Mathematics: Geometry - Angles

Angles and Vertices
An angle is formed when two lines or line segments meet at a common point. It may be a common starting point for a pair of segments or rays, or it may be the intersection of lines. Angles are represented by the symbol ∠.
The vertex is the point at which two segments or rays meet to form an angle. If the angle is formed by intersecting rays, lines, and/or line segments, the vertex is the point at which four angles are formed. The pairs of angles opposite one another are called vertical angles, and their measures are equal. A. acute angle is an angle with a degree measure less than 90°.
A right angle is an angle with a degree measure of exactly A. obtuse angle is an angle with a degree measure greater than 90° but less than 180°.
A straight angle is an angle with a degree measure of exactly 180°. This is also a semicircle.
A reflex angle is an angle with a degree measure greater than 180° but less than 360°.
A full angle is an angle with a degree measure of exactly 360°.

Relationships between Angles
Two angles whose sum is exactly 90° are said to be complementary.
The two angles may or may not be adjacent. In a right triangle, the two acute angles are complementary.
Two angles whose sum is exactly 180° are said to be supplementary.
The two angles may or may not be adjacent. Two intersecting lines always form two pairs of supplementary angles. Adjacent supplementary angles will always form a straight line.

Two angles that have the same vertex and share a side are said to be adjacent.


Vertical angles are not adjacent because they share a vertex but no common side.



When two parallel lines are cut by a transversal, the angles that are between the two parallel lines are interior angles.

In the diagram below, angles 3, 4, 5, and 6 are interior angles. that are outside the parallel lines are exterior angles. In the diagram below, angles 1, 2, 7, and 8 are exterior angles. that are in the same position relative to the transversal and a parallel line are corresponding angles. The diagram below has four pairs of corresponding angles: angles 1 and 5, angles 2 and 6, angles 3 and 7, and angles 4 and 8. Corresponding angles formed by parallel lines are congruent.
When two parallel lines are cut by a transversal, the two interior angles that are on opposite sides of the transversal are called alternate interior angles. In the diagram below, there are two pairs of alternate interior angles: angles 3 and 6, and angles 4 and 5. Alternate interior angles formed by parallel lines are congruent.
When two parallel lines are cut by a transversal, the two exterior angles that are on opposite sides of the transversal are called alternate exterior angles.

In the diagram below, there are two pairs of alternate exterior angles: angles 1 and 8, and angles 2 and 7.

Alternate exterior angles formed by parallel lines are congruent.



When two lines intersect, four angles are formed. The non-adjacent angles at this vertex are called vertical angles.

Vertical angles are congruent.
In the diagram,

and
.

 

P1. Find the measure of angles (a), (b), and (c) based on the figure with two parallel lines, two perpendicular lines and one transversal:

P1. (a) The vertical angle paired with (a) is part of a right triangle with the 40° angle. Thus the measure can be found:



The triangle formed by the supplementary angle to (b) is part of a triangle with the vertical angle paired with (a) and the given angle of 27°. Since
:

(c) As they are part of a transversal crossing parallel lines, angles (b) and (c) are supplementary. Thus