vertex to the midpoint of the opposite side.

Congruent Angles

Congruent Angles have the same angle (in degrees or radians). That is all. They don't have to point in the same direction. They don't have to be on similar sized lines.

Congruent complements Theorem

Congruent Complements Theorem If 2 angles are complementary to the same angle, then they are congruent to each other.

Congruent Supplements Theorem

Congruent Supplements Theorem If 2 angles are supplementary to the same angle, then they are congruent to each other.

Same-Side (Consecutive) Interior Angles

Consecutive Interior Angles. When two lines are crossed by another line (called the Transversal): The pairs of angles on one side of the transversal but inside the two lines are called Consecutive Interior Angles.

Converse of Isosceles Triangle Theorem

Converse of Isosceles Triangle Theorem. If two angles of a triangle are congruent , then the sides opposite to these angles are congruent.

Converse of the Perpendicular Bisector Theorem

Converse of the Perpendicular Bisector Theorem: 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 angle bisector of an angle, then it is equidistant

Converse of Perpendicular Bisector Theorem

Converse of the Perpendicular Bisector Theorem: 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 angle bisector of an angle, then it is equidistant from the sides of the angle.

Converse of Same-Side (Consecutive) Interior Angles Theorem

Converse of the Same-Side Interior Angles Postulate. If two lines and a transversal form same-side interior angles that are supplementary, then the lines are parallel. If two lines and a transversal form alternate exterior angles that are congruent, then the lines are

Converse of Triangle Proportionality Theorem

Converse of the Triangle Proportionality Theorem: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Corresponding Angles Postulate

Corresponding Angles Postulate. The Corresponding Angles Postulate states that, when two parallel lines are cut by a transversal , the resulting corresponding angles are congruent .

No Name Similarity Theorem"

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are. Similar to each other and the original triangle. ΔABC ~ ΔACD ~ ΔCBD. C. D. D. B. A. A. B. C. C. Page 2. Geometric Mean Altitude Theorem: In a right triangle, the altitude from the. to the. divides the hypotenuse into two.

Side-Side-Side Congruence Postulate (SSS)

If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent. The "included angle" in SAS is the angle formed by the two sides of the triangle being used. The "included side" in ASA is the side between the angles being used.

Angle Bisector Theorem

If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent. Recall that the criteria for our congruence postulates have called for three pairs of congruent parts between triangles.

Converse of Corresponding Angles Postulate

If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel. If two lines and a transversal form alternate interior angles that are congruent, then the lines are parallel.

Parallel Lines Theorem

If two lines are cut by a transversal so that the alternate exterior angles are congruent, then these lines are parallel. Example 3: If you are given a pair of alternate exterior angles that are congruent, then the two lines cut by the transversal are parallel.

Perpendicular Lines Theorem

If two lines are cut by a transversal so that the alternate exterior angles are congruent, then these lines are parallel. Example 3: If you are given a pair of alternate exterior angles that are congruent, then the two lines cut by the transversal are parallel.

Base of an Isosceles Triangle

In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles.

Legs of an Isosceles Triangle

In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles.

Incenter Theorem

Incenter Definition. # The incenter of the standard triangle Te(m,n) # It is the meeting point of the three angle bisectors (hence of any two) # Its existence is proved in the IncenterExists Theorem.

Isosceles Triangle Theorem

Isosceles Triangle Theorems. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. The converse of the base angles theorem, states that if two angles of a triangle are congruent, then sides opposite those angles are congruent.

Multiplication Property of Equality

Multiplicative Property of Equality. The formal name for the property of equality that allows one to multiply the same quantity by both sides of an equation. This, along with the additive property of equality, is one of the most commonly used properties for solving equations. Property: If a = b then a.

Parallel Lines Theorem (related to slope)

Parallel lines have the same slope and will never intersect. Parallel lines continue, literally, forever without touching (assuming that these lines are on the same plane). On the other hand, the slope of perpendicular lines are the negative reciprocals of each other, and a pair of these lines intersects at 90 degrees.

Reflexive Property of Equality

Reflexive pretty much means something relating to itself. The reflexive property of equality simply states that a value is equal to itself. Further, this property states that for all real numbers, x = x. ... Again, it states simply that any value or number is equal to itself.

Side-Angle-Side Similarity Theorem (SAS)

SAS means side, angle, side, and refers to the fact that two sides and the included angle of a triangle are known. The SAS Similarity Theorem states that if two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar.

Symmetric Property of Congruence

Segment Addition Postulate Point B is a point on segment AC, i.e. ... Addition Postulate If equal quantities are added to equal quantities, the sums are equal. Transitive Property If a = b and b = c, then a = c. Reflexive Property A quantity is congruent (equal) to itself. a = a Symmetric Property If a = b, then b = a.

Side-Angle-Side Congruence Postulate (SAS)

Side-Angle-Side Postulate. If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent

Symmetric Property of Equality

Symmetric Property of Equality. The following property: If if a = b then b = a. This is one of the equivalence properties of equality. See also. Reflexive property of equality, transitive property of equality, transitive property of inequalities.

Transitive Property of Congruence

That's transitivity. And if a = b and b < c, then a < c. ... Transitive Property (for three segments or angles): If two segments (or angles) are each congruent to a third segment (or angle), then they're congruent to each other.

Alternate Interior Angles Theorem

The Alternate Interior Angles Theorem states that, when two parallel lines are cut by a transversal , the resulting alternate interior angles are congruent .

Angle Addition Postulate

The Angle Addition Postulate states that: If point B lies in the interior of angle AOC, then. . The postulate describes that putting two angles side by side with their vertices together creates a new angle whose measure equals the sum of the measures of the two original angles.

Converse of Alternate Exterior Angles Theorem

The proof of this theorem is very similar to that of the Alternate Interior Angles Theorem. Converse of the Alternate Exterior Angles Theorem: If two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel.

Reflexive Property of Congruence

The reflexive property of congruence is used to prove congruence of geometric figures. This property is used when a figure is congruent to itself. Angles, line segments, and geometric figures can be congruent to themselves. Congruence is when figures have the same shape and size.

Substitution Property of Equality

The substitution property of equality, one of the eight properties of equality, states that if x = y, then x can be substituted in for y in any equation, and y can be substituted for x in any equation.

Subtraction Property of Equality

The subtraction property of equality tells us that if we subtract from one side of an equation, we also must subtract from the other side of the equation to keep the equation the same. ... It is the same with equations. To keep them the same, you have to do the same to both sides of the equation.

Right Angle Congruence Theorem

Theorem. If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent. Recall that the criteria for our congruence postulates have called for three pairs of congruent parts between triangles

Alternate Exterior Angles Theorem

This is where you get the alternate exterior angles theorem, which states that when you have a pair of parallel lines that are cut by a transversal, the alternate exterior angles are congruent. ... Remember, you will have congruent alternate exterior angles only when the two lines are parallel.

Triangle Inequality Theorem

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

Alternate Exterior Angles

When two lines are crossed by another line (called the Transversal): Alternate Exterior Angles are a pair of angles on the outer side of each of those two lines but on opposite sides of the transversal. In this example, these are two pairs of Alternate Exterior Angles: a and h.

Median

a line segment joining a vertex to the midpoint of the opposing side, bisecting it.

Perpendicular Bisector

a line that divides a line segment into two equal parts. It also makes a right angle with the line segment.

Right Triangle

a triangle with a right angle.

Angle-Angle-Side Congruence Postulate (AAS)

angle Angle Side Postulate. The Angle Angle Side postulate (often abbreviated as AAS) states that if two angles and the non-included side one triangle are congruent to two angles and the non-included side of another triangle, then these two triangles are congruent.

Converse of the Angle Bisector Theorem

angle Bisector Theorem Converse: If a point is in the interior of an angle and equidistant from the sides, then it lies on the bisector of the angle. ... A point is on the angle bisector of an angle if and only if it is equidistant from the sides of the triangle.

Division Property of Equality

division property of equality. It is a pretty simple property. It states that if you divide one side of an equation by a number, you also must divide the other side by the same number so that your equation stays the same.

Vertical Angles

each of the pairs of opposite angles made by two intersecting lines.

Supplementary Angles

either of two angles whose sum is 180°.

Equiangular

having equal angles.

Angle-Angle Similarity Postulate (AA)

he AA similarity postulate and theorem makes it even easier to prove that two triangles are similar. In the interest of simplicity, we'll refer to it as the AA similarity postulate. The postulate states that two triangles are similar if they have two corresponding angles that are congruent or equal in measure.

Triangle (Angle) Sum Theorem

he Angle Sum Theorem gives an important result about triangles, which is used in many algebra and geometry problems. We give the proof below. Theorem: The sum of the measures of the interior angles of a triangle is .

Exterior Angle

he angle between a side of a rectilinear figure and an adjacent side extended outward.

Distributive Property

he distributive property tells us how to solve equations in the form of a(b + c). The distributive property is sometimes called the distributive law of multiplication and division. ... Then we need to remember to multiply first, before doing the addition!

Angle-Side-Angle Congruence Postulate (ASA)

he included side means the side between two angles. In other words it is the side 'included between' two angles. In which pair of triangles pictured below could you use the Angle Side Angle postulate (ASA) to prove the triangles are congruent?

Equilateral Triangle

n geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.

Triangle Angle Bisector Theorem

n geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.

If two angles of a triangle are not congruent, then the linger side is opposite the larger angle

orollary to the triangle exterior angle theorem. the measure of the exterior angle of a triangle is greater than the measure of each remote interior angle. triangle side comparison theorem. if two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. triangle angle comparison theorem. if two ...

Segment Addition Postulate

segment Addition Postulate Defined. The segment addition postulate states that if we are given two points on a line segment, A and C, a third point B lies on the line segment AC if and only if the distances between the points meet the requirements of the equation AB + BC = AC.

Corresponding Angles

the angles that occupy the same relative position at each intersection where a straight line crosses two others. If the two lines are parallel, the corresponding angles are equal.

Same-Side (Consecutive) Interior Angles Theorem

the consecutive interior angles theorem states that when the two lines are parallel, then the consecutive interior angles are supplementary to each other. Supplementary means that the two angles add up to 180 degrees.

Converse of Pythagorean Theorem

the converse of the Pythagorean Theorem is also true. ... Pythagorean Theorem Converse: If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle

If two lines are parallel to the same line, the lines are parallel to each other

If two lines are parallel to the same line, then they are parallel to each other. Coplanar Right Angles Slopes are opposite reciprocals Page 2 In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

If two lines are perpendicular to the same transversal, then they are parallel

If two lines are parallel to the same line, then they are parallel to each other. In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.

If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular.

If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. If two sides of two adjacent acute angles are perpendicular, then the angles are complementary. ... If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

Pythagorean Inequalities Theorem

The Pythagorean Inequality is a generalization of the Pythagorean Theorem, which states that in a right triangle with sides of length we have . This Inequality extends this to obtuse and acute triangles. The inequality says: For an acute triangle with sides of length , .

Triangle Midsegment Theorem

The Triangle Midsegment Theorem: "In a triangle, the segment joining the midpoints of any two sides will be parallel to the third side and half its length." Consider the triangle below: Construct a line through C that is parallel to AB.

The acute angles of a right triangle are complementary

The acute angles of a right triangle are complementary. given: triangle ABC WITH RIGHT ANGLE C PROVE: ANGLE A AND ANGLE B are complementary. m(A) + m(B) = 90, hence they're complementary.

Converse of Alternate Interior Angles Theorem

The converse of this theorem, which is basically the opposite, is also a proven statement: if two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.

Perpendicular Transversal Theorem

The converse perpendicular transversal theorem tells you in a plane, if two lines are perpendicular to the same line, then they are parallel. These theorems are used to help you prove that two angles are congruent or that two lines are parallel.

Exterior Angle Theorem

The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate.

Side-Angle Inequality Theorem

Triangle Inequalities: Sides and Angles. ... There are two important theorems involving unequal sides and unequal angles in triangles. They are: Theorem 36: If two sides of a triangle are unequal, then the measures of the angles opposite these sides are unequal, and the greater angle is opposite the greater side.

Triangle Proportionality Theorem

Triangle Proportionality Theorem. If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally.

Two-Transversals Proportionality Corollary

Triangle Proportionality Theorem. Interactive help to prove the triangle proportionality theorem. A transversal is a line that intersects two or several lines. Theorem: A transversal that is parallel to one of the sides in a triangle divides the other two sides proportionally.

Circumcenter

he triangle can have, the circumcenter is the point where the perpendicular bisectors of a triangle intersect. The circumcenter is also the center of the triangle's circumcircle - the circle that passes through all three of the triangle's vertices.

Third Angles Theorem

hird Angle Theorem. If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles is also congruent

Hypotenuse-Leg Congruence Theorem (HL)

hypotenuse leg theorem, or HL theorem. This theorem states that 'if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.' This is kind of like the SAS, or side-angle-side postulate.

Perpendicular Lines theorem (related to slope)

if two lines have the same slope, then the lines are nonvertical parallel lines. If two lines are perpendicular and neither one is vertical, then one of the lines has a positive slope, and the other has a negative slope. Also, the absolute values of their slopes are reciprocals.

Incenter

incenter is always inside the triangle. It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle.

Orthocenter

is the point where the three altitudes of a triangle intersect. A altitude is a perpendicular from a vertex to its opposite side.

Hypotenuse

the longest side of a right triangle, opposite the right angle.

Perpendicular Bisector Theorem

the perpendicular bisector theorem - the theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the segment's endpoints. ... The converse states that if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Transitive Property of Equality

transitive Property of Equality. The following property: If a = b and b = c, then a = c. One of the equivalence properties of equality. Note: This is a property of equality and inequalities. (Click here for the full version of the transitive property of inequalities.)

If two sides of a triangle are not congruent, then the larger angle is opposite the longer side

two sides of one triangle are congruent to two sides of another, and the included angles are not congruent then the longer third side is opposite the larger included angle. 21 more rows

Vertical Angles Theorem

when two right triangles are congruent to one another. The LA theorem, or leg-acute, and LL theorem, or leg-leg, are useful shortcuts for proving congruence.