UNIT: 1.1 - Tools of Geometry LESSON: 1.2a – Angles 4/27/2017 9/17/15 CC Geometry UNIT: 1.1 - Tools of Geometry LESSON: 1.2a – Angles MAIN IDEA: Students will be able to use information to determine the measure of angles and explain relationships between angles. HOMEWORK: Textbook Page 41 #’s 9-10 Page 51 #’s 19-26 Take home quiz due Monday 9/21

Angle and Points ray vertex ray 4/27/2017 Angle and Points An Angle is a figure formed by two rays with a common endpoint, called the vertex. ray vertex ray Angles can have points in the interior, in the exterior or on the angle. A E D B C Points A, B and C are on the angle. D is in the interior and E is in the exterior. B is the vertex.

4/27/2017 Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide Using 3 points: vertex must be the middle letter This angle can be named as Using 1 point: using only vertex letter * Use this method is permitted when the vertex point is the vertex of one and only one angle. Since B is the vertex of only this angle, this can also be called . A C B

Naming an Angle - continued 4/27/2017 Naming an Angle - continued Using a number: A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named as . A B 2 C * The “1 letter” name is unacceptable when … more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present.

Example Therefore, there is NO in this diagram. 4/27/2017 Example K is the vertex of more than one angle. Therefore, there is NO in this diagram. There is

4 Types of Angles Acute Angle: Right Angle: Obtuse Angle: 4/27/2017 4 Types of Angles Acute Angle: an angle whose measure is less than 90. Right Angle: an angle whose measure is exactly 90 . Obtuse Angle: an angle whose measure is between 90 and 180. Straight Angle: an angle that is exactly 180 .

4/27/2017 Adding Angles When you want to add angles, use the notation m1, meaning the measure of 1. If you add m1 + m2, what is your result? m1 + m2 = 58. m1 + m2 = mADC also. Therefore, mADC = 58.

Angle Addition Postulate 4/27/2017 Angle Addition Postulate Postulate: The sum of the two smaller angles will always equal the measure of the larger angle. Complete: m  ____ + m  ____ = m  _____ MRK KRW MRW

Example: Angle Addition 4/27/2017 Example: Angle Addition K is interior to MRW, m  MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK. First, draw it! 3x + x + 6 = 90 4x + 6 = 90 – 6 = –6 4x = 84 x = 21 3x x+6 Are we done? mMRK = 3x = 3•21 = 63º

4/27/2017 Angle Bisector An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles. Example: Since 4   6, is an angle bisector. 5 3

Congruent Angles Definition: 4/27/2017 Congruent Angles Definition: If two angles have the same measure, then they are congruent. Congruent angles are marked with the same number of “arcs”. The symbol for congruence is  3 5 Example: 3   5.

Example Draw your own diagram and answer this question: 4/27/2017 Example Draw your own diagram and answer this question: If is the angle bisector of PMY and mPML = 87, then find: mPMY = _______ mLMY = _______

Complementary Angles Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. Example: These two angles are complementary. Their sum is 90˚.                                                        58° + 32° = 90°

Complementary Angles These two angles can be "pasted" together to form a right angle! Adjacent complementary angles.

Supplementary Angles Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. Example: These two angles are supplementary. The sum of their measures is 180˚                                                                        139° +41° = 180 °

Linear Pair of Angles Special Supplementary Angles Two angles that share a vertex and a side to form a line. Two angles that can be "pasted" together to form a straight line!

Vertical Angles Opposite angles formed by intersecting lines .      For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Angle BEC and angle AED are also vertical angles. Vertical angles are congruent - have the same degree measurement.           110 70 70 110

Review State whether the following are acute, right, or obtuse. 1. 3. 5. acute obtuse right 2. 4. ? acute obtuse ?

Complementary and Supplementary Find the missing angle. 1. Two angles are complementary. One measures 65 degrees. 2. Two angles are supplementary. One measures 140 degrees. Answer : 25 Answer : 40

Complementary and Supplementary Find the missing angle. You do not have a protractor. Use the clues in the pictures. 2. 1. x x 55 165 x = 180° – 165° x = 90° – 55° x = 15° x = 35 °

Vertical Angles Find the missing angle. x = 58 x 58

More drawings F E 20 70 90 Box in the corner indicates a right angle. J 20 H

Final Drawing Find the measure of each missing angle B C 68 60 52 A G

Exit Ticket Explain why angle 1 is congruent to angle 3.