1. 1-6: Exploring Angles Expectation:
G1.1.6: Recognize Euclidean geometry as an axiom system. Know the key axioms and understand the meaning of and distinguish between undefined terms (e.g., point, line, and plane), axioms, definitions, and theorems.
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1.6: Exploring Angles

2. Ray
Defn: A ray with endpoint D containing E and F, denoted DE or DF, is the union of DE and all points F, such that E is between D and F. F E D
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1.6: Exploring Angles

3. What do you mean? AB : The distance between A and B.
AB: The segment with endpoints A and B.
AB: The ray with endpoint A, containing B
AB: The line through A and B.
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1.6: Exploring Angles

4. Opposite Rays
Opposite Rays: AB and AC are opposite rays iff A, B and C are _____________ and A is between B and C. C A B
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1.6: Exploring Angles

5. Angles
Defn: An angle is the union of two _____ with the same endpoint, but no other points in common. The common endpoint is the __________ of the angle and all other points on the rays make up the _______ of the angle.
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1.6: Exploring Angles

6. Angle Y is the vertex and the sides are YX and YZ. X Y Z 4/26/2017
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7. _____ is the symbol for an angle.
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8. 3 ways to name an angle
1. If there is only one angle, we may name the angle by its vertex. A
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1.6: Exploring Angles

9. 3 ways to name an angle
2. We can place the vertex between a point from each side of the angle. Z X Y
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1.6: Exploring Angles

10. 3 ways to name an angle
3. If there are 2 or more angles, we can use numbers. 2 1 3 4
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11. Angles and the Plane Angles split the plane into 3 parts: interior
exterior
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1.6: Exploring Angles

12. Measure of an Angle
Defn: The measure of ∠ABC, written _______, is a real number between 0 and 180.
Technically, an angle ________ have measure of 0 or 180.
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1.6: Exploring Angles

13. Protractor Postulate
Given AB and any real number r, s.t. 0<r<180, there are exactly 2 rays AC such that m∠CAB = r (one on each side of AB).
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1.6: Exploring Angles

14. Given AB and let r = 45 C1 45° C2 45° A B 4/26/2017
1.6: Exploring Angles

15. Angle Addition Postulate
If R is in the interior of ∠ABC, then m∠ABR + m∠RBC = __________.
If m∠ABR+ m∠RBC=m∠ABC, then R is in the interior of _________.
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1.6: Exploring Angles

16. A R B C
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17. Determine the value of a if m∠WXZ = 59°.Y W Z
38° a X
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1.6: Exploring Angles

18. Determine the value of x if m∠WXZ = 102°.Y W Z
(3x + 9)°
(7x – 7)° X
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1.6: Exploring Angles

19. Acute, Right, Obtuse Angles
Defn: Acute, Right, Obtuse Angles: Let a be the measure of ∠A. Then angle A is:
a. Acute iff ________.
b. Right iff ________.
c. Obtuse iff ________.
d. Straight Angle iff _______.
e. Zero Angle if ________.
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1.6: Exploring Angles

20. Congruent Angles
Defn: ∠A is congruent to ∠B, written ∠A ≅ ∠B, iff ___________________.
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21. Angle Bisector
Defn: BD is the bisector of ∠ABC iff D is in the interior of ∠ABC and _____________.
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1.6: Exploring Angles

22. Solve for x if XY bisects ∠WXZ.
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1.6: Exploring Angles

23. Which of the following is true of a ray KL?
It extends infinitely in both directions.
Its endpoint is L.
It has length.
It contains exactly 2 points, K and L.
It is a portion of a line beginning with point K.
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1.6: Exploring Angles

24. The notation FG represents: The length of a line
The length of a segment
The length of a ray
Two points
A plane
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1.6: Exploring Angles

25. Assignment Pages 50 – 51, 17 – 25 (odds), 33 and 35 4/26/2017
1.6: Exploring Angles