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INVESTIGATION 1 TRANSVERSAL AND ANGLE RELATIONSHIPS OF PARALLEL & NON-PARALLEL LINES TRANSVERSAL A transversal is a line that intersects 2 or more coplanar lines at different points In the diagram, which line is the transversal? line t How many angles are formed? 8 CLASSIFYING THE ANGLES How could you describe the position of ∠1 in the top four angles? Upper left side of transversal What angle is in the upper left of the bottom four angles? ∠5 ∠1 & ∠5 are called corresponding angles CORRESPONDING ANGLES A pair of corresponding angles is any pair of angles that lie on the same side of a transversal and on the same side of the other 2 lines Besides ∠1 & ∠5, what are some other corresponding angles? ∠2 & ∠6 ∠3 & ∠7 ∠4 & ∠8 CLASSIFYING OTHER PAIRS OF ANGLES ∠3, ∠4, ∠5, & ∠6 are between lines m & n and are called interior angles What would be a good name for ∠1, ∠2, ∠7, & ∠8? exterior angles Angles on opposite sides of the transversal are called alternate angles ALTERNATE INTERIOR ANGLES A pair of alternate interior angles is any pair of nonadjacent angles that lie on opposite sides of the transversal and between the other 2 lines ∠3 & ∠6 ∠4 & ∠5 ALTERNATE EXTERIOR ANGLES A pair of alternate exterior angles is any pair of nonadjacent angles that lie on opposite sides of the transversal and outside the other 2 lines ∠1 & ∠8 ∠2 & ∠7 SAME-SIDE INTERIOR ANGLES (CONSECUTIVE INTERIOR) A pair of same-side interior angles is any pair of angles that lie on same side of the transversal and between the other 2 lines ∠3 & ∠5 ∠4 & ∠6 SAME-SIDE EXTERIOR ANGLES (CONSECUTIVE EXTERIOR) A pair of same-side exterior angles is any pair of angles that lie on same side of the transversal and outside the other 2 lines ∠1 & ∠7 ∠2 & ∠8 WHAT CAN BE SAID ABOUT THE LINES 𝑚 & 𝑛 IN FIGURE A THAT IS NOT TRUE IN FIGURE B? Figure A Figure B They are parallel IN THIS FIGURE, DO YOU THINK ∠1 ≅ ∠5? Yes What other angles appear to be congruent to ∠1? ∠4 & ∠8 Why is ∠1 ≅ ∠4? Vertical Angle Theorem Do lines m & n need to be parallel for that to be true? No WHAT ANGLES APPEAR TO BE CONGRUENT TO ∠7? ∠6, ∠3, & ∠2 Why is ∠7 ≅ ∠6? Vertical Angle Theorem Do lines m & n need to be parallel for that to be true? No IN THIS FIGURE, DO YOU THINK ∠1 ≅ ∠5? No Why or what is different from the previous example? Because the lines are not parallel What angle is congruent to ∠5 and why? ∠8 Vertical Angle Theorem POSTULATE 11: CORRESPONDING ANGLES POSTULATE If two parallel lines are cut by a transversal, then the corresponding angles are congruent. ∠1 ≅ ∠5 ∠2 ≅ ∠6 ∠3 ≅ ∠7 ∠4 ≅ ∠8 THEOREM 10-1: ALTERNATE INTERIOR ANGLES THEOREM If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. ∠3 ≅ ∠6 ∠4 ≅ ∠5 THEOREM 10-2: ALTERNATE EXTERIOR ANGLES THEOREM If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. ∠1 ≅ ∠8 ∠2 ≅ ∠7 OTHER RELATIONSHIPS OF ANGLES What is the relationship between ∠3 & ∠4? Why? They are supplementary Linear Pair Theorem If ∠4 ≅ ∠5 & ∠3 is supplementary to ∠4, then what is the relationship between ∠3 & ∠5? They are supplementary THEOREM 10-3: SAME-SIDE INTERIOR ANGLES THEOREM If two parallel lines are cut by a transversal, then the same-side interior angles are supplementary. ∠3 & ∠5 are supplementary ∠4 & ∠6 are supplementary THEOREM 10-4: SAME-SIDE EXTERIOR ANGLES THEOREM If two parallel lines are cut by a transversal, then the same-side exterior angles are supplementary. ∠1 & ∠7 are supplementary ∠2 & ∠8 are supplementary EXAMPLE: 𝑚∠8 = 110º FIND THE MEASURE OF THE OTHER ANGLES WHEN THE LINES ARE PARALLEL. JUSTIFY YOUR ANSWERS, 1. m∠7 = 70º, Linear Pair Theorem 2. m∠5 = 110º,Vertical Angle Theorem 3. m∠6 = 70º, Linear Pair Theorem 4. m∠4= 110º, Corresponding Angles Postulate 5. m∠2 = 70º, Same-side Exterior Angles Theorem 6. m∠1= 110º, Alternate Exterior Angles Theorem USE THE DIAGRAM TO IDENTIFY THE FOLLOWING 1. The alternate interior angles with transversal m. ∠2 & ∠9 or ∠4 & ∠10 2. ∠5 & ∠10 are same-side exterior angles for which transversal? transversal n 3. The corresponding angles with transversal p. ∠1 & ∠5 or ∠3 & ∠7 or ∠2 & ∠6 or ∠4 & ∠8 QUESTIONS/REVIEW Lines do not need to be parallel to identify the type of angle pairs However Postulate 11 and Theorems 10-1 to 10-4 are only true when the lines are parallel Future lessons we will use the converse of these postulate and theorems to show/prove lines are parallel So it is important to learn these concepts now Imagine how hard it would have been to learn division if you did not know how to multiply first