Angles

Angle and Points ray vertex ray 4/19/2017 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 B is the vertex. E D Points A, B and C are on the angle. B C D is in the interior E is in the exterior.

Naming an Angle Vertex must be the middle letter 4/19/2017 Naming an Angle Using 3 points: Vertex must be the middle letter This angle can be named as Using 1 point: Using only vertex letter Using a number: A Use the notation m2, meaning the measure of 2. B C

Example K is the vertex of more than one angle. 4/19/2017 Example Name all the angles in the diagram below K is the vertex of more than one angle. Therefore, there is NO in this diagram.

Example Name the three angles in the diagram.

4 Types of Angles Acute Angle: Right Angle: Obtuse Angle: 4/19/2017 4 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 .

Angle Addition Postulate Same idea as the segment addition postulate 4/19/2017 Angle Addition Postulate Same idea as the segment 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

Example Fill in the blanks. m < ______ + m < ______ = m < _______

4/19/2017 Adding Angles If you add m1 + m2, what is your result? m1 + m2 = 58. Also… m1 + m2 = mADC Therefore, mADC = 58.

Example First, draw it! mMRK = 3x = 3•21 = 63º 4/19/2017 Example K is interior to MRW, m  MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK. First, draw it! 3x + x + 6 = 90 4x + 6 = 90 – 6 = –6 4x = 84 x = 21 3x x+6 Are we done? mMRK = 3x = 3•21 = 63º

Example Given that m< LKN = 145, find m < LKM and m < MKN

Example Given that < KLM is a straight angle, find m < KLN and m < NLM

Example Given m < EFG is a right angle, find m < EFH and m < HFG

4/19/2017 Angle Bisector An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles. 5 3

3   5. Congruent Angles Definition: 4/19/2017 Congruent Angles Definition: If two angles have the same measure, then they are congruent. Congruent angles are marked with the same number of “arcs”. 3 5 3   5.

Example: is an angle bisector J T Which two angles are congruent? <JUK and < KUT or < 4 and < 6

Example: Given bisects < XYZ and m < XYW = . Find m < XYZ

Example: Given bisects < ABC. Find m < ABC