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- 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.
- 1.Supplement of congruent angles
- are congruent.
- 2.Complement of congruent angles
- are congruent.
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
- 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
- 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
- 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
- the perpendicular bisector of the
- other diagonal.
Theorem 8: Side, Side,
Side Congruency condition
- 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
- 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
- An exterior angle of a triangle
- is greater than either of the remote interior angles.
Theorem 11: Hypotenuse,
Acute Angle Congruence Condition
- 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
- it is equidistant from the sides
- of the angle.
Theorem 13 Given two non-congruent sides on a triangle,
- 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,
- the side opposite the greater
- angle is longer than the side opposite the smaller angle.
Theorem 15: The Triangle
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
- 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
- in the same plane and share a common side and their interiors have no points in
- the common side of two adjacent angles of equal measure.
- 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*.
- angles which sides form two pairs of opposite rays.
- angles: are angles which measure add up to 180*
- angles: are angles which measure add up to 90*
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
Congruence of 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
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.
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?
- 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
A kite is
1. a quadrilateral that has two pairs of congruent adjacent sides.
A kite with all sides congruent is
called a rhombus