Parallel Lines and Planes

Section Definitions

Parallel Lines - coplanar lines that do not intersect

Skew Lines - noncoplanar lines that do not intersect

Parallel Planes - Parallel planes do not intersect

THEOREM 3-1 If two parallel planes are cut by a third plane, then the lines of intersection are parallel.

Transversal - is a line that intersects each of two other coplanar lines in different points to produce interior and exterior angles

ALTERNATE INTERIOR ANGLES - two nonadjacent interior angles on opposite sides of a transversal

Alternate Interior Angles

ALTERNATE EXTERIOR ANGLES - two nonadjacent exterior angles on opposite sides of the transversal

Alternate Exterior Angles

Same-Side Interior Angles - two interior angles on the same side of the transversal

Same-Side Interior Angles

Corresponding Angles - two angles in corresponding positions relative to two lines cut by a transversal

Corresponding Angles

3 - 2 Properties of Parallel Lines

Postulate 10 If two parallel lines are cut by a transversal, then corresponding angles are congruent.

THEOREM 3-2 If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

THEOREM 3-3 If two parallel lines are cut by a transversal, then same-side interior angles are supplementary.

THEOREM 3-4 If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also.

Section Proving Lines Parallel

Postulate 11 If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel

THEOREM 3-5 If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.

THEOREM 3-6 If two lines are cut by a transversal and same- side interior angles are supplementary, then the lines are parallel.

THEOREM 3-7 In a plane two lines perpendicular to the same line are parallel.

THEOREM 3-8 Through a point outside a line, there is exactly one line parallel to the given line.

THEOREM 3-9 Through a point outside a line, there is exactly one line perpendicular to the given line.

THEOREM 3-10 Two lines parallel to a third line are parallel to each other.

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

Section Angles of a Triangle

 Triangle – is a figure formed by the segments that join three noncollinear points

 Scalene triangle – is a triangle with all three sides of different length.

 Isosceles Triangle – is a triangle with at least two legs of equal length and a third side called the base

 Angles at the base are called base angles and the third angle is the vertex angle

 Equilateral triangle – is a triangle with three sides of equal length

 Obtuse triangle – is a triangle with one obtuse angle (>90°)

 Acute triangle – is a triangle with three acute angles (<90°)

 Right triangle – is a triangle with one right angle (90°)

 Equiangular triangle – is a triangle with three angles of equal measure.

 Auxillary line – is a line (ray or segment) added to a diagram to help in a proof.

THEOREM 3-11  The sum of the measures of the angles of a triangle is 180 

Corollary  A statement that can easily be proved by applying a theorem

Corollary 1  If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.

Corollary 2  Each angle of an equiangular triangle has measure 60°.

Corollary 3  In a triangle, there can be at most one right angle or obtuse angle.

Corollary 4  The acute angles of a right triangle are complementary.

THEOREM 3-12  The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.

Section Angles of a Polygon

 Polygon – is a closed plane figure that is formed by joining three or more coplanar segments at their endpoints, and

 Each segment of the polygon is called a side, and the point where two sides meet is called a vertex, and

 The angles determined by the sides are called interior angles.

 Convex polygon - is a polygon such that no line containing a side of the polygon contains a point in the interior of the polygon.

 Diagonal - a segment of a polygon that joins two nonconsecutive vertices.

THEOREM 3-13  The sum of the measures of the angles of a convex polygon with n sides is (n-2)180°

THEOREM 3-14  The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°

Regular Polygon A polygon that is both equiangular and equilateral.

To find the measure of each interior angle of a regular polygon

3 - 6 Inductive Reasoning

 Conclusion based on several past observations  Conclusion is probably true, but not necessarily true.

Deductive Reasoning  Conclusion based on accepted statements  Conclusion must be true if hypotheses are true.

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