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Geometry G Notes: Angle Notation. Name______________________________ Definition: An angle is formed by two rays that share a common endpoint. M X A 1. The point that the two rays intersect is called the ________________________. 2. The two rays are called the ______________ of the angle. 3. When naming angles, it is typical to use one or three letters. Sometimes one cannot use one letter. When using three letters, the _________________ must be the letter in the middle. Other times one uses numbers to name the angles as below. M C I X K 1 2 A R 4. Name an angle using one letter. _________ 5. Name three different angles. _________, __________, _________ 6. IRC can also be named in what two other ways? , An angle breaks up a plane into three regions: the exterior of the angle the interior of the angle points on the angle. M 7. Name the points on the interior of FAB 8. Name the points on FAB. , , , , , F , N A T B S R Y 1 2 Name______________________________________________________________________ Date__________________ Hour_____________ 3.2 Notes – Angle Measure Geometry G In geometry, angles are measured in units called ____________________. The symbol is _______. There are 4 different types of angles: Name of angle Definition Example ACUTE OBTUSE RIGHT STRAIGHT Example 1: Use a protractor to measure PRO . What kind of angle is it? O P R 3 Example 2: Find the measure of each angle and classify it. a) VDS b) SDL c) IDS d) SDE I V L E D S Example 3: Use a protractor to draw an angle having each measurement. Then classify each angle. a) 115 b) 25 Example 4: The measure of B is 138. Solve for x. (5x – 7) B 4 Geometry G Protractor Practice Worksheet 1 Name: ____________________________ Date: ____________ Goal: To measure angles using a protractor. When using a protractor, you must use it correctly. What are the two things you need to do when using a protractor? 1. _______________________________________________________________________________ 2. _______________________________________________________________________________ 5 Geometry G Protractor Practice Worksheet 2 Name: ____________________________ Date: ____________ Measure each angle to the nearest whole degree. You may have to extend the sides of your angle to do the measurement. 1. 2. 3. 4. 5. 6. 6 Draw each angle using a Protractor. 7. A 35 8. B 100 9. C 150 10. D 50 11. F 90 12. G 180 7 Review 3.1 – 2 1] B Name the angle in 4 ways: 2 A 2] B Interior: K Exterior: R 4] Use a protractor to measure each angle. ABG = EBC = ABF = FBC = ABD = FBD = C I On: 3] T What points are in the interior, exterior or on the angle? F E D G A B Us a protractor to draw an angle having each of the following measurements: 50 125 90 158 C 8 Name_____________________________________________________ Date________________ Hour_______ 3.3 Notes – The Angle Addition Postulate Geometry G Suppose mKNL = 110 and mLNM = 25. What would you do to find the mKNM? L M K N Suppose mMNK = 155 and mLNM is 30. What would you do to find the mLNK? L M K N Angle Addition Postulate For any ABC, if D is in the interior of ABC, then mABD + mDBC = mABC. Draw a diagram below to show this. Vocabulary A _______ that divides an angle into ____________ angles of equal ________________ is called the ___________________________________. 9 10 Geometry G Worksheet 3.3 Name________________________ SHOW ALL YOUR WORK!! 1. Find m1 if mCUB = 78. B 1 48 S 2. Find m2 if mWHI = 160. E U I T C 42 104 2 H W 3. mSOX = 160 m1 = x + 14 m2 = 3x – 10 Find m2 S W 1 2 X O B 4. mBEA = 71. Find mREA. 2x R (5x + 8) E A O 5. mWOV = 12x. Find mLOV. W 76 (5x + 1) V L 11 6. mFIE = 3x, mRIE = 42, mFIR = 5x Find mFIR. E F R I 7. mHAK = 4x – 2, mKAW = 2x – 5, and mHAW = 77. Find mHAK and mKAW. H K W A 8. US bisects BUL, mBUS = 2x + 10, and mSUL = 3x – 18. Find mBUL. B S U L T 9. mTRI = 3x – 5, mIRB = x + 27, and mTRB = 86. Does RI bisect TRB? I R B N 10. Find the measure of each angle. a. mNEO = _______ b. mDES = _______ c. mDEO = _______ d. mSEO = _______ D O 27 18 S E C 12 Section 3.4 – Adjacent Angles and Linear Pairs In the diagrams below, 1 and 2 are … T 1 1 2 2 H O L 2 1 E Not Adjacent Angles Adjacent Angles Not Adjacent Angles 1 2 K Adjacent Angles What can you conclude about Adjacent Angles? Adjacent Angles are angles that have a shared ____________ and the same ________________, but no interior points in common. Try these… Determine whether 1 and 2 are adjacent angles. 1 1 2 2 1 2 In the diagrams below, 1 and 2 are … 1 1 2 2 1 a linear pair a linear pair 2 not a linear pair What can you conclude about a Linear Pair? Linear Pair consists of 2 angles that are ________________ and their noncommon sides are _________________ ________________. 13 14 3-5 – Complementary and Supplementary Angles F A E L Two angles whose measures add up to ______ are called __________________ __________. They can also be called a _____________ __________ if together they form a straight angle. In the picture above, _______ and _______ are ______________________ ___________. Two angles whose measures add up to ______ are _____________________ ___________. B 50° 40° C R T D In the diagram above, ______ and ______ are _____________________ ___________. A Use the figure on the right to name each of the following. L 1. Name a pair of complementary angles. N 2. Name a pair of supplementary angles. Q M 3. Name a different pair of supplementary angles. P O 4. Name a linear pair. Find the measure of each angle 5. 6. 56° 7. 56° 15 Find the measure of each angle in the diagram. DAB is a right angle ADE is a right angle 1 = 53 m 1 = m 12 3 = 55 5 = 88 m4 = m9 ABE = 100 DEB = 80 D 3 4 12 11 7 E 8 C 5 6 2 A 1 9 10 B 16 Geometry G Worksheet 3.5 Name_________________________ Supplementary and Complementary Angles Find the measures of angles 1 through 22. Mark them in your diagram. 57 6 7 14 15 1 75 62 8 73 16 3 71 2 17 4 42 11 10 9 18 91 20 19 25 104 5 122 13 12 21 70 22 17 D 23) Find mDBC. x 8x B A C C 24) Find mDBC. D (4x – 20) x A B 25) 1 and 2 are complementary. m1 = 2x + 7 and m2 = 4x – 19. Find the measure of each angle. 26) 3 and 4 are supplementary. m3 = 5x + 22 and m4 = 7x + 2. Find the measure of each angle. 27) Use the diagram on the right to name: C a) two complementary angles B D b) a linear pair G E c) two adjacent angles A F 18 Name_________________________________________ Date________ Hour____ 3.6 – Vertical Angles Geometry G Vertical Angles: 3 4 1 2 THEOREM: Examples: 1) Find x, y, and z x 51 Y z 2) Given: m4 = (2x + 5) m5 = (x +30) Find: m6 4 5 6 3) Identify each pair of angles as adjacent, vertical, complementary, supplementary, and/or linear pair. a) 1 and 2 b) 3 and 4 2 3 1 c) 5 and 4 4 5 d) 3 and 5 19 4) Find x and y if CBD is congruent to FDG. 5) Find each of the following: a) x b) mLAT c) mTAO d) mPAO 20 Vocabulary Words: Complementary Angles Angle Bisector Adjacent Angles. Supplementary Angles Linear Pair Right Angles Vertical Angles ST bisects RSW 1. In the pictures above, FOH and GOH are called _____________________________. 2. FOH and GOH are also called ____________________________________________. 3. Further, FOH and GOH are _____________________________________________. 4. In the pictures above, ACB and DCE are called ______________________________. 5. In the pictures above, JPK and KPL are called ______________________________. 6. JPK and KPL are also called ___________________________________. 7. Name the vertical angle ACD to ___________________________________. 8. What do you know about RST and TSW? ___________________________________ 9. What do you call LPM? ___________________________________ 21 In the figure, GA and GD, and GB and GE are opposite rays. 10] Which angle forms a linear pair with DGC ? ________ 11] Do BGC and EGD form a linear pair? 12] Name two angles that are adjacent to CGD . ________ 13] Name two angles that form a linear pair with BGD . ________ ________ 14] Name three angles adjacent to AGB . ________ ________ 15] Do CGE and CGB form a linear pair? 16] Name the vertical angle to EGD . _______________ 17] Name another pair of vertical angles. ____________ and _______________ ________ ________ ________ ________ 22 Name: E D A G F C B Name: 1] a linear pair 2] a pair of supplementary angles 3] a pair of complementary angles 4] a pair of adjacent angles 5] a pair of vertical angles 6] two right angles Write each pair of angles that you named above into the proper column of the table below. Angle Relationships Equals Equals 180 Equals 90 23 24 Determine the relationship in the diagram. Are the angles complementary or is it a right angle? The angles add to 90. Are the angles supplementary or are they a linear pair? The angles add to 180. Do you have an angle bisector? The two angles are congruent. Do you have vertical angles? The two angles are congruent. Write the equation and then solve the equation. 1. 2. Equation: _______________________ Equation: _______________________ x = ______ x = ______ 3. 4. Equation: _______________________ Equation: _______________________ x = ______ x = ______ 25 5. 6. Equation: _______________________ Equation: _______________________ x = ______ x = ______ 7. 8. Equation: _______________________ Equation: _______________________ x = ______ x = ______ mACB ________ mABC _________ 26 3.7 – Perpendicularity Name____________________________ Date___________________ Hour_____ Geometry G NOTES Perpendicularity, _____________________, and __________ measurements go together. Definition: If lines, rays or segments form right angles, then they are perpendicular( ). What would be the converse of the definition? Examples: DE EF ab E a D F b What conclusions would I be able to make if given the following: AB BC 1) A 2) B C 27 Example 1: True or False? 1. PRN is acute. 2. 4 8 3. m5 + m6 = 90 4. QR PR 5. 7 is obtuse Example 2: Find x. Example 3: Find mDBC. 28 Geometry G Section 2.5. Worksheet 3 Name____________________________ Warm – Up: 1. 2. ST bisects RSW , mRST 27 mFOH ________ mTSW _________ mWSR _________ 3. 4. mJPK _________ mCBF 70 & BD bisects CBF. mJPL _________ mCBD ________ 5. AB CD CE bisects DCB mABC ________ mDCE ________ mACD ________ mDCB ________ 6. mDBC ________ mCBE ________ BC bisects DBE mDBE ________ 29 7. mABE _________ mDBC _________ 8. BD bisects ABC and mABD 32 mDBC _________ mABC _________ 9. 10. Given l p m1 ______ m2 ______ m4 ______ m5 ______ m6 ______ m7 ______ AB CD HE bisects CHB mBHG 32 m3 ______ m1 ________ m2 ________ m3 ________ m4 ________ m5 ________ m6 ________ m7 ________ 30 Determine the relationship in the diagram. Are the angles complementary or is it a right angle? The angles add to 90. Are the angles supplementary or are they a linear pair? The angles add to 180. Do you have a angle bisector? The two angles are congruent. Do you have vertical angles? The two angles are congruent. Write the equation and then solve the equation. 1. 2. Equation: _______________________ Equation: _______________________ x = ______ x = ______ 3. 4. Equation: _______________________ Equation: _______________________ x = ______ x = ______ mABC _______ mABC _______ 31 Geometry G Section 2.5. Worksheet 6 Name____________________________ 5. 6. Equation: _______________________ Equation: _______________________ x = ______ x = ______ mABC _______ mABC _______ Note: Picture is not drawn to scale. 7. BD bisects ABC Equation: _______________________ 8. Equation: _______________________ x = ______ x = ______ mABC _______ mEBD _______ 32 9. Equation: _______________________ x = ________ mAFD ________ mAFB ________ mCFD ________ 10. 11. Equation:_________________________ Equation:_________________________ x = ________ x = ________ mABC ________ mPQR ________ mABE ________ mRQT ________ 33 Geometry G Section 2.5. Worksheet 7 Name____________________________ Determine the relationship in the diagram. Are the angles complementary or is it a right angle? The angles add to 90. Are the angles supplementary or are they a linear pair? The angles add to 180. Do you have an angle bisector? The two angles are congruent. Do you have vertical angles? The two angles are congruent. Write the equation and then solve the equation. 1. 2. 8x (6x + 12) (7x + 10) Equation: _______________________ (16x + 4) (18x + 4) Equation: _______________________ x = ______ x = ______ 3. 4. (7x - 12) (16x + 4) 18x (5x + 18) Equation: _______________________ Equation: _______________________ x = ______ x = ______ mABC _______ mABC _______ 34