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Angle Pair Relationships

Learn how to identify, name, and classify special angle pairs including straight angles, opposite rays, and adjacent angles. Understand complementary and supplementary angles, as well as congruent angles, with examples and essential questions.

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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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