Lesson PROPERTIES OF TRIANGLES

Source code of 'PROPERTIES OF TRIANGLES'

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<H2>Properties of triangles</H2>

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.
<TABLE>
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 <TD>
The lesson title

<A HREF=http://www.algebra.com/algebra/homework/Points-lines-and-rays/Points-and-Straight-Lines-basics.lesson>Points and Straight Lines basics</A>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

<A HREF=http://www.algebra.com/algebra/homework/Angles/Angles-basics.lesson>Angles basics</A>

<A HREF=http://www.algebra.com/algebra/homework/Angles/Vertical-Angles-.lesson>Vertical angles</A>

<A HREF=http://www.algebra.com/algebra/homework/Angles/Parallel-lines.lesson>Parallel lines</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Sum-of-the-interior-angles-of-a-triangle.lesson>Sum of the interior angles of a triangle</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Congruence-tests-for-triangles.lesson>Congruence tests for triangles</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Problems-on-congruence-tests-for-triangles.lesson>Problems on congruence tests for triangles</A>

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Problems-on-congruence-tests-for-triangles-2.lesson>Problems on congruence tests for triangles-2</A>

The lesson title

<A HREF=http://www.algebra.com/algebra/homework/Triangles/How-to-draw-a-congruent-segment-and-a-congruent-angle-using-a-compass-and-a-ruler.lesson>How to construct a congruent segment and 
a congruent angle using a compass and a ruler</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/How-to-draw-a-parallel-line-passing-through-a-given-point-using-a-compass-and-a-ruler.lesson>How to construct a parallel line passing through 
a given point using a compass and a ruler</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/How-to-draw-a-congruent-triangle-using-a-compass-and-a-ruler.lesson>How to construct a triangle 
using a compass and a ruler</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Isosceles-triangles.lesson>Isosceles triangles</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/An-altitude-a-median-and-an-angle-bisector-in-the-isosceles-triangle.lesson>An altitude, a median and an angle bisector 
in the isosceles triangle</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Properties-of-the-sides-of-parallelograms.lesson>Properties of the sides of parallelograms</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/The-line-segment-joining-the-midpoints-of-two-sides-of-a-triangle.lesson>The line segment joining the midpoints 
of two sides of a triangle</A>

The lesson title

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Altitudes-in-an-isosceles-triangle.lesson>Altitudes in an isosceles triangle</A>

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Medians-in-an-isosceles-triangle.lesson>Medians in an isosceles triangle</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Angles-and-sides-inequality-theorems-for-triangles.lesson>Angles and sides inequality theorems 
for triangles</A>

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Angle-bisectors-in-an-isosceles-triangle.lesson>Angle bisectors in an isosceles triangle</A>

The lesson title

<A HREF=http://www.algebra.com/algebra/homework/Triangles/A-perpendicular-bisector-of-a-segment.lesson>A perpendicular bisector of a segment</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/An-angle-bisector-properties.lesson>An angle bisector properties</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/How-to-bisect-a-segment-using-a-compass-and-a-ruler.lesson>How to bisect a segment using a compass 
and a ruler</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/-HOW-TO-bisect-an-angle-using-a-compass-and-a-ruler.lesson>How to bisect an angle using a compass 
and a ruler</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Perpendicular-bisectors-of-a-triangle-sides-are-concurrent.lesson>Perpendicular bisectors of a triangle sides 
are concurrent</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Angle-bisectors-of-a-triangle-are-concurrent.lesson>Angle bisectors of a triangle are concurrent</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Altitudes-of-a-triangle-are-concurrent.lesson>Altitudes of a triangle are concurrent</A>

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Properties-of-diagonals-of-a-parallelogram.lesson>Properties of diagonals of a parallelogram</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Medians-of-a-triangle-are-concurrent.lesson>Medians of a triangle are concurrent</A>

The lesson title

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/Median-in-a-right-triangle-drawn-to-its-hypotenuse.lesson>Median drawn to the hypotenuse 
of a right triangle </A>

<A HREF=http://www.algebra.com/algebra/homework/Pythagorean-theorem/The-Pythagorean-Theorem.lesson>The Pythagorean Theorem</A>

<A HREF=http://www.algebra.com/algebra/homework/Pythagorean-theorem/More-proofs-of-the-Pythagorean-Theorem.lesson>More proofs of the Pythagorean Theorem</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Law-of-sines.lesson>Law of Sines</A>

<A HREF=http://www.algebra.com/algebra/homework/Triangles/Law-of-sines-the-Geometric-Proof.lesson>Law of sines - the Geometric Proof</A>

<A HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/Solve-triangles-using-Law-of-Sines.lesson>Solve triangles using Law of Sines</A>

The lesson title

<A HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/Proof-of-the-Law-of-Cosines-revisited.lesson>Proof of the Law of Cosines revisited</A>

<A HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/Solve-triangles-using-Law-of-Cosines.lesson>Solve triangles using Law of Cosines</A>

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/On-what-segments-the-angle-bisector-divides-the-side-of-a-triangle.lesson>On what segments the angle bisector 
divides the side of a triangle</A>

<A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-length-of-a-median-of-a-triangle.lesson>The length of a median of a triangle</A>

......
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 <TD>
The property

The length of any side of a triangle does not exceed&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
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.

&nbsp;&nbsp;&nbsp;&nbsp;If a triangle has two congruent sides, then 
&nbsp;&nbsp;&nbsp;&nbsp;the angles opposite to these sides are congruent.
(In an isosceles triangle the angles at the base are congruent).
&nbsp;&nbsp;&nbsp;&nbsp;If a triangle has two congruent angles, then 
&nbsp;&nbsp;&nbsp;&nbsp;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,  
&nbsp;&nbsp;&nbsp;&nbsp;then the opposite sides are of equal length in pairs.
If in a quadrilateral two opposite sides are parallel and of  
&nbsp;&nbsp;&nbsp;&nbsp;equal length, then two other opposite sides are 
&nbsp;&nbsp;&nbsp;&nbsp;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 
&nbsp;&nbsp;&nbsp;&nbsp;of equal length.
If in a triangle the two altitudes are of equal length, then 
&nbsp;&nbsp;&nbsp;&nbsp;the triangle is isosceles.

If a triangle is isosceles, then the two medians are 
&nbsp;&nbsp;&nbsp;&nbsp;of equal length.
If in a triangle the two medians are of equal length, then 
&nbsp;&nbsp;&nbsp;&nbsp;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 
&nbsp;&nbsp;&nbsp;&nbsp;are of equal length.
If in a triangle the two angle bisectors are of equal length, 
&nbsp;&nbsp;&nbsp;&nbsp;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.

......
 </TD>
 <TD>
Under the topic

Points and Straight Lines basics&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

Angles, complementary, 
supplementary angles

Triangles

Geometry

Under the topic

Triangles

Under the topic

Geometry

Triangles

Geometry

Under the topic

Triangles

Geometry

Triangles

Under the topic

Geometry

Pythagorean theorem

Triangles

Trigonometry

Under the topic

Trigonometry

Geometry

......
 </TD>
 <TD>
In the section&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

Geometry

Word problems

In the section

Geometry

In the section

Word problems

Geometry

Word problems

In the section

Geometry

Word problems

Geometry

In the section

Word problems

Geometry

Algebra-II

In the section

Algebra-II

Word problems

......
 </TD>
 </TR>
</TABLE>

For the solutions of the typical word problems on triangles related to their sides and angles measures see the lessons 
&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-the-angles-of-triangles-Examples.lesson>HOW TO solve problems on the angles of triangles - Examples</A>,
&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-the-angles-of-isosceles-triangles-Examples.lesson>HOW TO solve problems on the angles of isosceles triangles - Examples</A>,
&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-the-triangle-sides-measures-Examples.lesson>HOW TO solve problems on the triangle sides measures - Examples</A>, and
&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/How-to-solve-problems-on-the-isosceles-triangle-sides-measures-Examples.lesson>HOW TO solve problems on the isosceles triangle sides measures - Examples</A> 
under the topic Geometry of the section Word problems in this site.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.