Learning Target

1. Learning Target 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, 2 4

3. Proving Vertical Angle Theorem 5 and 6 are a linear pair, GIVEN 6 and 7 are a linear pair 5 7 1 2 3 PROVE Statements Reasons 5 and 6 are a linear pair,Given 6 and 7 are a linear pair 5 and 6 are supplementary,Linear Pair Postulate 6 and 7 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DandBE,thenCF.

6. PROPERTIES OF PARALLEL LINES 1 2 POSTULATE POSTULATE 15Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2

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

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

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

10. PROPERTIES OF PARALLEL LINES jk THEOREMS ABOUT PARALLEL LINES THEOREM 3.7Perpendicular Transversal If a transversal is perpendicular to one of two parallellines, then it is perpendicular to the other.

11. Proving the Alternate Interior Angles Theorem 4 3 2 1 p || q GIVEN 1 2 PROVE Statements Reasons 1  3Corresponding Angles Postulate 3  2Vertical Angles Theorem 1  2Transitive property of Congruence Prove the Alternate Interior Angles Theorem. SOLUTION p || q Given

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

13. Using Properties of Parallel Lines m 4 =125° Corresponding Angles Postulate m 4 + (x + 15)° =180° Linear Pair Postulate 125° + (x + 15)° =180° Substitute. x =40° Subtract. PROPERTIES OF SPECIAL PAIRS OF ANGLES Use properties of parallel lines to findthe value of x. SOLUTION

14. Estimating Earth’s Circumference: History Connection 1 m 2 of a circle 50 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

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

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

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

18. Example Using the Third Angles Theorem Find the value of x. SOLUTION 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 Third Angles Theorem 60˚ = (2x+ 30)˚ Substitute. 30 = 2x Subtract 30 from each side. 15 = x Divide each side by 2.

19. Goal 2 Learning Target Proving Triangles are Congruent  ,  , and  RP MN PQ NQ QR QM PQR NQM So, all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. By the definition of congruent triangles,  . Decide whether the triangles are congruent. Justify your reasoning. SOLUTION Paragraph Proof From the diagram, you are given that all three corresponding sides are congruent. 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.

20. Example Proving Two Triangles are Congruent Prove that  . AEB DEC A B E || , AB DC D C E is the midpoint of BC and AD. GIVEN  . PROVE AEB DEC 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.  , AB DC

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

22. Goal 2 Proving Triangles are Congruent E B D A F C DEF DEF ABC DEF ABC ABC DEF JKL ABC JKL If  , then  . L If  and  , then . J K 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 4.4Properties of Congruent Triangles Reflexive Property of Congruent Triangles Every triangle is congruent to itself. Symmetric Property of Congruent Triangles Transitive Property of Congruent Triangles

23. Using the SAS Congruence Postulate Prove that AEBDEC. 1 2 1 2 Statements Reasons AE  DE, BE  CE Given 1  2Vertical Angles Theorem 3 AEBDEC SAS Congruence Postulate

24. Proving Triangles Congruent ARCHITECTURE You are designing the window shown in the drawing. You want to make DRAcongruent to DRG. You design the window so that DRAG and RARG. D A G R GIVEN DRAG RARG DRADRG PROVE MODELING A REAL-LIFE SITUATION Can you conclude that DRADRG? SOLUTION

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

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

27. Congruent Triangles in a Coordinate Plane d = (x2 – x1 )2+ (y2 – y1 )2 d = (x2 – x1 )2+ (y2 – y1 )2 BC = (–4 – (–7))2+ (5– 0)2 GH = (6 – 1)2+ (5– 2)2 = 32+ 52 = 52+ 32 = 34 = 34 Use the distance formula to find lengths BC and GH.

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