1. Learning Target #17 I can use theorems, postulates or definitions to prove that… a. vertical angles are congruent. b. When a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent, and same- side interior angles are supplementary.

2. Proving Vertical Angle Theorem THEOREM Vertical Angles Theorem Vertical angles are congruent 1 3,24

3. Proving Vertical Angle Theorem PROVE 5757 GIVEN 5 and6 are a linear pair, 6 and7 are a linear pair 1 2 3 StatementsReasons 5 and6 are a linear pair, Given 6 and7 are a linear pair 5 and6 are supplementary, Linear Pair Postulate 6 and7 are supplementary 5 7 Congruent Supplements Theorem

4. Third Angles Theorem

5. Goal 1 The Third Angles Theorem below follows from the Triangle Sum Theorem. THEOREM Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. If  A   D and  B   E, then  C   F.

6. P ROPERTIES OF P ARALLEL L INES POSTULATE POSTULATE 15 Corresponding Angles Postulate 1 2 1 2 If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.

7. P ROPERTIES OF P ARALLEL L INES THEOREMS ABOUT PARALLEL LINES THEOREM 3.4 Alternate Interior Angles 3 4 3 4 If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

8. P ROPERTIES OF P ARALLEL L INES THEOREMS ABOUT PARALLEL LINES THEOREM 3.5 Consecutive Interior Angles 5 6 m 5 + m 6 = 180° If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

9. P ROPERTIES OF P ARALLEL L INES THEOREMS ABOUT PARALLEL LINES THEOREM 3.6 Alternate Exterior Angles 7 8 7 8 If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.

10. P ROPERTIES OF P ARALLEL L INES THEOREMS ABOUT PARALLEL LINES THEOREM 3.7 Perpendicular Transversal j k If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.

11. Proving the Alternate Interior Angles Theorem Prove the Alternate Interior Angles Theorem. S OLUTION GIVEN p || q p || qGiven StatementsReasons 1 2 3 4 PROVE 1 2 1  3 Corresponding Angles Postulate 3  2 Vertical Angles Theorem 1  2 Transitive property of Congruence

12. Using Properties of Parallel Lines S OLUTION Given that m 5 = 65°, find each measure. Tell which postulate or theorem you use. Linear Pair Postulate m 7 = 180° – m 5 = 115° Alternate Exterior Angles Theorem m 9 = m 7 = 115° Corresponding Angles Postulate m 8 = m 5 = 65° m 6 = m 5 = 65° Vertical Angles Theorem

13. Using Properties of Parallel Lines Use properties of parallel lines to find the value of x. S OLUTION Corresponding Angles Postulate m 4 = 125° Linear Pair Postulate m 4 + (x + 15)° = 180° Substitute. 125° + (x + 15)° = 180° P ROPERTIES OF S PECIAL P AIRS OF A NGLES Subtract. x = 40°

14. Over 2000 years ago Eratosthenes estimated Earth’s circumference by using the fact that the Sun’s rays are parallel. When the Sun shone exactly down a vertical well in Syene, he measured the angle the Sun’s rays made with a vertical stick in Alexandria. He discovered that Estimating Earth’s Circumference: History Connection m 2 1 50 of a circle

15. Estimating Earth’s Circumference: History Connection m 2 1 50 of a circle Using properties of parallel lines, he knew that m 1 = m 2 He reasoned that m 1 1 50 of a circle

16. The distance from Syene to Alexandria was believed to be 575 miles Estimating Earth’s Circumference: History Connection m 1 1 50 of a circle Earth’s circumference 1 50 of a circle 575 miles Earth’s circumference 50(575 miles) Use cross product property 29,000 miles How did Eratosthenes know that m 1 = m 2 ?

17. Estimating Earth’s Circumference: History Connection How did Eratosthenes know that m 1 = m 2 ? S OLUTION Angles 1 and 2 are alternate interior angles, so 1  2 By the definition of congruent angles, m 1 = m 2 Because the Sun’s rays are parallel,

18. Example Using the Third Angles Theorem Find the value of x. SOLUTIO N In the diagram,  N   R and  L   S. From the Third Angles Theorem, you know that  M   T. So, m  M = m  T. From the Triangle Sum Theorem, m  M = 180˚– 55˚ – 65˚ = 60˚. m  M = m  T 60˚ = (2 x + 30)˚ 30 = 2 x 15 = x Third Angles Theorem Substitute. Subtract 30 from each side. Divide each side by 2.

19. Goal 2 SOLUTIO N Paragraph Proof From the diagram, you are given that all three corresponding sides are congruent. , NQPQ , MNRP  QMQR and Because  P and  N have the same measures,  P   N. By the Vertical Angles Theorem, you know that  PQR   NQM. By the Third Angles Theorem,  R   M. Decide whether the triangles are congruent. Justify your reasoning. So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles, . PQRNQM Proving Triangles are Congruent Learning Target

20. Example Proving Two Triangles are Congruent AB C D E ||, DCAB , DCAB E is the midpoint of BC and AD. Plan for Proof Use the fact that  AEB and  DEC are vertical angles to show that those angles are congruent. Use the fact that BC intersects parallel segments AB and DC to identify other pairs of angles that are congruent. GIVEN PROVE . AEBDEC Prove that . AEBDEC

21. Example Proving Two Triangles are Congruent StatementsReasons  EAB   EDC,  ABE   DCE  AEB   DEC E is the midpoint of AD, E is the midpoint of BC , DE AE  CEBE Given Alternate Interior Angles Theorem Vertical Angles Theorem Given Definition of congruent triangles Definition of midpoint ||, DCAB DCAB  SOLUTION  AEBDEC AB C D E Prove that . AEB DEC

22. Goal 2 You have learned to prove that two triangles are congruent by the definition of congruence – that is, by showing that all pairs of corresponding angles and corresponding sides are congruent. THEOREM Theorem 4.4 Properties of Congruent Triangles Reflexive Property of Congruent Triangles D E F A B C J K L Every triangle is congruent to itself. Symmetric Property of Congruent Triangles Transitive Property of Congruent Triangles If , then . ABCDEF ABC If  and , then . JKLABCDEF ABCJKL Proving Triangles are Congruent

23. 1 Using the SAS Congruence Postulate Prove that  AEB  DEC. 2 3  AEB   DEC SAS Congruence Postulate 21 AE  DE, BE  CEGiven 1  2Vertical Angles Theorem Stateme nts Reasons

24. D G A R Proving Triangles Congruent M ODELING A R EAL- L IFE S ITUATION PROVE  DRA  DRG S OLUTION A RCHITECTURE You are designing the window shown in the drawing. You want to make  DRA congruent to  DRG. You design the window so that DR AG and RA  RG. Can you conclude that  DRA   DRG ? GIVEN DR AG RA RG

25. 2 3 4 5 6 SAS Congruence Postulate  DRA   DRG 1 Proving Triangles Congruent Given DR AG If 2 lines are, then they form 4 right angles. DRA and DRG are right angles. Right Angle Congruence Theorem DRA  DRG Given RA  RG Reflexive Property of Congruence DR  DR StatementsReasons D GAR GIVEN PROVE  DRA  DRG DR AG RA RG

26. Congruent Triangles in a Coordinate Plane AC  FH AB  FG AB = 5 and FG = 5 S OLUTION Use the SSS Congruence Postulate to show that  ABC   FGH. AC = 3 and FH = 3

27. Congruent Triangles in a Coordinate Plane d = (x 2 – x 1 ) 2 + ( y 2 – y 1 ) 2 = 3 2 + 5 2 = 34 BC = (– 4 – (– 7)) 2 + (5 – 0 ) 2 d = (x 2 – x 1 ) 2 + ( y 2 – y 1 ) 2 = 5 2 + 3 2 = 34 GH = (6 – 1) 2 + (5 – 2 ) 2 Use the distance formula to find lengths BC and GH.

28. Congruent Triangles in a Coordinate Plane BC  GH All three pairs of corresponding sides are congruent,  ABC   FGH by the SSS Congruence Postulate. BC = 34 and GH = 34