Triangles

• Meaning of triangles

• Triangle - Wikipedia, the free encyclopedia

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C ...

• Geometry: Congruent Triangles - Math for Morons Like Us

Math for Morons Like Us - Geometry: Congruent Triangles ... Parallel Lines Congruent Tri. Congruent R. Tri.

• Triangle Geometry -- Triangles

A triangle is a polygon with three sides. There are of course triangles of different shapes and sizes. The applet below show triangles of many shapes and sizes.

Triangle (geometry)

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Introduction; Plane Triangles; Spherical Triangles

I

Introduction

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Triangle (geometr

y), geometric figure consisting of three points, called v

ertices, connected by three sides. In Euclidean plane ge

ometry, the sides are

straight line segments (see Fig. 1

). In spherical geometry, the sides are arcs of great circl

es (see Fig. 2). See Geometry; Trigonometry. The term triangle is sometimes used to d

escribe a geometric fi

gure having three vert

ices and sides that are arbitrary curves (see

Fig. 3).

II Plane Triangles Print this section A Euclidean plane triangle has three interior angles (the word interior is often omitted), each formed by two adjacent sides, as ÐCAB in Fig. 1, and six exterior angles, each formed by a side and the extension of an adjacent side, as ÐFEG in Fig. 4. A capital letter is customarily used to designate a vertex of a triangle, the angle at that vertex, or the measure of the angle in angular units; the corresponding lower case letter designates the side opposite the angle or its length in linear units. A side or its length is also designated by naming the endpoints; thus, in Fig. 1, the side opposite angle A is a or BC.

An angle A is acute if 0° < a =" 90°;">

A triangle is scalene if no two of its sides are equal in length, as in Fig. 1; isosceles if two sides are equal, as in Fig. 6; and equilateral if all three sides are equal, as in Fig. 7. The side opposite the right angle in a right triangle, HK in Fig. 5, is called the hypotenuse; the other two sides are called legs. The two equal sides of an isosceles triangle are also called legs, and the included angle is called the vertex angle. The third side is called the base, and the adjacent angles are called the base angles.

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If two sides of a triangle are of unequal size, the angles opposite are of unequal size; the larger side is opposite the larger angle. In the same way, if two angles are unequal, the sides opposite are unequal. Therefore, the three angles of a scalene triangle are of different sizes; the base angles of an isosceles triangle are equal; and the three angles of an equilateral triangle are equal (an equilateral triangle is also equiangular). The side opposite a right angle or an obtuse angle is the longest side of the triangle. see Trigonometry: Law of Sines for the exact relation between the sizes of the angles and sides of a triangle.

In any triangle, an altitude is a line through a vertex perpendicular to the opposite side or its extension, for example, DX in Fig. 8a and 8b. A median is the line determined by a vertex and the midpoint of the opposite side, AN in Fig. 9. An interior angle bisector is the line through a vertex bisecting the interior angle at the vertex, AR in Fig. 10; similarly, an exterior angle bisector bisects the exterior angle at the vertex, AV in Fig. 10. A perpendicular bisector is a line perpendicular to a side at its midpoint, HK in Fig. 11. The terms altitude, median, and interior angle bisector are also often applied to the line segment determined by the vertex and the intersection with the opposite side or its extension.

The three altitudes of a triangle meet in a common point called the orthocenter, O in Fig. 8a and 8b. The three medians meet in a point called the centroid, M in Fig. 9. The centroid divides a median through it into two segments, and the segment having a vertex as endpoint is always twice as long as the other segment. The three angle bisectors meet in a point called the incenter (I in Fig. 10), and the three exterior angle bisectors meet in three points called the excenters, U, V, W of Fig. 10. The incenter and the three excenters are centers of circles tangent to the sides or side extensions of the triangle. The three perpendicular bisectors meet in a point called the circumcenter, H in Fig. 11, which is the center of a circle passing through the three vertices.

If a, b, c are the three sides of a triangle, and h_a is the altitude through vertex A, the semiperimeter s is given by the formula s = y (a + b + c), and the area K by the formula K = yah_a = ÅÆÇÈÇÉÊ. Many other formulas interrelate the various parts of a triangle.

• Vertical angle Theorem Proofs

Vertical Angle Theorem: If two angles are vertical angles, then they have equal measures.

Given:

Statements/Conclusion	Justification
1) BEA and CED are vertical angles.	1) Given
2) and are opposite rays. and are opposite rays.	2) Definition of vertical angles
3) 1 and 3 are a linear pair. 2 and 3 are a linear pair.	3) Definition of linear pair
4) 1 and 3 are supplementary angles. 2 and 3 are supplementary angles.	4) Linear Pair Theorum
5) m1 + m3 = 180 degrees. m2 + m3 = 180 degrees.	5) Definition of supplementary angles
6) m1 + m3 = m2 + m3	6) Substitution Property
7) m3 = m3*	7) Reflexive Property
8) m1 = m2	8) Subtraction Property of Equality

* Step 7 is usually not necessary and is only included for clarification.

· Postulates

Angle Side Angle Postulate

Proving Congruent Triangles with ASA

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

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Theorems and Postulates for proving triangles congruent
Hypotenuse Leg Theorem | Side Side Side | Side Angle Side | Angle Side Angle | Angle Angle Side |isosceles triangle proofs|CPCTC | indirect proof| quiz on all theorems/postulates

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Example of Angle Side Angle Proof

ABC XYZ

• Two angles and the included side are congruent
• CAB = ZXY (angle)
• AB = XY (side)
• ACB = XZY (angle)
• Therefore, by the Angle Side Angle postulate (ASA), the triangles are congruent.

Included Side

The included side means the side between two angles. In other words it is the side 'included between' two angles.

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Side Angle Side Postulate

Proving Congruent Triangles with SAS

The Side Angle Side postulate (often abbreviated as SAS) states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent.

Example 1

ABC XYZ
• Two sides and the included angle are congruent
  • AC = ZX (side)
  • ACB = XZY (angle)
  • CB = ZY (side)
• Therefore, by the Side Angle Side postulate, the triangles are congruent.

Included Angle

The included angle means the angle between two sides. In other words it is the angle 'included between' two sides.

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Side Side Side Postulate

Proving Congruent Triangles with SSS

The Side Side Side postulate states that if three sides of one triangle are congruent to three sides of another triangle, then these two triangles are congruent.
Example 1

ABC XYZ
• Alll 3 sides are congruent
  • ZX = CA (side)
  • XY = AB (side)
  • YZ = BC (side)
• Therefore, by the Side Side Side postulate, the triangles are congruent.

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Angle Angle Side Postulate

Proving Congruent Triangles with AAS

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 angle of another triangle, then these two triangles are congruent.

Example of Angle Angle Side Proof (AAS)

ABC XYZ
• Two angles and a non-included side are congruent
  • CAB = ZXY (angle)
  • ACB = XZY (angle)
  • AB = XY (side)
• Therefore, by the Angle Angle Side postulate (AAS), the triangles are congruent.

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Hypotenuse Leg Theorem

Proving Congruent Right Triangles

The hypotenuse leg theorem states that any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg are congruent triangles.
Example 1
Given AB = XZ, AC = ZY, ACB = ZYX = 90°
Prove ABC XYZ

• ABC and XZY are right triangles since they both have a right angle
• AB = XZ (hypotenuse) reason: given
• AC = ZY (leg) reason: given
• ABC XYZ by the hypotenuse leg theorem which states that two right triangles are congruent if their hypotenuses are congruent and a corresponding leg is congruent.

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The following proof simply shows that it does not matter which of the two (corresponding) legs in the two right triangles are congruent.

Example 2
Given AB = XZ, CB = XY, ACB = ZYX = 90°
Prove ABC XYZ

• ABC and XZY are right triangles since they both have a right angle
• AB = XZ (hypotenuse) reason: given
• CB = XY (leg) reason: given
• ABC XYZ by the hypotenuse leg theorem which states that two right triangles are congruent if their hypotenuses are congruent and a corresponding leg is congruent.

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CPCTC

Corresponding parts of congruent triangles are congruent

CPCTC is a short hand acronym for the phrase

'corresponding parts of congruent triangles are congruent'

What does CPCTC mean? It means that once we know that two triangles are congruent, we know that all corresponding sides and angles are congruent!

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How to prove triangles are congruent with SSS, SAS, AAS, and other postulates

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Remember the meaning of congruent triangles. Two triangles are congruent if all 3 sides and 3 angles of one triangle equal all 3 sides and all 3 angles of the other triangle. CPCTC takes advantage of this fact. The general method is to first prove that 2 triangles are congruent and then use that knowledge to prove that a certain pair of corresponding sides or angles are congruent.

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Isosceles Triangle Proofs

Theorems and practice

Proofs involving isosceles triangles often require special consideration because an isosceles triangle has several distinct properties that do not apply to normal triangles. Therefore, when you are trying to prove that two triangles are congruent, and one or both triangles, are isosceles you have a few theorems that you can use to make your life easier.

Isosceles Triangle

An isosceles triangle has two congruent sides and two congruent angles. The congruent angles are called the base angles and the other angle is known as the vertex angle. BAC and BCA are the base angles of the triangle picture on the left. The vertex angle is ABC

Isosceles Triangle Theorems

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

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