Angles, Triangle Centers and Key Geometry Concepts

Angles and Angle Measurement

Angle: An angle is the geometric figure formed by two rays (or lines) that start from a common point called the vertex.

Measuring Angles

Degrees: The magnitude of an angle can be measured in degrees (°). In the sexagesimal system the circle is divided into 360 equal parts; each part is one degree.

Radians: The radian is the angle subtended by an arc equal in length to the radius of the circle. Radians provide the circular or radian system of measurement.

Positive and Negative Angles

Positive angle: Measured when the rotation from the initial side to the terminal side is counterclockwise.

Angle of elevation: An angle measured from the horizontal line upward.

Angle of depression: An angle measured from the horizontal line downward.

Parallel Lines and a Transversal

When two parallel lines are cut by a transversal, eight angles are formed. Several pairs of these angles are equal or supplementary depending on their positions.

If a line intersects two parallel lines, alternate interior angles are those located between the parallels on opposite sides of the transversal. The angles 2 and 3 are equal.

Consecutive interior (same-side interior) angles
Consecutive interior angles are non-adjacent, lie on the same side of the transversal and inside the parallels. They are supplementary:

A + A' = 180°

Same-side (internal) angles:
Angles that lie on the same side of the transversal and inside the parallel lines are supplementary (their sum is 180°).

Pythagorean Theorem

Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.

Triangle Centers and Notable Points

STRAIGHT AND NOTEWORTHY POINTS OF A TRIANGLE

Perpendicular bisectors: The perpendicular bisectors of a triangle are the lines perpendicular to each side at its midpoint. The point where the three perpendicular bisectors intersect is the circumcenter, the center of the circumcircle.

Altitudes: The altitudes of a triangle are the lines dropped from each vertex perpendicular to the opposite side. The point where the three altitudes intersect is the orthocenter.

Angle bisectors: The angle bisectors of a triangle are the lines that divide each angle into two equal angles. The point where the three angle bisectors intersect is the incenter, the center of the inscribed circle.

Medians: The medians of a triangle are the lines that go from each vertex to the midpoint of the opposite side. The point where the three medians intersect is the centroid.

• Circumcenter: Intersection of perpendicular bisectors — equidistant from the vertices.
• Incenter: Intersection of angle bisectors — equidistant from the sides (center of inscribed circle).
• Centroid: Intersection of medians — center of mass; it divides each median in a 2:1 ratio.
• Orthocenter: Intersection of altitudes.

Draw two triangles, one isosceles and one equilateral, and determine the circumcenter, the orthocenter, the incenter and the centroid.