4.1

CPCTC: Corresponding Parts of Congruent Triangles are Congruent.

"Corresponding parts of congruent triangles are congruent" (CPCTC) is a succinct statement of a theorem regarding congruent trigonometry, defined as triangles either of which is an isometry of the other. ... CPCTC is useful in proving various theorems about triangles and other polygons.

- Perpendicular Bisector:

A perpendicular bisector is a special kind of segment, ray, or line that. (1) intersects a given segment at a 90° angle, and. (2) passes through the given segment's midpoint. Segment CD is the perpendicular bisector to segment AB. We derive two important theorems from the characteristics of perpendicular bisectors.

AAS Triangle Congruence Theorem:

AAS Postulate (Angle-Angle-Side) If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. In order to use this postulate, it is essential that the congruent sides not be included between the two pairs of congruent angles.

ASA Triangle Congruence Theorem:

ASA Postulate (Angle-Side-Angle) If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. In a sense, this is basically the opposite of the SAS Postulate.

SSS Triangle Congruence Theorem:

Congruent Triangles - Three sides equal (SSS) Definition: Triangles are congruent if all three sides in one triangle are congruent to the corresponding sides in the other. There are five ways to test that two triangles are congruent. This is one of them (SSS).

Isosceles Triangle Theorem:

The Base Angles Theorem. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Converse of the Base Angles Theorem.

Triangle Sum Theorem:

The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.

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.

Equilateral Triangle Theorem:

The following corollaries of equilateral triangles are a result of the Isosceles Triangle Theorem: (1) A triangle is equilateral if and only if it is equiangular. (2) Each angle of an equilateral triangle has a degree measure of 60. The congruent sides of the triangle imply that all the angles are congruent.

SAS Triangle Congruence Theorem:

Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles. SSS (side, side, side) SSS stands for "side, side, side" and means that we have two triangles with all three sides equal. ... SAS (side, angle, side) ... ASA (angle, side, angle) ... AAS (angle, angle, side) ... HL (hypotenuse, leg)

perpendicular bisector

a line, ray, or segment that intersects a segment at its midpoint, forming angles of 90° When creating a kite, the longer vertical stick is attached to the midpoint of the shorter horizontal one. These sticks intersect at right angles. The vertical stick is the perpendicular bisector of the horizontal one. In a square, the diagonals are perpendicular bisectors of each other. The perpendicular bisectors of a triangle are concurrent, intersecting at the circumcenter.

equilateral polygon

a polygon in which all sides are equal The lengths of all the sides of an equilateral figure are equal in measure. A square is equilateral as all its sides are equal. The two triangles forming a star in the center of the Israeli flag are equilateral triangles. All regular polygons are equilateral.

triangle sum theorem

a proven statement that the sum of the measures of the interior angles of a triangle is 180° The triangular sum theorem is used to find the measure of an unknown interior angle of a triangle if the measures of the other two interior angles of the triangle are known. In a sail boat, if the measures of any two interior angles of the triangular sail are known, then the measure of the unknown interior angle can be found using the triangle sum theorem.

isosceles triangle

a three-sided polygon with two sides equal in length In an isosceles triangle, two sides are equal in length, and the angles opposite them are equal in measure. A pennant used to cheer a sports team is usually in the shape of an isosceles triangle. When the two legs of a right triangle are equal in length, it is an isosceles right triangle.

equiangular polygon

a two-dimensional closed shape in which all angles are congruent All regular polygons are equiangular. A square and a rectangle are equiangular polygons, since their interior angles are right angles. A yield traffic sign is an equiangular triangle, since its interior angles are congruent.

remote interior angle

an angle inside a polygon, not adjacent to the exterior angle When viewed from the side, the different parts of a folding deck chair intersect to form a triangle. One of the sides of this triangle can be extended to form an exterior angle. The angles inside the triangle that are not adjacent to the exterior angle are the remote interior angles.

interior angles

any of four angles located between a pair of lines cut by a transversal or an angle on the inside of a closed figure Alternate interior angles formed by two parallel lines intersected by a transversal are congruent. Same side interior angles formed by two parallel lines intersected by a transversal are supplementary. The angle between any two adjacent sides of a closed figure is an interior angle.

exterior angle

any of the four angles located outside a pair of lines, intersected by a transversal or an angle on the outside of a closed figure made by extending an adjacent side Alternate exterior angles formed by two parallel lines intersected by a traversal are congruent, while exterior angles on the same line are supplementary. The angle between a side of a polygon and the line extended from an adjacent side is an exterior angle of the polygon. The sum of all the exterior angles of any polygon is 360°.

HL Triangle Congruence Theorem:

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.'

Angle Bisector:

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.

interior (of angle / circle / polygon)

the region that contains the points that lie between the sides of an angle or inside a closed figure The stars on the U.S. flag are inside a blue rectangle. They belong to the interior of the rectangle. Similarly, the numbers on a circular clock lie in its interior.