Section 1
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Reflexive Property Equality
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Cards (129)
Section 1
(50 cards)
Reflexive Property Equality
a=a (Reflexive)
(The Vertical Angle Theorem)
Vertical angles are congruent.
A median of a triangle
is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Definition: A bisector of a segment
is a line, ray, or segment that intersects the segment at its midpoint.
Postulate (The Angle Addition Postulate)
If C lies in the interior of ∠AOB, then m∠AOC+m∠COB=m∠AOB
Postulate: The Angle-Side-Angle Postulate (ASA)
If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Definition: The midpoint of a segment
is the point that divides the segment into two congruent parts.
(The Angle Bisector Theorem)
If ray BD is the bisector of ∠ABC, then 2m∠ABD=m∠ABC and m∠ABD=1/2m∠ABC.
(The Right Angle Theorem)
All right angles are congruent.
(The Supplement Theorem)
If two angles are supplements of congruent angles or of the same angle, then the two angles are congruent. SCAC
Distributive Property Equality
a(b+c)=ab+ac (Distrib. Prop.)
Transitive Property of Equality
If a=b and b=c, then a=c (Transitive)
Postulate: The Side-Angle-Side Postulate (SAS)
Given △ABC and △DEF, if AB⎯⎯⎯⎯⎯⎯⎯≅DE⎯⎯⎯⎯⎯⎯⎯⎯, AC⎯⎯⎯⎯⎯⎯⎯⎯≅DF⎯⎯⎯⎯⎯⎯⎯⎯, and ∠BAC≅∠EDF, then △ABC≅△DEF.
Theorem 1.8: If the exterior sides of two adjacent, acute angles are perpendicular, then the adjacent angles are......
If the exterior sides of two adjacent, acute angles are perpendicular, then the adjacent angles are complementary. Another version of this theorem is "If A is in the interior of right angle ∠XYZ, then ∠XYAand ∠ZYA are complementary."
The measure of angle ABC is denoted
m∠ABC. Note we are distinguishing between the angle as a set of points and its measure. We will write m∠1=90, not ∠1=90.
examples of how to incorporate auxiliary lines into proofs
Connecting two points: Drawing a median: Drawing an angle bisector: Extending a line segment:
Postulate: The Angle-Angle-Side Postulate (AAS)
f two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
Definition: Complementary angles
are two angles whose measures sum to 90.
Definition: The bisector of an angle
is a ray or segment in the interior of the angle that divides the angle into two congruent adjacent angles.
Substitution Property of Equality
If a=b, then either may be substituted for the other. (Substitution)
Definitions: An angle
a plane figure formed by two different rays that have the same endpoint. The rays are called the sides of the angle and the common endpoint is called the vertex.
Vertex angle
the last angle
base angles
The congruent angles of an isosceles triangle
Mulitplication Property Equality
If a=b and c=d, then ac=bd (Mult. Prop.)
Postulate (The Protractor Postulate)
To every angle there corresponds a real number x between 0 and 180, not including 0, but including 180; i.e., 0<x<180.
Collinear points
are points that lie on a line.
Definition: Supplementary angles
are two angles whose measures sum to 180.
Coplanar points
are points that all lie on some plane
Postulate (The Line Postulate
Through any two points there is exactly one line.
Postulate (The Ruler Postulate):
The set of real numbers and the points on a line can be put into a one-to-one correspondence. A scale can be obtained by assigning 0 to one point and 1 to another point.
The Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Definition: Congruent segments
are segments that have the same length.
Postulate: The Side-Side-Side Postulate (SSS)
Given △ABC and △DEF, if line AB ≅ line DE, line BC ≅ line EF, and line AC≅ line DF, then △ABC≅△DEF.
Postulate (The Segment Addition Postulate)
If B is between A and C, then AB+BC=AC.
Reflexive Property of Congruence
line AB≅line AB (Reflexive)
(The Midpoint Theorem)
If M is the midpoint of line AB, then 2AM=AB and AM=1/2AB.
Definition An isosceles triangle
a triangle that has two of its sides congruent. The congruent sides are called the legs of the triangle.
Definition: Perpendicular line
are two lines that form right angles.
Theorem 1.6: If two lines are perpendicular, then the adjacent angles....
If two lines are perpendicular, then the adjacent angles they form are congruent. This is more frequently expressed as "Adjacent angles formed by perpendicular lines are congruent."
Adjacent angles
are two angles in a plane that have a common vertex and a common side but no common interior points.
Postulate (The Angle Addition Postulate) 2
Postulate: If C does not lie on line AOB, then m∠AOC+m∠COB=180.
Definitions: Vertical angles
are two angles whose sides form pairs of opposite rays.
(The Complement Theorem)
If two angles are complements of congruent angles or of the same angle, then the two angles are congruent CCAC
An altitude
is a segment drawn from a vertex of the triangle perpendicular to the line that contains the opposite side; the endpoints of an altitude are one of the vertices of the triangle and a point on the opposite side (extended, if necessary).
The Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Theorem 1.7: If two lines form congruent adjacent angle, then the lines .....
If two lines form congruent adjacent angles, then the lines are perpendicular. Note that this is the converse of Theorem 1.6.
Definition: Two lines are perpendicular...
if and only if they form right angles.
Corollary 2.3: Every equilateral triangle is
equiangular.
Addition Property Equality
If a=b and c=d, then a+c=b+d (Add. Prop.)
Transitive Property of Congruence
If line AB≅ line CD, and line CD≅ line CD≅ line EF, then line AB≅ line EF (Transitive)
Section 2
(50 cards)
Theorem 3.4: AIC → P
Given two coplanar lines and a transversal, if a pair of alternate interior angles are congruent, then the lines are parallel.
Corollary 2.4: Every equiangular triangle is
equilateral
Theorem 3.13: If three parallel lines cut off congruent segments on one transversal, then they cut off
congruent segments on every transversal.
Theorem 4.5: If both pairs of opposite sides of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
Theorem 4.1: Opposite sides of a parallelogram are
congruent.
Theorem 3.10: The sum of the measures of the interior angles of a triangle is
180
Theorem 4.3: The diagonals of a parallelogram
bisect each other.
Postulate: The Perpendicular Postulate
In a plane, through a point on a line, there is one and only one line perpendicular to the given line.
Theorem 2.12: The Perpendicular Bisector Characterization Theorem (PBC)
In a plane, the perpendicular bisector of a segment is the set of all points that are equidistant from the endpoints of the segment.
For any two lines in space, one of the following conditions must exist:
1) the lines must intersect in a single point and therefore must be coplanar; 2) the lines do not intersect and are not coplanar, in which case they are called skew lines; 3) the lines do not intersect but are coplanar, in which case they are parallel lines.
Theorem 3.12: The No-Choice Theorem
If two angles of one triangle are congruent, respectively, to two angles of a second triangle, then the third angles are congruent. If two angles of one triangle are congruent, respectively, to two angles of a second triangle, then the third angles are congruent.
Theorem 3.1: Exterior Angle Inequality
The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.
Theorem 3.20: The Triangle Inequality
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Postulate: The Parallel Postulate
Given a point not on a line, there is exactly one line through the given point and parallel to the given line.
Theorem 2.14: In a plane, if each of two points is equidistant from the endpoints of a segment, then the line they lie on is the
perpendicular bisector of the segment.
Theorem 2.16: The Angle Bisector Characterization Theorem (ABC)
The bisector of an angle is the set of all points in the interior of the angle that are equidistant from the sides of the angle.
Theorem 2.11: Halves of congruent segments are
congruent.
Proving Parallelograms
Both pairs of opposite sides are parallel. Both pairs of opposite sides are congruent. One pair of opposite sides is both parallel and congruent. Both pairs of opposite angles are congruent. The diagonals bisect eachother.
Theorem 2.15: The Hypotenuse-Leg Theorem (HL)
If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent.
Theorem 3.19: If two angles of a triangle are not congruent,
then the sides opposite those angles are not congruent, and the longer side is opposite the larger angle.
Theorem 2.13: If a point is not on a line, then there is a line through the point which is
perpendicular to the given line.
Theorem 2.8: The angle bisectors of the base angles of an isosceles triangle are
congruent.
Theorem 3.11: The Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle.
Theorem 3.3: In a plane, if two lines m and n are perpendicular to a third line ℓ,
then the lines m and n are parallel to each other.
Corollary:
If a line through the midpoint of one side of a triangle is parallel to a second side of the triangle, then it passes through the midpoint of the third side.
If at least two of the sides of the triangle are congruent, then the triangle is an
isosceles triangle
Theorem 4.2: Opposite angles of a parallelogram are
congruent.
Characteristics of Parallelograms
Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. The diagonals bisect each other. Consecutive angles are supplementary.
Definition: Parallel
Two lines are parallel if and only if they are coplanar and do not intersect.
Indirect Proof
Assume opposite of the Prove Solve for contradictions
Theorem 3.18: If two sides of a triangle are not congruent,
then the angles opposite those sides are not congruent, and the larger angle is opposite the longer side.
If all three sides are congruent, then the triangle is an
equilateral triangle.
Theorem 3.5: CAC → P
Given two coplanar lines and a transversal, if a pair of corresponding angles are congruent, then the lines are parallel.
the sum of the exterior angles
is 360
Definition: A parallelogram
is a quadrilateral in which both pairs of opposite sides are parallel.
Theorem 4.4: Consecutive angles of a parallelogram are
supplementary.
Theorem 3.2: If two distinct coplanar lines m and n are parallel to a third line ℓ, then lines m and n are
parallel to each other.
Theorem 2.7: The medians drawn to the legs of an isosceles triangle are
congruent.
Theorem 2.5: The bisector of the vertex angle of an isosceles triangle is
perpendicular to the base.
Definition: Transversal
A line that intersects two different lines in two distinct points is called a transversal.
Theorem 2.6: The median drawn from the vertex angle of an isosceles triangle is
the angle bisector of the vertex angle.
Theorem 3.7: P → CAC
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Theorem 3.9: If two parallel lines are cut by a transversal, then same-side interior angles are
supplementary.
Definition: perpendicular bisector
In a plane, the perpendicular bisector of a segment is the line that is perpendicular to the segment at its midpoint.
A diagonal of a polygon is
a segment that joins nonconsecutive vertices of the polygon.
Theorem 3.14: The sum of the measures of the interior angles of a convex polygon with n sides is
(n−2)⋅180
Theorem 3.6: Given two coplanar lines and a transversal, if a pair of same-side interior angles are supplementary,
then the lines are parallel.
Theorem 2.9: The altitudes drawn to the legs of an isosceles triangle are
congruent.
Theorem 2.10: Halves of congruent angles are
congruent.
Theorem 3.8: P → AIC
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Section 3
(29 cards)