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Angle Pair Relationships

Angle Pair Relationships. Angle Pair Relationship Essential Questions. How are special angle pairs identified?. 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.

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Angle Pair Relationships

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  1. Angle Pair Relationships

  2. Angle Pair Relationship Essential Questions How are special angle pairs identified?

  3. 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

  4. S vertex T Angles – sides and vertex There is another case where two rays can have a common endpoint. angle This figure is called an _____. Some parts of angles have special names. side vertex The common endpoint is called the ______, and the two rays that make up the sides ofthe angle are called the sides of the angle. side R

  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. Angles C A 1 B ABC 1 B CBA BA and BC 1) Name the angle in four ways. 2) Identify the vertex and sides of this angle. vertex: Point B sides:

  8. 2) What are other names for ? 3) Is there an angle that can be named ? 1 XWZ YWX XWY or 1 2 W Angles 1) Name all angles having W as their vertex. X W 1 2 Y Z No!

  9. 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.

  10. 40° 110° 90° 50° 75° 130° Angle Measure Classify each angle as acute, obtuse, or right. Acute Obtuse Right Obtuse Acute Acute

  11. 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. ____

  12. 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

  13. 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 ____

  14. 1 2 1 2 Z D X Adjacent Angles and Linear Pairs of Angles Determine whether 1 and 2 are adjacent angles. No. Yes. In this example, the noncommon sides of the adjacent angles form a ___________. straight line linear pair These angles are called a _________

  15. 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.

  16. 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.

  17. 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

  18. 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.

  19. 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

  20. 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

  21. 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

  22. Congruent Angles measure Recall that congruent segments have the same ________. Congruent angles _______________ also have the same measure.

  23. 50° 50° B V Congruent Angles Two angles are congruent iff, they have the same ______________. degree measure B  V iff mB = mV

  24. 1 2 X Z Congruent Angles To show that 1 is congruent to 2, we use ____. arcs To show that there is a second set of congruent angles, X and Z, we use double arcs. This “arc” notation states that: X  Z mX = mZ

  25. 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

  26. 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

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

  28. 130° x° Vertical Angles Find the value of x in the figure: The angles are vertical angles. So, the value of x is 130°.

  29. 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.

  30. 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°

  31. 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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