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-
Postulate 5
- 1.
- That, if a straight line falling
- on two straight lines makes the interior angles on the same side less than two
- right angles, the two straight lines, if produced indefinitely, meet on that
- side on which are angles less than the two right angles.
-
Theorem 1
- 1.Supplement of congruent angles
- are congruent.
- 2.Complement of congruent angles
- are congruent.
-
Theorem 2
1. Vertical angles are congruent.
-
Theorem 3: The
Isosceles Triangle Theorem
- 1. If two sides of a triangle are
- congruent, then the angles opposite these sides are congruent
or
- 2.The base angles of an isosceles
- triangle are congruent
-
Theorem 4: The Angle,
Side, Angle condition
- 1. Given a one to one correspondence
- between the vertices of two triangles, if two angles and the included side of
- on triangle are congruent to the corresponding parts of the second triangle,
- the two triangles are congruent.
-
Theorem 5 The median of the base of an isosceles triangle
is
- 1.
- perpendicular bisector as well as
- the angle bisector of the angle of the triangle.
-
Theorem 6 Every point on the perpendicular bisector of a segment
is
- 1.
- equidistant from the endpoints of
- the segment.
-
Theorem 7The diagonal of a kite connecting the vertices
where the congruent sides intersect bisects the angles at these vertices and is
- 1.
- the perpendicular bisector of the
- other diagonal.
-
Theorem 8: Side, Side,
Side Congruency condition
- 1.
- Given a one to one correspondence
- among the vertices of two triangles, if the three sides of one triangle are
- congruent to the corresponding sides of the second triangle, then the triangles
- are congruent.
-
Theorem 9: Hypotenuse Leg
Congruence Condition
- 1.
- If the hypotenuse and a leg of
- one triangle are congruent to the hypotenuse and a leg of another right
- triangle, then the triangles are congruent.
-
Theorem 10:The Exterior
Angle Theorem
- 1.
- An exterior angle of a triangle
- is greater than either of the remote interior angles.
-
Theorem 11: Hypotenuse,
Acute Angle Congruence Condition
- 1.
- If the hypotenuse and an acute
- angle of one right triangle are congruent to the hypotenuse and an acute angle
- of another right triangle, then the triangles are congruent.
-
Theorem 12 A point is on the angle bisector of an angle IFF
- 1.
- it is equidistant from the sides
- of the angle.
-
Theorem 13 Given two non-congruent sides on a triangle,
- 1.
- the angle opposite the longer
- side is greater than the angle opposite the shorter side.
-
Theorem 14 Given two non-congruent angles in a triangle,
- 1.
- the side opposite the greater
- angle is longer than the side opposite the smaller angle.
-
Theorem 15: The Triangle
Inequality
1.The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
-
Theorem 16: SAS
Inequality Theorem
- 1.
- If in ∆ABC and ∆ XYZ we have AB = XY, AC = XZ, m.ang. A > m.ang. X, then BC > YZ, and conversely if BC > YZ, then m.ang. A > m.ang. X.
-
Theorem 17 If two line in the same plane are each perpendicular
to a third line in that plane
1. then they are parallel.
-
Theorem 18 If two lines are cut by a transversal and a pair of corresponding angles is congruent (or a pair of alternate interior angles is congruent),
1. then the lines are parallel.
-
Theorem 19 If two parallel lines are cut by a transversal,
- 1. then a pair of corresponding
- angles is congruent.
-
Theorem 20 Two lines in a plane are parallel IFF
- 1. a pair of corresponding angles
- formed by a transversal is congruent.
-
Theorem 21 Two lines in a plane are parallel IFF
- 1. a pair of alternate interior
- angles formed by a transversal are congruent.
-
Theorem 22 Two lines are parallel IFF
- 1. a pair of interior angles on the
- same side of a transversal is supplementary.
-
Theorem 23 The sum of the measures
- 1. The sum of the measures of the
- interior angles of a triangle is 180*.
-
Definition 3.9 For any three points A,B, and C,
- 1. we say that B is between A and C, and we write A-B-C, IFF A, B, and C are
- distinct, collinear points, and AB+ BC = AC.
-
Definition 3.10: Adjacent Angles
- 1.
- Lie
- in the same plane and share a common side and their interiors have no points in
- common.
-
Definition 3.11:
Angle bisector
- 1.
- Is
- the common side of two adjacent angles of equal measure.
-
Definition 3.12:
Linear pair
- 1.
- Is
- formed by two adjacent angles in which the non-common sides are opposite rays.
- The sum of the measures of two angles in a linear pair is 180*.
-
Definition 3.13:
Vertical angles
- 1.
- are
- angles which sides form two pairs of opposite rays.
-
Definition 3.14
Supplementary/Complementary
- 1.
- Supplementary
- angles: are angles which measure add up to 180*
- 2.
- Complementary
- angles: are angles which measure add up to 90*
-
Definition 3.15
1. When 2 lines intersect they form four angles.
2. When a line intersects a segment at its midpoint and is perpendicular to thesegment
- 1. if one of the angles is a right angle,
- then all the others are right angles, Then the lines are said to be
- perpendicular. We write m ┴ n.
- 2., it is called the perpendicular bisector of the segment. M is the
- midpoint of AB. Then AM = MB or line AM cong. line MB
-
Definition 4.1:
Congruence of Triangles
- 1.
- Triangles
- are congruent if there is a one to one correspondence between their vertices so
- that the corresponding sides are congruent and corresponding angles are
- congruent.
-
Definition 4.4
Acute and Obtuse triangles
1. A triangle is acute if all of its angles are acute.
2.A triangle is obtuse if one of its angles is obtuse.
-
Definition 5.1
Line segment connecting the vertex of a triangle with the midpoint of the opposite side is?
The segment from a vertex of a triangle
perpendicular to the line containing the opposite side is?
- 1.
- The
- segment connecting the vertex of a triangle with the midpoint of the opposite
- side is called a median.
- The segment from a
- vertex of a triangle perpendicular to the line containing the opposite side is
- an altitude of the triangle
-
Definition 6.1
A kite is
1. a quadrilateral that has two pairs of congruent adjacent sides.
-
Definition 6.2
A kite with all sides congruent is
called a rhombus
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