Angle 1 and angle 2 are supp

Prove: AC is not perpendicular to DE

Statements	Reasons
B is the midpoint of AC and DE	Given
Angle 1 and angle 2 are supp	Given
The vertical angles of 1 and 2 are also supp.	Definition of vertical angles
AC is not perpendicular to DE	Definition of intersecting lines

Theorem 1-2

Through a line and a point not on the line there is exactly one plane:

Given: B is non collinear to line AC

Prove: B is coplanar to AC

Statements	Reasons
B is non collinear to AC	Given
B is on the same plane as AC	Definition of coplanar
B is coplanar to AC	substitution

Theorem 1-3

If 2 lines intersect, then exactly one plane contains the lines

Given: AB and CD intersect

Prove: AB and CD are coplanar

Statements	Reasons
AB and CD intersect	Given
AB and CD form 2 lines	Postulate 6
AB and CD are coplanar	Postulate 8

Theorem 2-1

Known as the midpoint theorem where a midpoint of a line splits the line into 2 halves.

Given: B is collinear to AC

Prove: B is the midpoint of AC

Statements	Reasons
B is collinear to AC	Given
AB + BC= AC	Postulate 2
B is equidistant from A and C	Def. of a midpoint
B is the midpoint of AC	substitution

Theorem 2-2

Known as the angle bisector theorem where a bisector of an angle splits that angle into 2 halves.

Given: DB bisects angle ABC

Prove: angle ABD is congruent to angle CBD

Statements	Reasons
DB bisects angle ABC	Given
ABD + CBD = ABC	Postulate 4
ABD = CBD	Definition of angle bisector
½ of ABC = ABD or CBD	Angle bisector theorem

Theorem 2-3

Vertical angles are congruent

Given: line AB intersects line CD

Prove: 1 = 2

Statements	Reasons
Line AB intersects line CD	Given
AB and CD form opposite = angles	Def. of vertical angles
1 and 2 are opposite angles	substitution
1 = 2	Theorem 2-3

Theorem 2-4

If 2 lines are perpendicular, then they form congruent adjacent angles

Given: AB is perpendicular to CD

Prove: 1 + 2 are both 90 degrees

Statements	Reasons
AB is perpendicular to CD	Given
1 = 3	Theorem 2-3
1 = 2	Def. of adjacent angles
1 + 2 are both 90 degrees	Theorem 2-4

Theorem 2-5

(Converse of 2-4)

Theorem 2-6

If The exterior sides of 2 adjacent acute angles are perpendicular, then the angles are complementary.

Given: AB is perpendicular to CD

EB bisects ABC

Prove: ABE + EBC = 90

Statements	Reasons
AB is perpendicular to CD with E bisecting them	Given
ABE = EBC	Def. of angle bisector
ABC = 90 degrees	Def. of perpendicular lines
ABE + EBC = 90	transitive
ABE + EBC = ABC	Theorem 2-6 (substitution)

Theorem 2-7

If two angles are supplements of congruent angles (or the same angle), then the 2 angles are congruent

Given:?

Theorem 2-8

Same as 2-7, but with complementary angles

Theorem 3-1

If 2 parallel planes are cut by a 3^rd plane, then the lines of the intersection are paralell.

Theorem 3-2

Дата добавления: 2015-11-16