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Warm up 9/2/14

Warm up 9/2/14. Answer the question and draw a picture if you can: 1) What is a right angle? 2) What is an acute angle? 3) What is an obtuse angle? 4) What is a vertical angle?. Angle Pair Relationships. Angle Pair Relationship Essential Questions.

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Warm up 9/2/14

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  1. Warm up 9/2/14 • Answer the question and draw a picture if you can: • 1) What is a right angle? • 2) What is an acute angle? • 3) What is an obtuse angle? • 4) What is a vertical angle?

  2. Angle Pair Relationships

  3. Angle Pair Relationship Essential Questions How do I prove geometric theorems involving lines, angles? • Standard: MCC9-12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

  4. Z Y XY and XZ are ____________. X Straight Angles Opposite rays ___________ are two rays that are part of a the same line and have only their endpoints in common. opposite rays The figure formed by opposite rays is also referred to as a ____________. A straight angle measures 180 degrees. straight angle

  5. S vertex SRT R TRS 1 T Naming Angles There are several ways to name this angle. 1) Use the vertex and a point from each side. or side The vertex letter is always in the middle. 2) Use the vertex only. 1 R side If there is only one angle at a vertex, then theangle can be named with that vertex. 3) Use a number.

  6. D 2 F DEF 2 E FED E Angles Symbols:

  7. A A A obtuse angle 90 < m A < 180 acute angle 0 < m A < 90 right angle m A = 90 Angle Measure Once the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle.

  8. 40° 110° 90° 50° 75° 130° Angle Measure Classify each angle as acute, obtuse, or right. Acute Obtuse Right Obtuse Acute Acute

  9. A B D C Adjacent Angles When you “split” an angle, you create two angles. The two angles are called _____________ adjacent angles adjacent = next to, joining. 2 1 1 and 2 are examples of adjacent angles. They share a common ray. Name the ray that 1 and 2 have in common. ____

  10. Adjacent Angles J 2 common side R M 1 1 and 2 are adjacent with the same vertex R and N Adjacent angles are angles that: A) share a common side B) have the same vertex, and C) have no interior points in common

  11. Adjacent Angles B 2 1 1 2 G N L 1 J 2 Determine whether 1 and 2 are adjacent angles. No. They have a common vertex B, but _____________ no common side Yes. They have the same vertex G and a common side with no interior points in common. No. They do not have a common vertex or ____________ a common side The side of 1 is ____ The side of 2 is ____

  12. Linear Pairs of Angles D A B 2 1 C Note: Two angles form a linear pair if and only if (iff): A) they are adjacent and B) their noncommon sides are opposite rays 1 and 2 are a linear pair.

  13. In the figure, and are opposite rays. H T E 3 A 4 2 1 C ACE and 1 have a common side the same vertex C, and opposite rays and M Linear Pairs of Angles 1) Name the angle that forms a linear pair with 1. ACE 2) Do 3 and TCM form a linear pair? Justify your answer. No. Their noncommon sides are not opposite rays.

  14. Vertical Angles When two lines intersect, ____ angles are formed. four There are two pair of nonadjacent angles. vertical angles These pairs are called _____________. 1 4 2 3

  15. Vertical Angles Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines. Vertical angles: 1 and 3 1 4 2 2 and 4 3

  16. Vertical Angles Vertical angles are congruent. n m 2 1  3 3 1 2  4 4

  17. 130° x° Vertical Angles Find the value of x in the figure: The angles are vertical angles. So, the value of x is 130°.

  18. Vertical Angles Find the value of x in the figure: The angles are vertical angles. (x – 10) = 125. (x – 10)° x – 10 = 125. 125° x = 135.

  19. Complementary and Supplementary Angles E D A 60° 30° F B C Two angles are complementary if and only if (iff) The sum of their degree measure is 90. mABC + mDEF = 30 + 60 = 90

  20. E D A 60° 30° F B C Complementary and Supplementary Angles If two angles are complementary, each angle is a complement of the other. ABC is the complement of DEF and DEF is the complement of ABC. Complementary angles DO NOT need to have a common side or even the same vertex.

  21. Complementary and Supplementary Angles I 75° 15° H P Q 40° 50° H S U V 60° T 30° Z W Some examples of complementary angles are shown below. mH + mI = 90 mPHQ + mQHS = 90 mTZU + mVZW = 90

  22. Complementary and Supplementary Angles D C 130° 50° E B F A If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles. Two angles are supplementary if and only if (iff) the sum of their degree measure is 180. mABC + mDEF = 50 + 130 = 180

  23. Complementary and Supplementary Angles I 75° 105° H Q 130° 50° H S P U V 60° 120° 60° Z W T Some examples of supplementary angles are shown below. mH + mI = 180 mPHQ + mQHS = 180 mTZU + mUZV = 180 and mTZU + mVZW = 180

  24. 52° 52° A B Congruent Angles Suppose A  B and mA = 52. Find the measure of an angle that is supplementary to B. 1 B + 1 = 180 1 = 180 – B 1 = 180 – 52 1 = 128°

  25. G D 1 2 A C 4 B 3 E H Congruent Angles 1) If m1 = 2x + 3 and the m3 = 3x + 2, then find the m3 x = 17; 3 = 37° 2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC x = 29; EBC = 121° 3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4 x = 16; 4 = 39° 4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1 x = 18; 1 = 43°

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