Geometry Angles formed by Parallel Lines and Transversals CONFIDENTIAL

Give an example of each angle pair. Warm Up Give an example of each angle pair. 1) Alternate interior angles 2) Alternate exterior angles 3)Same side interior angles CONFIDENTIAL

Parallel, perpendicular and skew lines When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed. 1 2 3 4 5 6 7 8 There are several special pairs of angles formed from this figure. Vertical pairs: Angles 1 and 4  Angles 2 and 3  Angles 5 and 8  Angles 6 and 7 CONFIDENTIAL

Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3 Supplementary pairs: Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3 Angles 5 and 6 Angles 6 and 8 Angles 7 and 8 Angles 5 and 7 1 2 3 4 5 6 7 8 Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs. There are three other special pairs of angles. These pairs are congruent pairs. CONFIDENTIAL

Corresponding angle postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2 3 4 5 6 7 8 p q t 1 3 2 4 5 7 6 8 CONFIDENTIAL

Using the Corresponding angle postulate Find each angle measure. A) m( ABC) 800 x0 B C A x = 80 corresponding angles m( ABC) = 800 CONFIDENTIAL

subtract x from both sides B) m( DEF) (2x-45)0 = (x+30)0 corresponding angles x – 45 = 30 subtract x from both sides x = 75 add 45 to both sides m( DEF) = (x+30)0 = (75+30)0 = 1050 CONFIDENTIAL

Now you try! 1) m( DEF) R x0 S 1180 Q CONFIDENTIAL

Remember that postulates are statements that are accepted without proof. Since the Corresponding Angles postulate is given as a postulate, it can be used to prove the next three theorems. CONFIDENTIAL

Alternate interior angles theorem If two parallel lines are cut by a transversal, then the two pairs of Alternate interior angles are congruent. Hypothesis Conclusion 1 2 4 3 1 3 2 4 CONFIDENTIAL

Alternate exterior angles theorem If two parallel lines are cut by a transversal, then the two pairs of Alternate exterior angles are congruent. Hypothesis Conclusion 5 6 8 7 5 7 6 8 CONFIDENTIAL

Same-side interior angles theorem If two parallel lines are cut by a transversal, then the two pairs of Same-side interior angles are supplementary. Hypothesis Conclusion 1 2 4 3 m 1 + m 4 =1800 m 2 + m 3 =1800 CONFIDENTIAL

Alternate interior angles theorem 1 2 3 m l Given: l || m Prove: 2 3 Proof: 1 3 l || m Given Corresponding angles 2 3 2 1 Vertically opposite angles CONFIDENTIAL

Finding Angle measures Find each angle measure. A) m( EDF) 1250 B C A x0 D E F x = 1250 m( DEF) = 1250 Alternate exterior angles theorem CONFIDENTIAL

R T B) m( TUS) S U 36x = 180 x = 5 m( TUS) = 23(5)0 = 1150 13x0 23x0 Same-side interior angles theorem 36x = 180 Combine like terms x = 5 divide both sides by 36 m( TUS) = 23(5)0 Substitute 5 for x = 1150 CONFIDENTIAL

2) Find each angle measure. Now you try! 2) Find each angle measure. B C E D (2x+10)0 A (3x-5)0 CONFIDENTIAL

By the Alternative Exterior Angles Theorem, (25x+5y)0 = 1250 A treble string of grand piano are parallel. Viewed from above, the bass strings form transversals to the treble string. Find x and y in the diagram. (25x+5y)0 (25x+4y)0 1200 1250 By the Alternative Exterior Angles Theorem, (25x+5y)0 = 1250 By the Corresponding Angles Postulates, (25x+4y)0 = 1200 (25x+5y)0 = 1250 - (25x+4y)0 = 1200 y = 5 25x+5(5) = 125 x = 4, y = 5 Subtract the second equation from the first equation Substitute 5 for y in 25x +5y = 125. Simplify and solve for x. CONFIDENTIAL

3) Find the measure of the acute angles in the diagram. Now you try! 3) Find the measure of the acute angles in the diagram. (25x+5y)0 (25x+4y)0 1200 1250 CONFIDENTIAL

Find each angle measure: Assessment Find each angle measure: 1) m( JKL) 2) m( BEF) (7x-14)0 (4x+19)0 G A B C F D H E 1270 x0 K J L CONFIDENTIAL

Find each angle measure: 1 (3x+9)0 6x0 A B C D Y X E Z 4) m( CBY) CONFIDENTIAL

Find each angle measure: 5) m( KLM) 1150 Y0 K M L 6) m( VYX) Y X W Z (2a+50)0 V 4a0 CONFIDENTIAL

State the theorem or postulate that is related to the measures of the angles in each pair. Then find the angle measures: 1 2 3 4 5 7) m 1 = (7x+15)0 , m 2 = (10x-9)0 8) m 3 = (23x+15)0 , m 4 = (14x+21)0 CONFIDENTIAL

Parallel, perpendicular and skew lines Let’s review Parallel, perpendicular and skew lines When a transversal cuts (or intersects) parallel lines several pairs of congruent and supplementary angles are formed. 1 2 3 4 5 6 7 8 There are several special pairs of angles formed from this figure. Angles 1 and 4  Angles 2 and 3  Angles 5 and 8  Angles 6 and 7 Vertical pairs: CONFIDENTIAL

Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3 Supplementary pairs: Angles 1 and 2 Angles 2 and 4 Angles 3 and 4 Angles 1 and 3 Angles 5 and 6 Angles 6 and 8 Angles 7 and 8 Angles 5 and 7 1 2 3 4 5 6 7 8 Recall that supplementary angles are angles whose angle measure adds up to 180°. All of these supplementary pairs are linear pairs. There are three other special pairs of angles. These pairs are congruent pairs. CONFIDENTIAL

Corresponding angle postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2 3 4 5 6 7 8 p q t 1 3 2 4 5 7 6 8 CONFIDENTIAL

Using the Corresponding angle postulate Find each angle measure. A) m( ABC) 800 x0 B C A x = 80 corresponding angles m( ABC) = 800 CONFIDENTIAL

subtract x from both sides B) m( DEF) (2x-45)0 = (x+30)0 corresponding angles x – 45 = 30 subtract x from both sides x = 75 add 45 to both sides m( DEF) = (x+30)0 = (75+30)0 = 1050 CONFIDENTIAL

Alternate interior angles theorem If two parallel lines are cut by a transversal, then the two pairs of Alternate interior angles are congruent. Hypothesis Conclusion 1 2 4 3 1 3 2 4 CONFIDENTIAL

Alternate exterior angles theorem If two parallel lines are cut by a transversal, then the two pairs of Alternate exterior angles are congruent. Hypothesis Conclusion 5 6 8 7 5 7 6 8 CONFIDENTIAL

Same-side interior angles theorem If two parallel lines are cut by a transversal, then the two pairs of Same-side interior angles are supplementary. Hypothesis Conclusion 1 2 4 3 m 1 + m 4 =1800 m 2 + m 3 =1800 CONFIDENTIAL

Alternate interior angles theorem 1 2 3 m l Given: l || m Prove: 2 3 Proof: 1 3 l || m Given Corresponding angles 2 3 2 1 Vertically opposite angles CONFIDENTIAL

You did a great job today! CONFIDENTIAL