Ch. 1-3: Measuring Angles SOL: G4 Objectives: Measure and classify angles. Identify special angle pairs. Use the special angle pairs to find angle measures.

Angle formed by two noncollinear rays that have a common endpoint. A B Noncollinear rays Common Endpoint C A

Sides of an Angle The rays that make the angle Side AB Side AC A B C A Vertex of an Angle The common endpoint Vertex A

Symbols When we name an angle, the vertex point is always in the middle Note the vertex is the middle point listed The following are the different ways we can name this angle:  A,  BAC,  CAB,  4 A B C 4

Interior of an Angle Any point inside the angle Interior Exterior of an Angle Any point outside the angle Exterior

Equal vs. Congruent Just like with segments, there is only one way to measure an angle. To measure an angle, we find how many degrees there are from one side to the other. (360° in a circle) ∠ A ≅ ∠ C m ∠ A = m ∠ C = 70° A B C 70° 40°

Example 1: Use the diagram to answer a, b, and c. a.) Name all angles that have B as a vertex. b.) Name the sides of  5. c.) Write another name for  6. ∡ ABG, ∡ ABD, ∡ DBE or ∡ DBF, ∡ EBG or ∡ FBG, ∡ 5, ∡ 6, ∡ 7 BG and BE or BF ∡ DBE or ∡ DBF

Measuring Angles To measure an angle, you use a protractor. The protractor has two scales running from 0 to 180 degrees in opposite directions. These are the scales we use to determine the measure of the angle Place the center point of the protractor on the vertex Align the 0 on either side of the scale with one side of the angle. (Paying attention to which direction the angle is opening

Example 2: Find the measure of  PQR. P Q R Since QP is aligned with the 0 on the outer scale, use the outer scale to find that QR intersects the scale at 65 degrees. 65°

Classify Angles by Angle Measure Measures 90  Written as m  A = 90  Right Angle Acute Angle This symbol means, right angle, perpendicular A B Measures less than 90  Written as m  B < 90 

Obtuse Angle Straight Angle (Line) Measures greater than 90  Written as m  C > 90  C ABC Classify Angles by Angle Measure Measures 180 

Example 3: Measure the angle and classify it. 12°, Acute

Example 4: Measure the angle and classify it. 99°, Obtuse

Example 5: Measure the angle and classify it. 70°, Acute

Congruent Angles Angles that have the same measure Symbols:  NMP   QMR

Angle Bisector An angle bisector is a segment, line, or ray that splits an angle into two congruent angles. In the picture, ray BD bisects ∠ ABC. Therefore, we know ∠ ABD ≅ ∠ DBC.

Postulate 1.8: Angle Addition Postulate If point B is in the interior of ∠ AOC, then m ∠ AOB + m ∠ BOC = m ∠ AOC

Example 6: Apply the angle addition postulate. A D C B 23° 41° What is the m ∡ ABC? E H G F If m ∠ EFG = 23°, what is the m ∠ EFH? 11° If m ∠ KJL = 117°, what is the m ∠ KJM? K M L J 68° m ∡ ABD + m ∡ CBD = m ∡ ABC 23° + 41° = 64° 64° 12° m ∡ EFG - m ∡ GFH = m ∡ EFH 23° - 11° = 12° 49° m ∡ KJL - m ∡ LJM = m ∡ KJM 117° - 68° = 49°

Example 7: If m ∠ RQT = 155°, what are the m ∠ RQS and m ∠ SQT? m ∡ RQS + m ∡ TQS = m ∡ RQT (4x – 20) + (3x + 14) = 155° 7x – 6= 155° + 6 7x = 161° 7 x = 23 m ∡ RQS = 4x – 20 = 4(23) – 20 = 72° m ∡ TQS = 3x + 14 = 3(23) + 14 = 83° Check: 72° + 83° = 155°

Example 8: ∠ DEF is a straight angle. What are the m ∠ DEC and m ∠ CEF? m ∡ DEC + m ∡ FEC = m ∡ DEF (11x – 12) + (2x + 10) = 180° 13x – 2 = 180° x = 182° 13 x = 14° m ∡ DEC = 11x – 12 = 11(14) – 12 = 142° m ∡ FEC = 2x + 10 = 2(14) + 10 = 38° Check: 142° + 38° = 180°