G3: Angles

Angle and Points ray vertex ray 5/24/2019 Angle and Points An Angle is a figure formed by two rays with a common endpoint, called the vertex. ray vertex ray Angles can have points in the interior, in the exterior or on the angle. A E D B C Points A, B and C are on the angle. D is in the interior and E is in the exterior. B is the vertex.

5/24/2019 Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide Using 3 points: vertex must be the middle letter This angle can be named as Using 1 point: using only vertex letter * Use of this method is permitted when the vertex point is the vertex of one and only one angle. Since B is the vertex of only this angle, this can also be called . A C B

Naming an Angle - continued 5/24/2019 Naming an Angle - continued Using a number: A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named as . A B 2 C * The “1 letter” name is unacceptable when … more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present.

Example Therefore, there is NO in this diagram. 5/24/2019 Example K is the vertex of more than one angle. Therefore, there is NO in this diagram. There is

5 Types of Angles Acute Angle: Right Angle: Obtuse Angle: 5/24/2019 5 Types of Angles Acute Angle: an angle whose measure is less than 90. Right Angle: an angle whose measure is exactly 90 . Obtuse Angle: an angle whose measure is between 90 and 180. Straight Angle: an angle that is exactly 180 . Reflex Angle: an angle whose measure is greater than 180.

5/24/2019 Adding Angles When you want to add angles, use the notation m1, meaning the measure of 1. If you add m1 + m2, what is your result? m1 + m2 = 58. m1 + m2 = mADC also. Therefore, mADC = 58.

Angle Addition Postulate 5/24/2019 Angle Addition Postulate Postulate: The sum of the two smaller angles will always equal the measure of the larger angle. Complete: m  ____ + m  ____ = m  _____ MRK KRW MRW

Congruent Angles Definition: 5/24/2019 Congruent Angles Definition: If two angles have the same measure, then they are congruent. Congruent angles are marked with the same number of “arcs”. The symbol for congruence is  3 5 Example: 3   5.

5/24/2019 Angle Bisector An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles. Example: Since 4   6, is an angle bisector. 5 3

Adjacent Angles Definition: A pair of angles with a shared vertex and common side but do not have overlapping interiors. Examples: 1 and 2 are adjacent. 3 and 4 are not. 1 and ADC are not adjacent. 4 3 Adjacent Angles( a common side ) Non-Adjacent Angles

Complementary Angles Definition: A pair of angles whose sum is 90˚ Examples: Adjacent Angles ( a common side ) Non-Adjacent Angles

Supplementary Angles Definition: A pair of angles whose sum is 180˚ Examples: Adjacent supplementary angles Non-Adjacent Angles

Linear Pair Postulate Postulate: If two angles form a linear pair, then they are supplementary  

Vertical Angles Congruence Theorem Vertical angles are non-adjacent angles formed by intersecting lines. Theorem: If two lines intersect then vertical angles are congruent. Examples:

What’s “Important” in Geometry? 4 things to always look for ! 180˚ 90˚ 360˚ Most of the rules (theorems) and vocabulary of Geometry are based on these 4 things. . . . and Congruence

HW pg 29 #6, 25, 40 pg 39 #17, 31, 39, 42 pg 128 #12, 28 Lesson 1-4: Angles