1-6: Exploring Angles Expectation:

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


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.


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


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.


Angle

YZ

X Y is the vertex and the sides are YX and YZ.

_____ is the symbol for an angle.



3 ways to name an angle

1. If there is only one angle, we may name the angle by its vertex.

A



2. We can place the vertex between a point from each side of the angle.

X

Y

Z



3. If there are 2 or more angles, we can use numbers.

12

34


Angles and the Plane

Angles split the plane into 3 parts:

interior

exterior


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.


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).


Given AB and let r = 45

A B

C1

45°

C2

45°


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 _________.


B

A

C

R


Determine the value of a if m∠WXZ = 59°.

38° aZ

YW

X


Determine the value of x if m∠WXZ = 102°.

(3x + 9)° Z

YW

X

(7x – 7)°


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 ________.


Congruent Angles

Defn: ∠A is congruent to ∠B, written ∠A ≅ ∠B, iff

___________________.


Angle Bisector

Defn: BD is the bisector of ∠ABC iff D is in the interior of ∠ABC and _____________.


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.


The notation FG represents:A. The length of a lineB. The length of a segmentC.The length of a rayD.Two pointsE. A plane



Assignment

Pages 50 – 51,17 – 25 (odds), 33 and 35

1-6: Exploring Angles Expectation:

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