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Angles

Explore the fundamental concepts of angles, including definitions, types such as acute, right, obtuse, and straight angles, and the angle addition postulate. Learn how to express angle measurements using three points with the vertex as the middle letter, a single point (vertex letter), or numerically. Discover how to create angles and find angle measures through examples and problem-solving scenarios. Understand the significance of angle bisectors and congruent angles, reinforced through clear illustrations and real-world applications.

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Angles

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  1. Angles

  2. 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.

  3. 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

  4. 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.

  5. Example Name the three angles in the diagram.

  6. 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 .

  7. Angle Addition PostulateSame 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

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

  9. 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.

  10. 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

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

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

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

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

  15. 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.

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

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

  18. Example: Given bisects < ABC. Find m < ABC

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