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Download Mth 97 Fall 2012 Sections 5.1 and 5.2 Section 5.1 – Indirect
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Transcript
Mth 97 Fall 2012 Sections 5.1 and 5.2 Section 5.1 – Indirect Reasoning and the Parallel Postulates Indirect reasoning or indirect proof assumes that the hypothesis of the result we are trying to prove and the negation (or opposite) of the conclusion. Then we show that this situation leads to a contradiction. Because the situation is not possible, we conclude that our assumption must be incorrect and the conclusion as given in the original statement must follow from the hypothesis. (See page 241.) Parallel Lines (vocabulary) ______________ lines are lines that do not lie in the same plane and do not intersect. ___________________ lines are lines that lie in the same plane and do not intersect. (same slope) A ________________________ is a line that intersects two other lines. t 1 2 3 l 4 5 6 7 8 m ____________________ __________________ angles are interior angles on opposite sides of a transversal. ____________________ __________________ angles are exterior angles on opposite sides of a transversal. _____________________________ angles are non-adjacent angles on the same side of the transversal, one in the interior and one in the exterior. Theorems that can be used to show that two lines are parallel Theorem 5.1 – If two lines cut by a transversal form a pair of congruent alternate interior angles, then the lines are parallel. (See sketch above) If 3 5 or 4 6 then Corollary 5.2 – If tw o lines are both perpendicular to a transversal, then the lines are parallel. 4 6 r s t The converse is also true. 1 Mth 97 Fall 2012 Sections 5.1 and 5.2 Corollary 5.3 – If two lines cut by a transversal form a pair of congruent corresponding angles with the transversal, then the lines are parallel. m n t Corollary 5.4 – If two lines cut by a transversal form a pair of supplementary interior angles on the same side of the transversal, then the lines are parallel. m n t Postulate 5.1 – The Parallel Postulate (for Euclidean Geometry) Given a line l and a point P not on l, there is only one line m containing P such that l m . P Theorem 5.5 and Corollaries 5.7 and 5.8 are the converses of Corollaries 5.1, 5.3 and 5.4 and are summarized below. If two lines are parallel, then… x y Alternate interior angles are congruent. 1 2 4 3 x Corresponding angles are congruent. If x Consecutive interior angles are supplementary. y 5 6 8 7 y and the m2 115 , find the measures of the rest of the angles. m1 m3 m4 m6 m7 m8 m5 Do ICA 8 2 Mth 97 Fall 2012 Sections 5.1 and 5.2 Section 5.2 – Important Theorems based on the Parallel Postulate Theorem 5.9 – Angle Sum in a Triangle Theorem – The sum of the angle measures in a triangle is 180°. Corollary 5.10 – The Exterior Angle Theorem – an exterior triangle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. 2 1 E 3 K Given: 4 G Prove: Statement Reason m3 m4 180 m1 m2 m3 180 m3 m4 m1 m2 m3 m4 m1 m2 1. 2. 3. 4. 1. 2. 3. 4. More Triangular Congruence Corollary 5.11 – AAS – If two angles and a side of one triangle are congruent to the corresponding angles and side in another triangle, then the triangles are congruent. N E D W A If Y then Theorem 5.12 – HA Congruence – If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent. R A D T If O G then Theorem 5.13 – Angle Bisector Theorem – A point is on the bisector of an angle if and only if it is equidistant from the sides of the angle. P is on the bisector of A if and only if PB = PC. Proof is on pages 253-254. Next: City Designer Project Directions 3