Geometry Chapter 3 Study Guide

Parallel Lines: Coplanar lines that do not intersect (Denoted //)
Skew Lines: Noncoplanar lines; are neither parallel nor intersecting
Parallel Planes: Planes that do not intersect (Denoted //)
Transversal: A line that intersects two or more coplanar lines in different points
Corresponding Angles Postulate: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Given: m // n
Conclusion:    1‗    3,    2‗    4,    5‗    7,    6‗    8
Corresponding Angles Converse: If two lines are cut by a transversal and corresponding angles are congruent, then the two lines are parallel.
Given:    1‗    3 or    2‗    4 or    5‗    7 or    6‗    8
Conclusion: m // n
Alternate Interior Angles Theorem:
If two parallel lines are cut by a transversal, then the pairs of alternate interior 
angles are congruent.
Given: m // n
Prove:     3 ‗     6,     2 ‗    7
Statements	Reasons	
•	m // n	Given	
•	   3‗    8	Vertical Angles Theorem	
•	   8‗    6	Corresponding Angles Postulate	
•	   3‗    6
Likewise,    2‗    7	Transitive Property of Congruence 	

Alternate Interior Angles Converse: 
If two lines are cut by a transversal, and alternate interior angles are congruent, then the two lines are parallel.
Given:    3‗    6
Prove: m // n
Statements	Reasons	
•	    3‗     6	Given	
•	    3‗    8	Vertical Angles Theorem	
•	    6‗    8	Transitive Property of Congruence	
•	m // n	Corresponding Angles Converse	

Alternate Exterior Angles Theorem:
If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
Given: m // n
Prove:    1‗    8,    5‗    4
Statements	Reasons	
•	m // n	Given	
•	   1‗    6	Vertical Angles Theorem	
•	   8‗    6	Corresponding Angles Postulate	
•	   1‗    8
Likewise,    5‗    4	Transitive Property of Congruence 	

Alternate Exterior Angles Converse:
If two lines are cut by a transversal and alternate exterior angles are congruent, then the lines are parallel.
Given:    1‗    8
Prove: m // n
Statements	Reasons	
•	    1‗     8	Given	
•	    3‗    8	Vertical Angles Theorem	
•	    1‗    3	Transitive Property of Congruence	
•	m // n	Corresponding Angles Converse	

Same-Side Interior Angles Theorem: 
If two parallel lines are cut by a transversal, then the pairs of same-side interior angles are supplementary.
Given: m // n
Prove:    2 is supplementary to    3,    6 is supp. to    7
Statements	Reasons	
•	m // n	Given	
•	   3 is supp. to    7	Angle Addition Post.	
•	   2‗   7	Alternate Interior Angles Thrm.	
•	   2 is supplementary to    3
Likewise,    6 is supp. to    7	Congruent Supp. Thrm.	

Same-Side Interior Angles Converse:
If two lines are cut by a transversal and same-side interior angles are supplementary, then the lines are parallel.
Given:    2 is supp. to    3
Prove: m // n
Statements	Reasons	
•	   2 is supp. to    3	Given	
•	   3 is supp. to    4	Angle Add. Post.	
•	   2‗    4	Congruent Supplements Theorem	
•	m // n	Corr. Angles Conv.	

Same-Side Exterior Angles Theorem:
If two parallel lines are cut by a transversal, then the pairs of same-side exterior angles are supplementary. 
Given: m // n
Prove:    1 is supp. to    4,    5 is supp. to    8
Statements	Reasons	
•	m // n	Given	
•	   1 is supp. to    5	Angle Addition Post.	
•	   1‗   8	Alternate Exterior Angles Thrm.	
•	   5 is supplementary to    8
Likewise,    1 is supp. to    4	Congruent Supp. Conv.	
Same-Side Exterior Angles Converse: 
If two lines are cut by a transversal and the same-side exterior angles are supplementary, then the lines are parallel. 
Given:    1 is supp. to    4
Prove: m // n
Statements	Reasons	
•	   1 is supp. to    4	Given	
•	   3 is supp. to    4	Angle Add. Post.	
•	   1‗    4	Congruent Supplements Theorem	
•	m // n	Corr. Angles Conv.	

Perpendicular Transversal Theorem:
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.
Given: p    m, m // n
Prove: p   n
Statements	Reasons	
•	p    m, m // n	Given	
•	   1 is a right angle	Definition of perpendicular	
•	   1‗   2	Corresponding Angles Post.	
•	   2 is a right angle	Right Angles Conv.	
•	p   n	Definition of perpendicular	

Parallel Postulate: Through a point outside a line, there is exactly one line parallel to the given line.
Perpendicular Postulate: Through a point outside a line, there is exactly one line perpendicular to the given line. 

Theorem: 
Two coplanar lines perpendicular to the same line are parallel. 
Given: a   c, b   c
Prove: a // b
Statements	Reasons	
•	a    c, b    c	Given	
•	   1 and    2 are right angles	Definition of perpendicular	
•	   1‗    2	Right angles thrm.	
•	a // b	Corr. Angles Conv.	

Theorem:
Two lines parallel to a third line are parallel to each other. 
Given: p // q, q //r
Prove: p // r
Statements	Reasons	
•	p // q, q // r	Given	
•	   1‗   2,    2‗    3	Corr. Angles Post.	
•	   1‗    3	Trans. Prop. Of Congruence	
•	p // r	Corr. Angles Conv.	

Triangle: A figure formed by three segments joining three noncollinear points 
	ABC is a triangle
Vertex: A point where two segments of a polygon intersect
	  ABC has vertices A, B, and C
Sides: The segments that form a polygon
	  ABC has sides AB, BC, and AC
Scalene Triangle: A triangle with no congruent sides
Isosceles Triangle: A triangle with at least two congruent sides
Equilateral Triangle: A triangle with all three sides congruent
Acute Triangle: A triangle with all three acute angles
Obtuse Triangle: A triangle with one obtuse angle
Right Triangle: A triangle with one right angle
Equiangular Triangle: A triangle with all three angles congruent
Corollary: A statement that can be proved easily by another theorem, or one that is very similar to another theorem
Triangle Sum Theorem: 
The sum of the measures of a triangle is 180.
Given:      ABC
Prove: m   A+m   B+m    C= 180

Statements	Reasons	
•	   ABC	Given	
•	Draw BD // AC	// Postulate	
•	   1        A,    2       C	Alt. Int. Angles Theorem	
•	m    1+m   B+m    2=180	Angle Add. Post.	
•	m   1=m   A, m   2=m   C	Def. of Congruent Angles	
•	m   A+m   B+m    C= 180	Subst. Prop. Of = (5 – 4)	

Third Angles Theorem:
If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
Given:    A      D,    C       F
Prove:    B       E

		

		

Exterior Angle of a Triangle Theorem:
The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles.
Given:     ABC with ext.    1
Prove: m   1=m   A+m   B

		

		

Theorem: 
Each angle of an equiangular triangle has measure 60.
Given: Equiangular    ABC
Prove: m    A=m   B=m   C= 60

		

		

		

Theorem:
The acute angles of a right triangle are complementary.
Given:
Prove:

		

		

Polygon: A closed figure made of line segments, and only two segments meet at a vertex
Convex Polygon: Extending each side does not go into the polygon OR all diagonals are in the interior of the polygon
Regular Polygon: A polygon that is both equilateral and equiangular
Interior Angles Theorem: 
ISum = (n-2)180

Exterior Angles Theorem:
ESum = 360
ISum+ESum= 180n
(n-2)180+ ESum=180n
180n-360+ ESum=180n
-360+ ESum=0
ESum=360