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

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

Properties of Parallel Lines

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

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

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

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.

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 1 2

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 3 4

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 5 + m 6 = 180° 5 6

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 7 8

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

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

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

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 4 = 125° Linear Pair Postulate m 4 + (x + 15)° = 180° Substitute. 125° + (x + 15)° = 180° Subtract. x = 40°

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

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

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

Proving Triangles Congruent D 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

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

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 = 3 2 + 5 2 = 5 2 + 3 2 = 34 = 34

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

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

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 1 = m 2 He reasoned that m 1 1 50 of a circle

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 1 = m 2 ?

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