1-6: exploring angles expectation:
DESCRIPTION
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 4/26/2017 1.6: Exploring AnglesTRANSCRIPT
05/03/23 1.6: Exploring Angles
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.
1.6: Exploring Angles05/03/23
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.
D
EF
1.6: Exploring Angles05/03/23
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 BAB: The line through A and B.
1.6: Exploring Angles05/03/23
Opposite Rays
Opposite Rays: AB and AC are opposite rays iff A, B and C are _____________ and A is between B and C.
B A C
1.6: Exploring Angles05/03/23
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.
1.6: Exploring Angles05/03/23
Angle
YZ
X Y is the vertex and the sides are YX and YZ.
_____ is the symbol for an angle.
1.6: Exploring Angles05/03/23
1.6: Exploring Angles05/03/23
3 ways to name an angle
1. If there is only one angle, we may name the angle by its vertex.
A
1.6: Exploring Angles05/03/23
3 ways to name an angle
2. We can place the vertex between a point from each side of the angle.
X
Y
Z
1.6: Exploring Angles05/03/23
3 ways to name an angle
3. If there are 2 or more angles, we can use numbers.
12
34
1.6: Exploring Angles05/03/23
Angles and the Plane
Angles split the plane into 3 parts:
interior
exterior
1.6: Exploring Angles05/03/23
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.
1.6: Exploring Angles05/03/23
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).
1.6: Exploring Angles05/03/23
Given AB and let r = 45
A B
C1
45°
C2
45°
1.6: Exploring Angles05/03/23
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 _________.
1.6: Exploring Angles05/03/23
B
A
C
R
1.6: Exploring Angles05/03/23
Determine the value of a if m∠WXZ = 59°.
38° aZ
YW
X
1.6: Exploring Angles05/03/23
Determine the value of x if m∠WXZ = 102°.
(3x + 9)° Z
YW
X
(7x – 7)°
1.6: Exploring Angles05/03/23
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 ________.
1.6: Exploring Angles05/03/23
Congruent Angles
Defn: ∠A is congruent to ∠B, written ∠A ≅ ∠B, iff
___________________.
1.6: Exploring Angles05/03/23
Angle Bisector
Defn: BD is the bisector of ∠ABC iff D is in the interior of ∠ABC and _____________.
1.6: Exploring Angles05/03/23
Solve for x if XY bisects ∠WXZ.
8x+12 15x-37
X
Y ZZ
Which of the following is true of a ray KL?A. It extends infinitely in both directions.B. Its endpoint is L.C. It has length.D. It contains exactly 2 points, K and L.E. It is a portion of a line beginning with point K.
1.6: Exploring Angles05/03/23
The notation FG represents:A. The length of a lineB. The length of a segmentC.The length of a rayD.Two pointsE. A plane
1.6: Exploring Angles05/03/23
1.6: Exploring Angles05/03/23
Assignment
Pages 50 – 51,17 – 25 (odds), 33 and 35