Chapter 4. Congruent triangles

ANGLES

An angle is a set of points consisting of two rays, with a common endpoint called THE VERTEX of the angle. The rays are callled SIDES or LEGS of the angle

An angle can be classified according its measurement:

Acute angle: Measure between 0 to 90 degrees

Right angle: Measure 90 degrees

Obtuse angle: Measure between 90 to180 degrees

Straight angle: Measure 180 degrees

Reflex angle: Measure between180 to 360 degrees

In addition, an angle can be classified according its characteristics and relationship:

Adjacent angles: They have the same vertex and a common side, but they don’t share any interior point.

Linear pair angles: Two adjacent angles which sum 180 degrees

ANGLES

Vertical angles: Are the opposite pair of angles formed when two lines intersect. They are always congruent

Complementary angles: Angles adjacents or not, which sum 90 degrees

ANGLES

Supplementary angles: Are angles which sum is 180 degrees

ANGLES

Angles between parallel lines crossed by a transversal

-) Co-interior angles: 2-5, 3-8-) Co-exterior angles: 1-6, 4-7-) Vertical angles: 1-3, 2-4, 5-7, 6-8-) Corresponding angles: 1-5, 2-6, 4-8, 3-7-) Alternate interior angles: 2-8, 3-5-) Alternate exterior angles: 1-7, 4-6

ANGLES

Congruents angles

TRIANGLE

Definition:

3-sides geometric figure. The points of the intersection of the sides are called VERTEX.

TRIANGLES CLASSIFICATION: SidesEQUILATERAL TRIANGLE

The Equilateral triangle has three equal sides and three equal angles.

Each angle is 60°

TRIANGLES CLASSIFICATION : SidesISOSCELES TRIANGLE

The Isosceles has two equal sides forming two equal angles with the

base.

TRIANGLES CLASSIFICATION : SidesSCALENE TRIANGLE

The Scalene Triangle has no congruent sides. In other words, each

side must have a different length.

TRIANGLES CLASSIFICATION : AnglesACUTE TRIANGLE

The Acute Triangle has three acute angles (an acute angle measures l

less than 90°)

TRIANGLES CLASSIFICATION : AnglesOBTUSE TRIANGLE

The Obtuse Triangle has an obtuse angle (an obtuse angle has more

than 90°). In the picture the shaded angle is the obtuse angle that

distinguishes this triangle

Since the total degrees in any triangle is 180°, an obtuse triangle can

only have one angle that measures more than 90°.

TRIANGLES CLASSIFICATION : AnglesRIGHT TRIANGLE

The Right Triangle has one 90° angle.

TRIANGLES CLASSIFICATION

TRIANGLES

Some properties:

A. The sum of the interior angles of a triangle is 180 degrees.

B. An exterior angle is the angle formed by a side and the extension of one of its adjacent sides

In the graphic, 120 degree is an external angle

TRIANGLESB. The Sum of an exterior and an interior angle of any triangle is 180

degrees; so they are supplementary angles

C. The measure of an exterior angle of a triangle is equal to the sum of the two interior angles that are not adjacent to it.

In the graphic, 120 = x + 45

In the graphic, <y + 120 = 180degree => <y = 180 – 120 = 60 degreewhere, <y is an internal angle and 120 degree is an external angle

TRIANGLESD. The shortest side is opposite to the smallest angle, and the longest side is

opposite to the longest angle

E. Any side of a triangle is shorter than the sum of the measure of the length of the other two sides