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Parallel Lines and Planes
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Section 3 - 1 Definitions
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Parallel Lines - coplanar lines that do not intersect
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Skew Lines - noncoplanar lines that do not intersect
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Parallel Planes - Parallel planes do not intersect
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THEOREM 3-1 If two parallel planes are cut by a third plane, then the lines of intersection are parallel.
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Transversal - is a line that intersects each of two other coplanar lines in different points to produce interior and exterior angles
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ALTERNATE INTERIOR ANGLES - two nonadjacent interior angles on opposite sides of a transversal
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Alternate Interior Angles 2 1 4 3
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ALTERNATE EXTERIOR ANGLES - two nonadjacent exterior angles on opposite sides of the transversal
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Alternate Exterior Angles 6 5 8 7
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Same-Side Interior Angles - two interior angles on the same side of the transversal
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Same-Side Interior Angles 2 1 4 3
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Corresponding Angles - two angles in corresponding positions relative to two lines cut by a transversal
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Corresponding Angles 6 5 2 1 4 3 8 7
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3 - 2 Properties of Parallel Lines
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Postulate 10 If two parallel lines are cut by a transversal, then corresponding angles are congruent.
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THEOREM 3-2 If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
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THEOREM 3-3 If two parallel lines are cut by a transversal, then same-side interior angles are supplementary.
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THEOREM 3-4 If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also.
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Section 3 - 3 Proving Lines Parallel
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Postulate 11 If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel
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THEOREM 3-5 If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
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THEOREM 3-6 If two lines are cut by a transversal and same- side interior angles are supplementary, then the lines are parallel.
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THEOREM 3-7 In a plane two lines perpendicular to the same line are parallel.
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THEOREM 3-8 Through a point outside a line, there is exactly one line parallel to the given line.
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THEOREM 3-9 Through a point outside a line, there is exactly one line perpendicular to the given line.
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THEOREM 3-10 Two lines parallel to a third line are parallel to each other.
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Ways to Prove Two Lines Parallel 1. Show that a pair of corresponding angles are congruent. 2. Show that a pair of alternate interior angles are congruent 3. Show that a pair of same-side interior angles are supplementary. 4. In a plane show that both lines are to a third line. 5. Show that both lines are to a third line
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Section 3 - 4 Angles of a Triangle
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Triangle – is a figure formed by the segments that join three noncollinear points
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Scalene triangle – is a triangle with all three sides of different length.
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Isosceles Triangle – is a triangle with at least two legs of equal length and a third side called the base
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Angles at the base are called base angles and the third angle is the vertex angle
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Equilateral triangle – is a triangle with three sides of equal length
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Obtuse triangle – is a triangle with one obtuse angle (>90°)
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Acute triangle – is a triangle with three acute angles (<90°)
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Right triangle – is a triangle with one right angle (90°)
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Equiangular triangle – is a triangle with three angles of equal measure.
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Auxillary line – is a line (ray or segment) added to a diagram to help in a proof.
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THEOREM 3-11 The sum of the measures of the angles of a triangle is 180
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Corollary A statement that can easily be proved by applying a theorem
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Corollary 1 If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
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Corollary 2 Each angle of an equiangular triangle has measure 60°.
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Corollary 3 In a triangle, there can be at most one right angle or obtuse angle.
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Corollary 4 The acute angles of a right triangle are complementary.
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THEOREM 3-12 The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.
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Section 3 - 5 Angles of a Polygon
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Polygon – is a closed plane figure that is formed by joining three or more coplanar segments at their endpoints, and
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Each segment of the polygon is called a side, and the point where two sides meet is called a vertex, and
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The angles determined by the sides are called interior angles.
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Convex polygon - is a polygon such that no line containing a side of the polygon contains a point in the interior of the polygon.
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Diagonal - a segment of a polygon that joins two nonconsecutive vertices.
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THEOREM 3-13 The sum of the measures of the angles of a convex polygon with n sides is (n-2)180°
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THEOREM 3-14 The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°
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Regular Polygon A polygon that is both equiangular and equilateral.
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To find the measure of each interior angle of a regular polygon
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3 - 6 Inductive Reasoning
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Conclusion based on several past observations Conclusion is probably true, but not necessarily true.
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Deductive Reasoning Conclusion based on accepted statements Conclusion must be true if hypotheses are true.
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THE END
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