3.2 – Use Parallel Lines and Transversals Geometry Chapter 3 3.2 – Use Parallel Lines and Transversals

Warm-Up 1.) <1 𝑎𝑛𝑑<6 Alt. Exterior Angles 2.) <4 𝑎𝑛𝑑<7 Identify the angle pairs. 1.) <1 𝑎𝑛𝑑<6 Alt. Exterior Angles 2.) <4 𝑎𝑛𝑑<7 Alt. Interior Angles 3.) <3 𝑎𝑛𝑑<4 Corresponding Angles 4.) <2 𝑎𝑛𝑑<7 Consec. Interior Angles 5 1 2 3 7 8 4 6

Use Parallel Lines and Transversals Objective: Students will be able to use parallel lines and specific angle pairs to find angle measures. Agenda Postulates/Practice Theorems/Practice Proving Theorems

For the Postulates List Postulate 15 – Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 5 𝒎 𝒏 𝒕 𝒎 ∥𝒏 <𝟏≅ <𝟓

For the Theorems List 𝒎 ∥𝒏 <𝟑≅ <𝟕 Theorem 3.1– Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 7 3 𝒎 𝒏 𝒕 𝒎 ∥𝒏 <𝟑≅ <𝟕

Practice – Postulate 15/Thm 3-1 Use the diagram to find 𝑚<3 and 𝑚<5. 𝒎<𝟓=𝟏𝟒𝟎° (Alt. Int. <‘s) 𝒎<𝟑=𝟏𝟒𝟎° (Corr. <‘s)

For the Theorems List 𝒎 ∥𝒏 <𝟒≅ <𝟔 Theorem 3.2– Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 4 6 𝒎 𝒏 𝒕 𝒎 ∥𝒏 <𝟒≅ <𝟔

Practice – Thm 3-2 𝒎<𝟓=𝟖𝟕° (Alt. Int. <‘s) Use the diagram to find 𝑚<5 and 𝑚<6. 𝒎<𝟓=𝟖𝟕° (Alt. Int. <‘s) 𝒎<𝟔=𝟗𝟑° (Alt. Ext. <‘s)

For the Theorems List 𝒎 ∥𝒏 <𝟐 𝐚𝐧𝐝 <𝟖 𝐚𝐫𝐞 𝐬𝐮𝐩𝐩𝐥𝐞𝐦𝐞𝐧𝐭𝐚𝐫𝐲 Theorem 3.3– Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary. 2 8 𝒎 𝒏 𝒕 𝒎 ∥𝒏 <𝟐 𝐚𝐧𝐝 <𝟖 𝐚𝐫𝐞 𝐬𝐮𝐩𝐩𝐥𝐞𝐦𝐞𝐧𝐭𝐚𝐫𝐲 𝒎<𝟐+𝒎<𝟖=𝟏𝟖𝟎°

Practice – Thm 3-2 𝒎<𝟑=𝟏𝟑𝟓° (Alt. Ext. <‘s) Use the diagram to find 𝑚<3, 𝑚<4, and 𝑚<7. 𝒎<𝟑=𝟏𝟑𝟓° (Alt. Ext. <‘s) 𝒎<𝟒=𝟏𝟑𝟓° (Consec. Int. <‘s) 𝒎<𝟕=𝟒𝟓° (Corr. <‘s)

Example Equation: 115+𝑥+5=180 120+𝑥=180 𝒙=𝟔𝟎 115° 4 (𝑥+5)° Use the given diagram to find the value of x. Equation: 115+𝑥+5=180 120+𝑥=180 𝒙=𝟔𝟎 115° 4 (𝑥+5)°

Exit Ticket. 1.) Corresponding Angles Theorem <1≅ <2 Given the diagram, give a statement that can be made using the following postulate/theorem. 1.) Corresponding Angles Theorem <1≅ <2 2.) Alternate Exterior Angles Theorem <3≅ <8 3.) Consecutive Interior Angles Theorem 𝑚<5+𝑚<4=180° 4.) Alternate Interior Angles Theorem <4≅ <7

Proving Theorem 3-1 Given: 𝑚∥𝑛 Prove: <2≅ <3 Statements Reasons 1.𝑚∥𝑛 1. Given 2. <1≅ <2 2. Vertical Angle Congruence Theorem 3. <1≅ <3 3. Corresponding Angles Postulate 4. <2≅ <3 4. Transitive Property

Proving Theorem 3-2 Given: 𝑚∥𝑛 Prove: <1≅ <2 Statements Reasons 1.𝑚∥𝑛 1. Given 2. <2≅ <3 2. Vertical Angle Congruence Theorem 3. <1≅ <3 3. Corresponding Angles Postulate 4. <1≅ <2 4. Transitive Property

Proving Theorem 3-3 Given: 𝑚∥𝑛 Prove: <4 and<5 are supplementary Statements Reasons 1.𝑚∥𝑛 1. Given 2. <4≅ <6 2. Alternate Interior Angles Theorem 3. 𝑚<4=𝑚<6 3. Def. of Congruent Angles 4. <5 and<6 are supplementary 4. Linear Pair Postulate 5. 𝑚<5+𝑚<6=180° 5. Def. of Supplementary Angles 6. 𝑚<5+𝑚<4=180° 6. Substitution Property 7.<4 and<5 are supplementary 7. Def. of Supplementary Angles