Angles and Parallel Lines Geometry D – Section 3.2

Angles and Parallel Lines We are going to investigate the relationship of various angles created by two parallel lines and a transversal. Obtain a ½ sheet of graph paper and a protractor. Construct two || lines and a transversal similar to the image on the next slide.

Angles and Parallel Lines Extend your lines the full height and width of the paper. Pause for time to work!

Angles and Parallel Lines Label the angles as shown below. 1 2 3 4 5 6 7 8 Pause for time to work!

Angles and Parallel Lines Measure all angles using a protractor to the nearest degree. 1 2 3 4 5 6 7 8 Pause for time to work!

Angles and Parallel Lines Measure all angles using a protractor to the nearest degree. 127o 53o 1 2 Note: Your measurements may be different values but should be in the same pattern. 3 4 53o 127o 127o 5 6 53o 7 8 53o 127o

Angles and Parallel Lines Identify the relationship between the following angles? 127o 53o 1 2 3 4 53o 127o 127o 5 6 53o 7 8 53o From Chapter 2, the angles are linear pairs. 127o What can be said about the measures of the linear pairs? Linear pairs are supplementary (sum to 180o).

Angles and Parallel Lines Identify the relationship between the following angles? 127o 53o 1 2 3 4 53o 127o 127o 5 6 53o 7 8 53o From Chapter 2, the angles are vertical angles. 127o What can be said about the measures of the vertical angles? Vertical angles are congruent angles.

Angles and Parallel Lines Identify the relationship between the following angles? 127o 53o 1 2 3 4 53o 127o 127o 5 6 53o 7 8 53o The angles are corresponding angles. 127o What can be said about the measures of the corresponding angles? The measures are equal and the angles are congruent.

Angles and Parallel Lines Corresponding Angles Postulate – If two parallel lines are cut by a transversal, then each pair of corresponding angles are congruent.

Angles and Parallel Lines Identify the relationship between the following angles? 127o 53o 1 2 3 4 53o 127o 127o 5 6 53o 7 8 53o The angles are alternate interior angles. 127o What can be said about the measures of the alternate interior angles? The measures are equal and the angles are congruent.

Angles and Parallel Lines Alternate Interior Angles Theorem – If two parallel lines are cut by a transversal, then each pair of alternate interior angles are congruent. You will prove this theorem as a homework problem!

Angles and Parallel Lines Identify the relationship between the following angles? 127o 53o 1 2 3 4 53o 127o 127o 5 6 53o 7 8 53o The angles are alternate interior angles. 127o What can be said about the measures of the alternate interior angles? The measures add to 180o.

Angles and Parallel Lines Consecutive Interior Angles Theorem – If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary (sum to 180o). You will prove this theorem as a homework problem!

Angles and Parallel Lines Identify the relationship between the following angles? 127o 53o 1 2 3 4 53o 127o 127o 5 6 53o 7 8 53o The angles are alternate exterior angles. 127o What can be said about the measures of the alternate interior angles? The measures are equal and the angles are congruent.

Angles and Parallel Lines Alternate Exterior Angles Theorem – If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent. Prove: Statement Reason ? p || q, t is a transversal of p & q Given t Corresponding ‘s are 1 2 ? p 3 4 ? 5 6 Vertical ‘s are q 7 8 ? Transitive Property

Angles and Parallel Lines Perpendicular Transversal Theorem – In a plane, if a line is perpendicular to one of two perpendicular lines, then it is perpendicular to the other. t p q If t is perpendicular ( ) to p, then it is also perpendicular to q. You will prove this theorem as a homework problem!

Angles and Parallel Lines Applications – Gather into groups of not more than 3. Work the following problems in your group. Compare your answers to those provided.

Angles and Parallel Lines Given j || k, Applications – Make a sketch of the problem in your notes. Find the measure of 3 4 5 2 1. 43o 1 7 8 6 9 Corresponds with 1. 11 10 2. 24o 12 Alternate exterior with 14. 13 3. 156o Linear pair with 9. 180o – 24o = 156o 14

Angles and Parallel Lines Given j || k, Find the measure of Applications – 3 4 5 2 4. 137o 1 7 8 6 9 Linear pair with 3. 11 10 5. 156o 12 Vertical angle with 10. 13 6. 43o Vertical with 1. Alternate Interior of 3. 14

Angles and Parallel Lines Applications – Find the values of x and y in each figure. Find the measure of each given angle. Note: Figures are not drawn to scale. Given: Pause for time to work!

Angles and Parallel Lines Applications – Solution Given: Linear pairs are supplementary. (5x + 2) + (9x + 10) = 180o 14x + 12 = 180 14x = 168 x = 12 Linear Pair By corresponding angles, 62o 118o 3y – 1 = 62 62o 3y = 63 y = 21 and

Angles and Parallel Lines Applications – Find the values of x, y and z in each figure. (2z)o (3x–3)o (4y+2)o 66o Pause for time to work!

Angles and Parallel Lines is a corresponding angle with the angle of 66o. Applications – (3x – 3)o and 66o are linear pairs and sum to 180o. (3x – 3)o + 66o = 180o 3x + 63 = 180 3x = 117, x = 39 (2z)o (3x–3)o 66o (4y + 2)o and 66o are congruent alternate interior angles. (4y + 2)o = 66o 4y = 64, y = 16 (4y+2)o 66o (3x–3)o and (2z)o are congruent alternate interior angles. (3x–3)o = 3(39) – 3 = 114o (2z)o = 114o, z = 57

Angles and Parallel Lines Applications – Find the values of x, y and z in each figure. There are other ways of doing this problem correctly. If you worked it a different way, would you be willing to share how you did it? (2z)o (3x–3)o (4y+2)o 66o

Angles and Parallel Lines Applications – Find the measures of all the angles on the object if the measure of angle 1 is 30o. Pause for time to work!

Angles and Parallel Lines Applications – Find the measures of all the angles on the object if the measure of angle 1 is 30o. Perpendicular transversal theorem. Perpendicular lines intersect in 4 right (90o) angles. 90o 90o 90o 90o 90o

Angles and Parallel Lines Applications – Find the measures of all the angles on the object if the measure of angle 1 is 30o. Vertical Angle Vertical Angle Alternate interior angle with angle 1. 30o 150o 150o 30o 90o 30o Linear pairs are supplementary. 90o 30o Given 90o 90o 90o Vertical Angles

Angles and Parallel Lines Applications – Find the measures of all the angles on the object if the measure of angle 1 is 30o. 30o Vertical angles. 150o 150o 30o 30o 90o 60o All angles have been found! 60o 90o 30o 90o 90o Since the transversal is , these two angles must add to 90o using angle addition. 90o

Angles and Parallel Lines Assignment – 3.2 / 17-20, 24, 26, 30, 32, 36, 38, 40, 43, 48, 55, 57, 60, 62 Please return your protractor!!!! Thank you Mr. Matzke!!!!