# Lesson PROPERTIES OF TRIANGLES

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## Properties of triangles

For your convenience, this file consolidates my lessons on triangles in this site. The file contains short annotations to the lessons and the major properties of triangles.
The properties are presented with the links to the corresponding lessons. The lessons are listed in the logical order, which means that every given lesson refers to the
preceding ones only and does not refer to that follow. The list consolidates the relevant lessons that are located under different topics and even in the different sections
in this site. At the end of the list there are links to the lessons on word problems related to triangles.
 The lesson title Points and Straight Lines basics                           Angles basics Vertical angles Parallel lines Sum of the interior angles of a triangle Congruence tests for triangles Problems on congruence tests for triangles Problems on congruence tests for triangles-2 The lesson title How to construct a congruent segment and a congruent angle using a compass and a ruler How to construct a parallel line passing through a given point using a compass and a ruler How to construct a triangle using a compass and a ruler Isosceles triangles An altitude, a median and an angle bisector in the isosceles triangle Properties of the sides of parallelograms The line segment joining the midpoints of two sides of a triangle The lesson title Altitudes in an isosceles triangle Medians in an isosceles triangle Angles and sides inequality theorems for triangles Angle bisectors in an isosceles triangle The lesson title A perpendicular bisector of a segment An angle bisector properties How to bisect a segment using a compass and a ruler How to bisect an angle using a compass and a ruler Perpendicular bisectors of a triangle sides are concurrent Angle bisectors of a triangle are concurrent Altitudes of a triangle are concurrent Properties of diagonals of a parallelogram Medians of a triangle are concurrent The lesson title Median drawn to the hypotenuse of a right triangle The Pythagorean Theorem More proofs of the Pythagorean Theorem Law of Sines Law of sines - the Geometric Proof Solve triangles using Law of Sines The lesson title Proof of the Law of Cosines revisited Solve triangles using Law of Cosines On what segments the angle bisector divides the side of a triangle The length of a median of a triangle ...... The property The length of any side of a triangle does not exceed                         the sum of lengths of the other two sides. Angles - what is this? Comparing angles. Measuring angles. Adding angles. Types of angles. Supplementary angles. Complementary angles. Vertical angles - definitions and examples. Vertical angles are congruent. Proof of the theorem and examples. Angles formed by two straight lines and a transverse line. Postulates on parallel lines. The sum of the interior angles of a triangle is equal to 180°. Three postulates are presented for congruency of triangles (SAS, ASA and SSS) with the detailed discussion. Problems on congruence tests for triangles are presented. If two sides in one triangle are congruent to the two sides in another triangle and the median drawn to one of them in one triangle is congruent to the median drawn to the congruent side in the second triangle, then the triangles are congruent. The property The first lesson on construction problems. It explains what the construction problems are and how to draw a congruent segment and a congruent angle using a compass and a ruler. This lesson explains how to construct a parallel line passing through a given point using a compass and a ruler. This lesson explains how to construct a triangle using a compass and a ruler for three basic cases: 1) the triangle is given by one side and the two adjacent interior angles; 2) the triangle is given by two sides and the included angle; 3) the triangle is given by its three sides.     If a triangle has two congruent sides, then     the angles opposite to these sides are congruent. (In an isosceles triangle the angles at the base are congruent).     If a triangle has two congruent angles, then     the sides opposite to these angles are congruent. (If a triangle has two congruent angles it is isosceles). In an isosceles triangle the altitude drawn to the base coincides with the median and the angle bisector. If in a quadrilateral the opposite sides are parallel in pairs,     then the opposite sides are of equal length in pairs. If in a quadrilateral two opposite sides are parallel and of     equal length, then two other opposite sides are     parallel and of equal length too. The straight line connecting midpoints of two sides of a triangle is parallel to the third side and is half of its length. If a straight line bisects one side of a triangle and is parallel to its second side, then it bisects the third side of the triangle. The property If a triangle is isosceles, then the two altitudes are     of equal length. If in a triangle the two altitudes are of equal length, then     the triangle is isosceles. If a triangle is isosceles, then the two medians are     of equal length. If in a triangle the two medians are of equal length, then     the triangle is isosceles. In a triangle, the angle opposite to the longer side is greater than the angle opposite to the shorter side. In a triangle, the side opposite to the greater angle is longer than the side opposite to the smaller angle. If two triangles have two sides correspondingly congruent in pairs, and the third sides are not equal, then the greater included angle corresponds to the greater third side. If two triangles have two sides correspondingly congruent in pairs, and the third sides are not equal, then the greater third side corresponds to the greater included angle. If a triangle is isosceles, then the two angle bisectors     are of equal length. If in a triangle the two angle bisectors are of equal length,     then the triangle is isosceles. The property A point is equidistant from the endpoints of a line segment if and only if the point lies on the perpendicular drawn through the midpoint of the line segment A point is equidistant from the sides of an angle if and only if the point lies on the angle bisector This lesson is on the construction problems. It explains how to bisect a segment using a compass and a ruler and how to construct a perpendicular to the given segment. This lesson is on the construction problems. It explains how to bisect an angle using a compass and a ruler. Perpendicular bisectors of a triangle sides are concurrent Angle bisectors of a triangle are concurrent Altitudes of a triangle are concurrent In a parallelogram, diagonals bisect each other. If in a quadrilateral the diagonals bisect each other, then the quadrilateral is a parallelogram. Medians of a triangle are concurrent The property In a right triangle, the median drawn to the hypotenuse, has the measure half the hypotenuse. If in a triangle the median has the measure half the length of the side it is drawn to, then the triangle is a right triangle. In a right triangle, the median drawn to the hypotenuse divides the triangle in two isosceles triangles. The Pythagorean Theorem. The geometric proof is presented close to that by Euclid in his books "Elements". Two more proofs of the Pythagorean Theorem are presented. First one is based on the layout consideration. Second one uses similar triangles. Law of Sines is proved. The Trigonometric proof. Examples are presented showing how to apply the Law of Sines to solve triangles. Law of Sines - more Geometric proof. Examples on solving triangles with the use the Law of Sines. All possible cases are considered you can meet when solving triangles with the use the Law of Sines. The property Proof of the Law of Cosines Examples on solving triangles with the use the Law of Cosines. All possible cases are considered you can meet when solving triangles with the use the Law of Cosines. In a triangle, an angle bisector divides the opposite side in two segments proportional to the ratio of the length of the two other sides of the triangle. The formula is derived to calculate the length of the median via the lengths of the three sides of the triangle. Examples are presented showing how to use this formula. ...... Under the topic Points and Straight Lines basics       Angles, complementary, supplementary angles Angles, complementary, supplementary angles Angles, complementary, supplementary angles Triangles Triangles Triangles Geometry Under the topic Triangles Triangles Triangles Triangles Triangles Triangles Triangles Under the topic Geometry Geometry Triangles Geometry Under the topic Triangles Triangles Triangles Triangles Triangles Triangles Triangles Geometry Triangles Under the topic Geometry Pythagorean theorem Pythagorean theorem Triangles Triangles Trigonometry Under the topic Trigonometry Trigonometry Geometry Geometry ...... In the section       Geometry Geometry Geometry Geometry Geometry Geometry Geometry Word problems In the section Geometry Geometry Geometry Geometry Geometry Geometry Geometry In the section Word problems Word problems Geometry Word problems In the section Geometry Geometry Geometry Geometry Geometry Geometry Geometry Word problems Geometry In the section Word problems Geometry Geometry Geometry Geometry Algebra-II In the section Algebra-II Algebra-II Word problems Word problems ......

For the solutions of the typical word problems on triangles related to their sides and angles measures see the lessons
HOW TO solve problems on the angles of triangles - Examples,
HOW TO solve problems on the angles of isosceles triangles - Examples,
HOW TO solve problems on the triangle sides measures - Examples, and
HOW TO solve problems on the isosceles triangle sides measures - Examples
under the topic Geometry of the section Word problems in this site.

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