1. Introduction to Geometry

2. 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

3. 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

4.  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

5.  Supplementary angles: Are angles which sum is 180 degrees ANGLES

6. 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

7. TRIANGLE Definition: 3-sides geometric figure. The points of the intersection of the sides are called VERTEX.

8. TRIANGLES CLASSIFICATION: Sides EQUILATERAL TRIANGLE The Equilateral triangle has three equal sides and three equal angles. Each angle is 60°

9. TRIANGLES CLASSIFICATION : Sides ISOSCELES TRIANGLE The Isosceles has two equal sides forming two equal angles with the base.

10. TRIANGLES CLASSIFICATION : Sides SCALENE TRIANGLE The Scalene Triangle has no congruent sides. In other words, each side must have a different length.

11. TRIANGLES CLASSIFICATION : Angles ACUTE TRIANGLE The Acute Triangle has three acute angles (an acute angle measures l less than 90°)

12. TRIANGLES CLASSIFICATION : Angles OBTUSE 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°.

13. TRIANGLES CLASSIFICATION : Angles RIGHT TRIANGLE The Right Triangle has one 90° angle.

14. TRIANGLES CLASSIFICATION

15. 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

16. TRIANGLES B.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 degree where, <y is an internal angle and 120 degree is an external angle

17. TRIANGLES D.The shortest side is opposite to the smallest angle, and the longest side is opposite to the longest angle D.Any side of a triangle is shorter than the sum of the measure of the length of the other two sides

18. CONGRUENT TRIANGLES Triangles are congruent when they have exactly the same three sides and exactly the same three angles. These triangles are congruent: