1-4 Rays, Angles and Angle Measures Honors Geometry

Standard/Objectives: Objective 1: Students will understand geometric concepts and applications Objective 2: Students will use visualization, spatial reasoning, and geometric modeling to solve problems. Objective 3: Students will use angle postulates to Classify angles as acute, right, obtuse, or straight.

Ray Definition: An infinite set of points that has a definite starting point called the “ENDPOINT” and extends to infinity in ONE direction. SYMBOL: An arrow pointing to the right

Rays: Cont. Naming: named by using the endpoint first and any point on the ray. ray AB or AB A B

Example 1: Name ALL the rays & all the DIFFERENT rays: All = every ray every way Different = each ray one time A B C D E

Collinear Rays Definition: When the points of two rays ALL lie on the same line. Name ALL the rays: AD, AR, DR, RD, RA & DA Name all the DIFFERENT rays: AR, DR, RA & DA R D A

Opposite Rays Definition: Two rays that are collinear and have only their endpoints in common. rays BA & BC or BA & BC ALSO CALLED STRAIGHT ANGLES A B C

Angle Definition: a figure consisting of two of two NON-COLLINEAR rays with a common endpoint. Sides: the two rays with the common endpoint that form the angle. BA & BC Vertex: the common endpoint that makes the angle. point B

Angles: cont. Naming: ∠ Three points name an angle (the vertex & one point from each ray), the vertex must be the middle point. ∠CBA or ∠ABC A number or lowercase letter. ∠1 or ∠a The vertex. ∠B

Angles: cont. NOTE: if more than one angle share the same vertex then you CAN’T use the vertex to name the angle.

EXAMPLE 1 Name angles Name the three angles in the diagram. ∠ WXY, or ∠YXW ∠YXZ, or ∠ ZXY ∠ WXZ, or ∠ ZXW You should not name any of these angles ∠ X because all three angles have point X as their vertex.

Angles: cont. An angle separates a plane into three parts. INTERIOR: point F is on the interior of ∠ V EXTERIOR: point S is on the exterior of ∠V ON: point H is on ∠V H S F

Classifying Angles Angles are classified as acute, right, obtuse, and straight, according to their measures. Angles have measures greater than 0° and less than or equal to 180°.

1. Name all the angles in the diagram. Which angle is a right angle? GUIDED PRACTICE 1. Name all the angles in the diagram. Which angle is a right angle? ∠PQR , ∠ PQS, ∠RQS ; ∠ PQS is a right angle . ANSWER

2. Draw a pair of opposite rays. What type of angle do the rays form? GUIDED PRACTICE 2. Draw a pair of opposite rays. What type of angle do the rays form? ANSWER Straight Angle

Postulate: Protractor Postulate On a plane, given ray OB and a number between 0° & 180°, there is exactly one ray w/ endpoint O that extends to any side of ray OB such that the degree measure of the angle formed is “r°”. A r° O B

Measures of Angles: The measure of A is denoted by mA. The measure of an angle can be approximated using a protractor, using units called degrees(°). For instance, BAC has a measure of 50°, which can be written as mBAC = 50°. B C A

EXAMPLE 2 Measure and classify angles Use the diagram to find the measure of the indicated angle. Then classify the angle. a. ∠KHJ b. ∠GHK c. ∠GHJ d. ∠GHL SOLUTION A protractor has an inner and an outer scale. When you measure an angle, check to see which scale to use.

Postulate: Angle Addition Postulate If point D is in the interior of BAC, forming ray AD then: mBAD + mDAC = mBAC

Angle Addition Postulate EXAMPLE 3 Find angle measures ALGEBRA Given that m∠LKN =145 , Find m∠LKM & m∠MKN SOLUTION Write and solve an equation to find the value of x. m∠LKN = m∠ LKM + m∠MKN Angle Addition Postulate 145 = (2x + 10) + (4x – 3) o Substitute angle measures. 145 = 6x + 7 Combine like terms. 138 = 6x Subtract 7 from each side. 23 = x Divide each side by 6.

3. Given that ∠KLM is a straight angle, find m ∠ KLN and m∠ NLM. GUIDED PRACTICE Find the indicated angle measures. 3. Given that ∠KLM is a straight angle, find m ∠ KLN and m∠ NLM. ANSWERS 125°, 55°

GUIDED PRACTICE 4. Given that ∠EFG is a right angle, find m∠EFH and m ∠HFG. ANSWER 60°, 30°

Definition of a Congruent Angles  

Definition of a Angle Bisector When a ray lies on the INTERIOR of an angle and divides the angle into 2 congruent angles. BD is an angle bisector of <ABC. A D B C

Ex: If FH bisects EFG & mEFG=120o, find mEFH m∠GFH + m∠HFE = m∠EFG m∠GFH + m∠GFH = m∠EFG 2 (m∠GFH) = 120° m∠GFH = 60° E H F G

Angle Bisector Postulate   A D B C

Assignment: pp. 41-43 #1-10 all 12-35 all 45,46 58-64 even