1. Warm-Up #14, Wednesday, 3/9 3 1 2

2. Homework
* Triangle Proofs Worksheet #1, 2, 3, 4, 6, 10

3. Properties of Parallel Lines

6. Key Concepts, continued
Theorem
Angles supplementary to the same angle or to congruent angles are congruent.
If and , then
1.8.1: Proving the Vertical Angles Theorem

7. Key Concepts, continued (continued)
Theorem
Perpendicular Bisector Theorem
If a point lies on the perpendicular bisector of a segment, then that point is equidistant from the endpoints of the segment.
If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.
1.8.1: Proving the Vertical Angles Theorem

8. Key Concepts, continued
Theorem
If is the perpendicular bisector of , then DA = DC.
If DA = DC, then is the perpendicular bisector of
1.8.1: Proving the Vertical Angles Theorem

9. Midpoint is the middle point of a line segment
Midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

10. 1 2 1 2 PROPERTIES OF PARALLEL LINES POSTULATE
POSTULATE 15 Corresponding Angles Postulate
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. 1 2

11. 3 4 3 4 PROPERTIES OF PARALLEL LINES THEOREMS ABOUT PARALLEL LINES
THEOREM 3.4 Alternate Interior Angles
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. 3 4

12. 5 6 m 5 + m 6 = 180° PROPERTIES OF PARALLEL LINES
THEOREMS ABOUT PARALLEL LINES
THEOREM 3.5 Consecutive Interior Angles
If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
m m 6 = 180° 5 6

13. 7 8 7 8 PROPERTIES OF PARALLEL LINES THEOREMS ABOUT PARALLEL LINES
THEOREM 3.6 Alternate Exterior Angles
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. 7 8

14. j k PROPERTIES OF PARALLEL LINES THEOREMS ABOUT PARALLEL LINES
THEOREM 3.7 Perpendicular Transversal
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
j k

15. 1  3 Corresponding Angles Postulate
Proving the Alternate Interior Angles Theorem
Prove the Alternate Interior Angles Theorem.
SOLUTION
GIVEN
p || q
PROVE
Statements
Reasons
p || q Given 1
1  Corresponding Angles Postulate 2 3
3  Vertical Angles Theorem
1  Transitive property of Congruence 4

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

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

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

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

20. Proving Triangles Congruent
MODELING A REAL-LIFE SITUATION
Proving Triangles Congruent
ARCHITECTURE 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 ? D G A R
SOLUTION
GIVEN
DR AG
RA RG
PROVE
 DRA  DRG

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

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

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

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

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

26. 1 m 2 of a circle 50 m 1 = m 2 1 m 1 of a circle 50
Estimating Earth’s Circumference: History Connection
m 2 1 50
of a circle
Using properties of parallel lines, he knew that
m = m 2
He reasoned that
m 1 1 50
of a circle

27. 1 m 1 of a circle 50 1 of a circle 50 50(575 miles) 29,000 miles
Estimating Earth’s Circumference: History Connection
m 1 1 50
of a circle
The distance from Syene to Alexandria was believed to be 575 miles
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 = m ?

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