Geometry. 1.5 Measuring and Constructing Angles Day 1
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1 Geometry Day 1
2 1.5 warm up 1. Identify two planes that intersect at EF. 2. The cities on the map lie approximately in a straight line. Find the distance from Ann Arbor to Nashville. 3. Sketch DR in plane N and DT intersecting the plane.
3 Essential Question How can you measure and classify an angle? 1.4 Perimeter and Area in the Coordinate Plane
4 What You Will Learn Name, measure, and classify angles. Identify congruent angles. Use Angle Addition Postulate. Bisect angles.
5 What is an Angle? Endpoint Vertex Sides An ANGLE consists of two different rays that have the same endpoint. We call the endpoint the VERTEX of the angle. The rays are the sides of the angle.
6 Interior/Exterior R R is in the interior of A A S S is in the exterior of A
7 Naming Angles 1 A You can name an angle several different ways. ABC or CBA or just B (The middle letter is the vertex.) B C or 1. Important point: when there is more than one angle at the same vertex, you must use three letters.
8 Example 1 A lighthouse keeper measures the angles formed by the lighthouse at point M and three boats. Name three angles shown in the diagram.
9 Let s See What You Know Write three names for the angle.
10 Let s See What You Know Write three names for the angle.
11 Angle Measure The measure of ABC is denoted m ABC. Angles are measured in degrees, using the symbol. For example, if the measure of D is 24 degrees, we would write m D = 24. Angles are measured using a tool called a protractor.
12 Angle Classification A 0 m A 90 Acute
13 Angle Classification Use a square at the vertex to indicate this is a right angle. A m A 90 Right
14 Angle Classification A 90 m A 180 Obtuse
15 Angle Classification A m A 180 Straight
16 Congruent Angles Angles that have the same measure are congruent. The symbol for congruent is If TUF and LUK each measure 50, then they are congruent. Angles are congruent TUF LUK Measures are equal m TUF m LUK
17 Protractor Postulate (Postulate 1.3) The measure of an angle is a real number from 0 to 180. m AOB = a b or m AOB = b a a b
18 Example 2 Find the measure of each angle. Then classify each angle. a. m GHK = 125 obtuse b. m JHL = 90 right c. m LHK = 35 acute
19 Postulate 1.4 Angle Addition Postulate A B D C If B is in the interior of ADC, then m ADB + m BDC = m ADC
20 Example
21 Example 4
22 Essential Question How can you measure and classify an angle? 1.4 Perimeter and Area in the Coordinate Plane
23 Angle Bisector A D A ray that divides an angle into two angles that are congruent is an angle bisector. B C ABD CBD Use congruence marks to indicate congruent angles.
24 Example 5 B G 55 U M Think: Ray UG divides BUM into two equal parts, so each is 55. m BUM = 110 Ray UG is the angle bisector of BUM. Find m GUM and m BUG
25 Example 6 R G 43 U T m RUG = 43 Ray UG is the angle bisector of RUT. Find m GUT and m RUT Think: Ray UG divides RUT into two equal parts, so each is 43. The total is 86.
26 Example 7 RT bisects ARC. m ART = (7x + 12) m TRC = (4x + 30) Find the measure of each angle.
27 Your Turn B A F C Ray AF bisects CAB m FAB = 8x + 10 m FAC = 3x + 25 Find m BAF and m CAB.
28 Your Turn Solution B F 8x x + 25 C 8x + 10 = 3x x = 15 x = 3 A m BAF = 8(3) + 10 = 34 m CAB = 2(34) = 68
29 Essential Question When is a ray an angle bisector? 1.4 Perimeter and Area in the Coordinate Plane
30 Assignment 1.5 Day 1 #1, 2, 6 14 even, 17 20, odd 1.5 Day 2 #21 29 odd, odd, 52, 54 Both assignments are due Thursday.
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