OC 1.7/3.5 Proofs about Parallel and Perpendicular Lines

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1 (Segments, Lines & Angles) Date Name of Lesson 1.5 Angle Measure 1.4 Angle Relationships 3.6 Perpendicular Bisector (with Construction) 1.4 Angle Bisectors (Construct and Measurements of Angle Bisector) Quiz 3.1 Transversal Measurements 3.1 Parallel Lines with Transversal 3.2 Interior Angles Quiz 3.2 Exterior Angles 3.2 Corresponding Angles Class Activity Quiz 2.6 Algebraic Proofs 2.7, 2.8 Mini Proofs OC 1.7/3.5 Proofs about Parallel and Perpendicular Lines Proofs Activity Quiz Practice Test Unit Test 1

with an Angle Guided Practice Use the map of a high school shown to answer the following.

2 Ray (Segments, Lines & Angles) 1.5 Angle Measure Notes Opposite Rays Angle Sides Vertex Naming an Angle Points on a Plane with an Angle Guided Practice Use the map of a high school shown to answer the following. 1. Name all angles that have B as a vertex. 2. Name the sides of What is another name for GHL? 4. Name a point in the interior of DBK. 2

3 (Segments, Lines & Angles) Your Turn 5. Name all angles that have B as a vertex. 6. Name the sides of Write another name for 6. Degree Classify Angles right angle acute angle obtuse angle 3

4 (Segments, Lines & Angles) Guided Practice Classify each angle as right, acute, or obtuse. Then use a protractor to measure the angle to the nearest degree. 8. MJP 9. LJP 10. NJP Your Turn 11. TYV 12. WYT 13. TYU 4

Vertical Angles Guided Practice Name an angle

5 (Segments, Lines & Angles) 1.4 Angle Relationships Notes Special Angles Pairs Name and Definition Examples Nonexamples Adjacent Angles Linear Pair Vertical Angles Guided Practice Name an angle pair that satisfies each condition. 1. two acute adjacent angles 2. two obtuse vertical angles Your Turn 3. two angles that form a linear pair 4. two acute vertical angles 5

Find the measures of two supplementary angles if the measures of one angles is 6 less than five

6 (Segments, Lines & Angles) Angle Pair Relationships Vertical Angles Complementary Angles Supplementary Angles Linear Pair Guided Practice 5. Find the measures of two supplementary angles if the measures of one angles is 6 less than five times the measure of the other angle. Your Turn 6. Find the measures of two supplementary angles if the difference in the measures of the two angles is 18. 6

7 (Segments, Lines & Angles) Perpendicular Lines Guided Practice 7. Find x and y so that PR and SQ are perpendicular. Your Turn 8. Find x and y so that KO and HM are perpendicular. 7

(Segments, Lines & Angles) CAN be Assumed All points shown are coplaner G, H, J are collinear HM, HL, HK,GJ H is between G and J intersect at H L is in the interior of MHK GHM and MHL are adjacent

8 (Segments, Lines & Angles) CAN be Assumed All points shown are coplaner G, H, J are collinear HM, HL, HK,GJ H is between G and J intersect at H L is in the interior of MHK GHM and MHL are adjacent angles GHL and LHJ are a linear pair Interpreting Diagrams JHK and KHG are supplementary Guided Practice Determine whether each statement can be assumed from the figure. Explain. 9. KHL and GHM are complementary CANNOT be Assumed Perpendicular lines: HM HL Congruent angles JHK GHM JHK KHL KHL LHM Congruent segments GH HJ HJ HK HL HG HK HL 10. GHK and JHK are a linear pair 11. HL is perpendicular to HM Your Turn 12. m VYT = TYW and TYU are supplementary 14. VYW and TYS are adjacent angles 8

9 (Segments, Lines & Angles) 3.4 ext. Perpendicular Bisector Notes Bisector - Instructions to Construct a Perpendicular Bisector 1. Place your compass point on A and stretch the compass MORE THAN half way to point B, but not beyond B. 2. With this length, swing a large arc that will go BOTH above and below AB. (If you do not wish to make one large continuous arc, you may simply place one small arc above AB and one small arc below AB.) 3. Without changing the span on the compass, place the compass point on B and swing the arc again. The two arcs you have created should intersect. 4. With your straightedge, connect the two points of intersection. 5. This new straight line bisects AB. Label the point where the new line and AB cross as C. Guided Practice 1. Your Turn

10 (Segments, Lines & Angles) Construct a perpendicular bisector and then name all the relationships that we know about the figure, and what values we cannot state specific relationships about. Relationships What cannot be assumed 10

11 Angle Bisector 1.4 Angle Bisectors Notes Guided Practice 1. In the figure, KJ and KM are opposite rays, and KN bisects JKL. If m JKN = 8x 13 and m NKL = 6x + 11, find m JLN. Your Turn 2. In the figure, BA and BC are opposite rays, and BH bisects EBC. If m ABE = 2n + 7 and m EBF = 4n 13, find m ABE. 11

12 Steps to Bisecting an Angle 1. Start with angle PQR that we will bisect. 2. Place the compasses' point on the angle's vertex Q. 3. Adjust the compasses to a medium wide setting. The exact width is not important. 4. Without changing the compasses' width, draw an arc across each leg of the angle. 5. The compasses' width can be changed here if desired. Recommended: leave it the same. 6. Place the compasses on the point where one arc crosses a leg and draw an arc in the interior of the angle. 7. Without changing the compasses setting repeat for the other leg so that the two arcs cross. 8. Using a straightedge or ruler, draw a line from the vertex to the point where the arcs cross. This is the bisector of the angle PQR. Guided Practice Construct an angle bisector. 3. Your Turn

13 3.1 Transversal Measurements Notes Use a protractor to measure all of the angles below. Angle Measurements What kind of relationships did you discover? 13

14 Which examples do you think have parallel lines? Why? 14

15 3.1 Parallel Lines with Transversal Notes Transversal - in figure transversal is line. In the figure line r and line s are parallel which make specific rules about angles given below. Interior Angles lie in the region between two lines that are not the transversal. In figure interior angles are,,, and. Consecutive Interior (AKA Same Side Interior) Angles are. Consecutive Interior Angles in figure are and, and also and. Alternate Interior Angles are. Alternate Interior Angles in figure are and, and also and. Exterior Angles lie in the region not between the two lines that are not the transversal. In figure exterior angles are,,, and. Alternate Exterior Angles are. Alternate Exterior Angles in figure are and, and also and. Corresponding Angles lie on the same side of the transversal and on the same side of a line. Corresponding Angles are. Corresponding Angles in the figure are and, and, and, and also and. 15

16 Identify each pair of angles as corresponding, alternate interior, alternate exterior, or consecutive interior. 16

17 3.2 Interior Angles Notes Review: Interior Angles lie in the region between two lines that are not the transversal. In figure interior angles are,,, and. Consecutive Interior (AKA Same Side Interior) Angles are. Consecutive Interior Angles in figure are and, and also and. Alternate Interior Angles are. Alternate Interior Angles in figure are and, and also and. If m 4 = 70 find the following: m 3 = m 5 = m 6 = If m B = 143 find the following: m C = m E = m H = If m 5 = 37 find the measure of the other interior angles m = m = m = 17

18 For each of the following find the value of the variable and the measures of the two angles. 18

19 3.2 Exterior Angles Notes Review: Exterior Angles lie in the region not between the two lines that are not the transversal. In figure exterior angles are,,, and. Alternate Exterior Angles are. Alternate Exterior Angles in figure are and, and also and. Use the information provided to find the numbered angle measure. If m 1 = 60 find the following: m 2 = m 7 = m 8 = If m 6 = 87 find the measure of the other exterior angles m = m = m = 19

20 For each of the following find the value of the variable and the angle measure. 20

21 3.2 Corresponding Angles Notes Review: Corresponding Angles lie on the same side of the transversal and on the same side of a line. Corresponding Angles are. Corresponding Angles in the figure are and, and, and, and also and. If m 4 = 55 find m 8 = If m 2 = 35 find m 6 = If m B = 162 find m F = If m G = 76 find m C = If m 7 = 37 find the measure of the corresponding angle m = m 5 = 139 find the measure of the corresponding angle m = 21

22 For each of the following find the value of the variable and the measures of the two angles. 22

23 Property Name Addition Property of Equality 2.6 Algebraic Proofs Notes Property Description If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a c = b c. Multiplication Property of Equality If a = b, then ac = bc. Division Property of Equality If a = b and c 0, then a c = b c. Reflexive Property of Equality a = a Symmetric Property of Equality If a = b, then b = a. Transitive Property of Equality Substitution Property of Equality If a = b and b = c, thena = c. If a = b, then b can be substituted for a in any expression. Guided Practice 1. Prove that if 5(x + 4) = 70, then x = 18. Write justification for each step. Your Turn 2. Solve 2(5 3y) 4(y + 7) = 92. Write a justification for each step. 23

24 Guided Practice 3. Your Turn

25 2.7, 2.8 Mini Proofs Notes A mini-geometry proof deals with knowing definitions for geometry terms and using them to show why something is the way it is. A few more properties to know for proofs. You will also need to know definitions from previous lessons. Name Description Segment Addition Postulate Reflexive Property of Congruence Symmetric Property of Congruence Transitive Property of Congruence Angle Addition Postulate Supplement Theorem Complement Theorem Congruent Supplements Theorem Congruent Complements Theorem If two angles form a linear pair, then they are supplementary. If two noncommon sides of two adjacent angles form a right angle, then the angles are complimentary. Vertical Angles Theorem Guided Practice Your Turn 3. 25

26 Guided Practice Your Turn

27 OC 1.7/3.5 Proofs about Parallel and Perpendicular Lines Notes 2 PROOF Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles have the same measure. Given: p q Prove: m 3 = m 5 Complete the proof by writing the missing reasons. Choose from the following reasons. You may use a reason more than once. Statements 1. p q 1. Reasons 2. 3 and 6 are supplementary m 3 + m 6 = and 6 are a linear pair and 6 are supplementary m 5 + m 6 = m 3 + m 6 = m 5 + m m 3 = m 5 8. REFLECT 2a. Suppose m 4 = 57 in the above figure. Describe two different ways to determine m 6. 2b. In the above figure, explain why 1, 3, 5, and 7 all have the same measure. 2c. In the above figure, is it possible for all eight angles to have the same measure? If so, what is that measure? 27

28 3 PROOF Corresponding Angles Theorem If two parallel lines are cut by a transversal, then the pairs of corresponding angles have the same measure. Given: p q Prove: m 1 = m 5 Complete the proof by writing the missing reasons. Statements 1. p q 1. Reasons 2. m 3 = m m 1 = m m 1 = m 5 4. REFLECT 3a. Explain how you can you prove the Corresponding Angles Theorem using the Same-Side Interior Angles Postulate and a linear pair of angles. In the diagram, suppose p q and line t is perpendicular to line p. Can you conclude that line t is perpendicular to line q? Explain. 28

Date Name of Lesson Assignments & Due Dates

Date Name of Lesson Assignments & Due Dates Basic Terms Points, Lines and Planes Constructions (Copy Angle and Segment) Distance Formula Activity for Distance Formula Midpoint Formula Quiz Angle Measure