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2-8 STUDY GUIDE & INTERVENTION – PROVEN ANGLES
Basic Relationships in Angles
Pairs of Angles:
Adjacent angles are two angles that lie in the same plane and have a common vertex and a common side, but no common interior points.
Adjacent angles are the angles formed by sharing a pair of parallel lines. Adjacent angles are always equal and have a sum of 180 degrees, because two lines form one right angle when they meet at two points.
Adjacent angles can be used to represent pairs of angles that have specific relationships with each other. s. A pair of adjacent angles with noncommon sides that are opposite rays is called a linear pair. Vertical angles are two nonadjacent angles formed by two intersecting lines.
Adjacent angles are angles which measure 90 degrees. adjacent angles are always adjacent, that is they have a common side and the angle opposite another angle is adjacent to it.
A right angle is the intersection of a right triangle and the horizontal axis. It exists when one side of the right triangle is the length of one side of its adjacent circle. The adjacent angles are those whose sum is 180°.
An angle is a special type of geometric object. Adjacent angles are generally two angles that are directly opposite from one another. For example, if you have two right angles then one is adjacent to the other.
Adjacent angles are angles that share common sides and vertices. They add up to right angles, or 180 degrees when added together.
QUICK HINTS ON FINDING ADJACENT ANGLES:
The angle opposite any given acute angle is called an adjacent angle will measure 90 degrees.
Adjacent angles are two non-collinear angles of a polygon which share a side.
Given that they share a side, the pattern of adjacent angles is predictable.
Example of Adjacent angle is shown below.
Proving Angle Relationships
Supplementary angles are supplementary or “helping” angles. These are the angles which define a polygon, and they can be identified by looking at how they lie on the perimeter of their enclosing polygon. The sum of supplementary angles of a polygon is always greater than 180°.
A supplementary angle is an angle that is formed by two lines. When an angle is supplementary, it can only be measured using two straight lines: the straight line that forms the angle’s vertex and the straight line outside of which the vertex lies.
When you have supplementary angles (or additional arcs), the sum of their measures equals 180°. You can use a protractor to help you measure the measure of additional arcs, and in this case, we will not draw those lines of measure.
Complementary angles are angles formed when two rays meet at a point, or the lines that define two segments that share exactly the same endpoint. The complementary angle is sometimes called an alternate interior angle or reverse exterior angle.
A complementary angle is an angle formed by two straight lines in a plane. It is equal to the angle whose measure is quadrilateral root of 2 (cubic root of -1). We also say that two angles are complementary when they have a dot product of zero. Rotations so as to maximize either cosine or sine produce complementary rotations, as pictured below.
A complementary angle is a pair of angles with a sum of 180°. The sum of two angles that are supplementary is equal to 90°. The difference between two angles is the opposite side and adjacent angle.
Difference between supplementary and complementary angles:
Complementary angles have equal measures, while supplementary angles do not. You can combine these two concepts by giving an example of how they are different, and how they are the same.
Supplementary and Complementary Angles:
There are two basic postulates for working with angles. The Protractor Postulate assigns numbers to angle measures, and the Angle Addition Postulate relates parts of an angle to the whole angle.
Protractor Postulate: Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180.
Angle Addition Postulate: R is in the interior of ∠PQS if and only if m∠PQR + m∠RQS = m∠PQS.
The two postulates can be used to prove the following two theorems.
Supplement Theorem: If two angles form a linear pair, then they are supplementary angles.
Example: If ∠1 and ∠2 form a linear pair, then m∠1 + m∠2 = 180.
Complement Theorem: If the noncommon sides of two adjacent angles form a right angle,
then the angles are complementary angles.
Congruent and Right Angles
The Reflexive Property of Congruence, Symmetric Property of Congruence, and Transitive Property of Congruence all hold true for angles.
The following theorems also hold true for angles.
- Congruent Supplements Theorem Angles supplement to the same angle or congruent angles are congruent.
- Congruent Compliments Theorem Angles compliment to the same angle or to congruent angles are congruent.
Vertical Angles Theorem
If two angles are vertical angles, then they are congruent.
Theorem 2.9 Perpendicular lines intersect to form four right angles.
Theorem 2.10 All right angles are congruent.
Theorem 2.11 Perpendicular lines form congruent adjacent angles.
Theorem 2.12 If two angles are congruent and supplementary, then each angle is a right angle.
Theorem 2.13 If two congruent angles form a linear pair, then they are right angles.