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Warm-Up #14, Wednesday, 3/9 1 2 3 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 mÐ1+ mÐ2 = 180 and mÐ2 + mÐ3 = 180, then Ð1@ Ð3. 6 1.8.1: Proving the Vertical Angles Theorem Key Concepts, 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. (continued) 7 1.8.1: Proving the Vertical Angles Theorem Key Concepts, continued Theorem If is the perpendicular bisector of AC, then DA = DC. If DA = DC, then is the perpendicular bisector of AC. 1.8.1: Proving the Vertical Angles Theorem 8 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. 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 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 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. 5 6 m 5+m 6 = 180° 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 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 Proving the Alternate Interior Angles Theorem Prove the Alternate Interior Angles Theorem. SOLUTION GIVEN p || q PROVE 1 Statements 2 Reasons 1 p || q Given 2 1 3 3 2 Vertical Angles Theorem 4 1 Transitive property of Congruence 3 2 Corresponding Angles Postulate 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° m 7 = 180° – m m 8 = m 5 = 65° Corresponding Angles Postulate m 9 = m 7 = 115° Alternate Exterior Angles Theorem Vertical Angles Theorem 5 = 115° Linear Pair Postulate PROPERTIES OF SPECIAL PAIRS OF ANGLES Using Properties of Parallel Lines Use properties of parallel lines to find the value of x. SOLUTION m m 4 = 125° 4 + (x + 15)° = 180° 125° + (x + 15)° = 180° x = 40° Corresponding Angles Postulate Linear Pair Postulate Substitute. Subtract. Proving Two Triangles are Congruent Prove that AEB DEC . A B E SOLUTION D C Statements Reasons AB || DC , AB DC 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 AE DE , BE CE Definition of midpoint AEB DEC Definition of congruent triangles Example Using the SAS Congruence Postulate Prove that AEB DEC. 1 2 3 1 Statements Reasons AE DE, BE CE Given 1 2 AEB DEC 2 Vertical Angles Theorem SAS Congruence Postulate 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 SOLUTION GIVEN PROVE DR AG RA RG DRA A DRG R G Proving Triangles Congruent GIVEN PROVE DR AG RA RG DRA D DRG A Statements R G Reasons Given 1 DR AG 2 DRA and DRG are right angles. If 2 lines are , then they form 4 right angles. 3 DRA 4 RA RG Given 5 DR DR Reflexive Property of Congruence 6 DRA DRG SAS Congruence Postulate DRG Right Angle Congruence Theorem 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= BC = (x 2 – x1 ) 2 + ( y2 – y1 ) 2 (– 4 – (– 7)) 2 + (5 – 0 ) 2 d= GH = (x 2 – x1 ) 2 + ( y2 – y1 ) 2 (6 – 1) 2 + (5 – 2 ) 2 = 32 + 52 = 52 + 32 = 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 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 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 1 50 of a circle Earth’s circumference 575 miles Earth’s circumference 50(575 miles) Use cross product property 29,000 miles How did Eratosthenes know that m 1=m 2? Estimating Earth’s Circumference: History Connection How did Eratosthenes know that m 1=m 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 2?