Chapter 5 Summary

Theorems

Th. 30. Exterior Angle Inequality

The measure of an exterior angle of a △ is greater than the measure of either remote interior angle.

If...then parallel theorems

Th	If...	then...
31	Two lines cut by a transversal have alternate interior angles congruent	the lines are parallel.
32	Two lines cut by a transversal have alternate exterior angles congruent
33	Two lines cut by a transversal have corresponding angles congruent
34	Two lines cut by a transversal have same side interior angles supplementary
35	Two lines cut by a transversal have same side exterior angles supplementary
36	Two coplanar lines are parallel to a third line

If parallel then...theorems

Th.	If...	then...
37	two parallel lines are cut by a transversal	alternate interior angles are congruent
38		any pair of angles formed are either congruent or supplementary
39		alternate exterior angles are congruent.
40		corresponding angles are congruent.
41		same side interior angles are supplementary.
42		same side exterior angles are supplementary.
43	in a plane, if a line is perpendicular to one of two parallel lines	it is perpendicular to the other.
44	two lines are parallel to a third line	they are parallel to each other. (Transitivity of parallelism.)

Properties of quadrilaterals

Prop.	If it's a...	then...
1	parallelogram	Opposite sides ∥ (def)
2		opposite sides ≅
3		opposite angles ≅
4		diagonals bisect each other
5		all pairs consecutive ∠s are supplementary
1	rectangle	all parallelogram props (def)
2		All angles are right angles
3		diagonals are ≅
1	kite	disjoint pairs of sides are congruent (def)
2,3		One diagonal ⊥ bis the other (half property)
4		One diagonal bisects a pair of opposite angles (half property)
5		one pair of opposite angles are congruent (half property)
1	rhombus	All parallelogram properties hold (def)
2		All kite properties hold (and half properties become full properties)
3		all sides ≅
4		diagonals bisect the angles (full prop)
5		diagonals ⊥ bis each other (full prop)
6		Diagonals divide rhombus into four congruent right triangles
1	square	All rectangle properties hold (def)
2		All rhombus properties hold (def)
3		Diagonals form isosceles right triangles
1	isosceles trapezoid	legs ≅ (def)
2		bases parallel (def)
3,4		lower, upper base angles congruent
5		diagonals congruent
6		any lower base angle is supplementary to any upper base angle

Proving it's a special quadrilateral

Prop	If...	then it's a ...
1	both pairs of opposite sides are parallel	parallelogram (def)
2	both pairs of opposite sides are congruent
3	one pair of opposite sides are both parallel and congruent
4	diagonals bisect each other
5	both pairs of opposite angles of a quadrilateral are congruent
1	a parallelogram contains at least one right angle	rectangle
2	the diagonals of a parallelogram are congruent
3	all four angles of a quadrilateral are right angles
1	two disjoint pairs of consecutive sides of a quadrilateral are ≅ (def)	kite
2	A diagonal of a quadrilateral is the perpendicular bisector of the other
1	a parallelogram contains a pair of consecutive sides that are ≅ (def)	rhombus
2	either diagonal of a parallelogram bisects two angles of the parallelogram
3	the diagonals of a quadrilateral are perpendicular bisectors of each other
1	a rectangle is also a rhombus	square
1	the nonparallel sides of a trapezoid are ≅ (def)	isosceles trapezoid
2	the lower or upper base angles of a trapezoid are ≅
3	the diagonals of a trapezoid are ≅