is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Back
Definition: A bisector of a segment
Front
is a line, ray, or segment that intersects the segment at its midpoint.
Back
Postulate (The Angle Addition Postulate)
Front
If C lies in the interior of ∠AOB, then m∠AOC+m∠COB=m∠AOB
Back
Postulate: The Angle-Side-Angle Postulate (ASA)
Front
If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Back
Definition: The midpoint of a segment
Front
is the point that divides the segment into two congruent parts.
Back
(The Angle Bisector Theorem)
Front
If ray BD is the bisector of ∠ABC, then 2m∠ABD=m∠ABC and m∠ABD=1/2m∠ABC.
Back
(The Right Angle Theorem)
Front
All right angles are congruent.
Back
(The Supplement Theorem)
Front
If two angles are supplements of congruent angles or of the same angle, then the two angles are congruent.
SCAC
Back
Distributive Property Equality
Front
a(b+c)=ab+ac
(Distrib. Prop.)
Back
Transitive Property of Equality
Front
If a=b and b=c, then a=c
(Transitive)
Back
Postulate: The Side-Angle-Side Postulate (SAS)
Front
Given △ABC and △DEF, if AB⎯⎯⎯⎯⎯⎯⎯≅DE⎯⎯⎯⎯⎯⎯⎯⎯, AC⎯⎯⎯⎯⎯⎯⎯⎯≅DF⎯⎯⎯⎯⎯⎯⎯⎯, and ∠BAC≅∠EDF, then △ABC≅△DEF.
Back
Theorem 1.8:
If the exterior sides of two adjacent, acute angles are perpendicular, then the adjacent angles are......
Front
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."
Back
The measure of angle ABC is denoted
Front
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.
Back
examples of how to incorporate auxiliary lines into proofs
Front
Connecting two points:
Drawing a median:
Drawing an angle bisector:
Extending a line segment:
Back
Postulate: The Angle-Angle-Side Postulate (AAS)
Front
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.
Back
Definition: Complementary angles
Front
are two angles whose measures sum to 90.
Back
Definition: The bisector of an angle
Front
is a ray or segment in the interior of the angle that divides the angle into two congruent adjacent angles.
Back
Substitution Property of Equality
Front
If a=b, then either may be substituted for the other.
(Substitution)
Back
Definitions: An angle
Front
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.
Back
Vertex angle
Front
the last angle
Back
base angles
Front
The congruent angles of an isosceles triangle
Back
Mulitplication Property Equality
Front
If a=b and c=d, then ac=bd
(Mult. Prop.)
Back
Postulate (The Protractor Postulate)
Front
To every angle there corresponds a real number x between 0 and 180, not including 0, but including 180; i.e., 0<x<180.
Back
Collinear points
Front
are points that lie on a line.
Back
Definition: Supplementary angles
Front
are two angles whose measures sum to 180.
Back
Coplanar points
Front
are points that all lie on some plane
Back
Postulate (The Line Postulate
Front
Through any two points there is exactly one line.
Back
Postulate (The Ruler Postulate):
Front
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.
Back
The Converse of the Isosceles Triangle Theorem
Front
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Back
Definition: Congruent segments
Front
are segments that have the same length.
Back
Postulate: The Side-Side-Side Postulate (SSS)
Front
Given △ABC and △DEF, if line AB ≅ line DE,
line BC ≅ line EF, and line AC≅ line DF, then △ABC≅△DEF.
Back
Postulate (The Segment Addition Postulate)
Front
If B is between A and C, then AB+BC=AC.
Back
Reflexive Property of Congruence
Front
line AB≅line AB
(Reflexive)
Back
(The Midpoint Theorem)
Front
If M is the midpoint of line AB,
then 2AM=AB and AM=1/2AB.
Back
Definition An isosceles triangle
Front
a triangle that has two of its sides congruent. The congruent sides are called the legs of the triangle.
Back
Definition: Perpendicular line
Front
are two lines that form right angles.
Back
Theorem 1.6:
If two lines are perpendicular, then the adjacent angles....
Front
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."
Back
Adjacent angles
Front
are two angles in a plane that have a common vertex and a common side but no common interior points.
Back
Postulate (The Angle Addition Postulate) 2
Front
Postulate: If C does not lie on line AOB, then m∠AOC+m∠COB=180.
Back
Definitions: Vertical angles
Front
are two angles whose sides form pairs of opposite rays.
Back
(The Complement Theorem)
Front
If two angles are complements of congruent angles or of the same angle, then the two angles are congruent
CCAC
Back
An altitude
Front
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).
Back
The Isosceles Triangle Theorem
Front
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
Back
Theorem 1.7:
If two lines form congruent adjacent angle, then the lines .....
Front
If two lines form congruent adjacent angles, then the lines are perpendicular.
Note that this is the converse of Theorem 1.6.
Back
Definition: Two lines are perpendicular...
Front
if and only if they form right angles.
Back
Corollary 2.3: Every equilateral triangle is
Front
equiangular.
Back
Addition Property Equality
Front
If a=b and c=d, then a+c=b+d
(Add. Prop.)
Back
Transitive Property of Congruence
Front
If line AB≅ line CD,
and line CD≅ line CD≅ line EF,
then line AB≅ line EF
(Transitive)
Back
Section 2
(50 cards)
Theorem 3.4: AIC → P
Front
Given two coplanar lines and a transversal, if a pair of alternate interior angles are congruent, then the lines are parallel.
Back
Corollary 2.4: Every equiangular triangle is
Front
equilateral
Back
Theorem 3.13: If three parallel lines cut off congruent segments on one transversal, then they cut off
Front
congruent segments on every transversal.
Back
Theorem 4.5: If both pairs of opposite sides of a quadrilateral are congruent,
Front
then the quadrilateral is a parallelogram.
Back
Theorem 4.1: Opposite sides of a parallelogram are
Front
congruent.
Back
Theorem 3.10: The sum of the measures of the interior angles of a triangle is
Front
180
Back
Theorem 4.3: The diagonals of a parallelogram
Front
bisect each other.
Back
Postulate: The Perpendicular Postulate
Front
In a plane, through a point on a line, there is one and only one line perpendicular to the given line.
Back
Theorem 2.12: The Perpendicular Bisector Characterization Theorem (PBC)
Front
In a plane, the perpendicular bisector of a segment is the set of all points that are equidistant from the endpoints of the segment.
Back
For any two lines in space, one of the following conditions must exist:
Front
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.
Back
Theorem 3.12: The No-Choice Theorem
Front
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.
Back
Theorem 3.1: Exterior Angle Inequality
Front
The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.
Back
Theorem 3.20: The Triangle Inequality
Front
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Back
Postulate: The Parallel Postulate
Front
Given a point not on a line, there is exactly one line through the given point and parallel to the given line.
Back
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
Front
perpendicular bisector of the segment.
Back
Theorem 2.16: The Angle Bisector Characterization Theorem (ABC)
Front
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.
Back
Theorem 2.11: Halves of congruent segments are
Front
congruent.
Back
Proving Parallelograms
Front
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.
Back
Theorem 2.15: The Hypotenuse-Leg Theorem (HL)
Front
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.
Back
Theorem 3.19: If two angles of a triangle are not congruent,
Front
then the sides opposite those angles are not congruent, and the longer side is opposite the larger angle.
Back
Theorem 2.13: If a point is not on a line, then there is a line through the point which is
Front
perpendicular to the given line.
Back
Theorem 2.8: The angle bisectors of the base angles of an isosceles triangle are
Front
congruent.
Back
Theorem 3.11: The Exterior Angle Theorem
Front
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.
Back
Theorem 3.3: In a plane, if two lines m and n are perpendicular to a third line ℓ,
Front
then the lines m and n are parallel to each other.
Back
Corollary:
Front
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.
Back
If at least two of the sides of the triangle are congruent, then the triangle is an
Front
isosceles triangle
Back
Theorem 4.2: Opposite angles of a parallelogram are
Front
congruent.
Back
Characteristics of Parallelograms
Front
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
The diagonals bisect each other.
Consecutive angles are supplementary.
Back
Definition: Parallel
Front
Two lines are parallel if and only if they are coplanar and do not intersect.
Back
Indirect Proof
Front
Assume opposite of the Prove
Solve for contradictions
Back
Theorem 3.18: If two sides of a triangle are not congruent,
Front
then the angles opposite those sides are not congruent, and the larger angle is opposite the longer side.
Back
If all three sides are congruent, then the triangle is an
Front
equilateral triangle.
Back
Theorem 3.5: CAC → P
Front
Given two coplanar lines and a transversal, if a pair of corresponding angles are congruent, then the lines are parallel.
Back
the sum of the exterior angles
Front
is 360
Back
Definition: A parallelogram
Front
is a quadrilateral in which both pairs of opposite sides are parallel.
Back
Theorem 4.4: Consecutive angles of a parallelogram are
Front
supplementary.
Back
Theorem 3.2: If two distinct coplanar lines m and n are parallel to a third line ℓ, then lines m and n are
Front
parallel to each other.
Back
Theorem 2.7: The medians drawn to the legs of an isosceles triangle are
Front
congruent.
Back
Theorem 2.5: The bisector of the vertex angle of an isosceles triangle is
Front
perpendicular to the base.
Back
Definition: Transversal
Front
A line that intersects two different lines in two distinct points is called a transversal.
Back
Theorem 2.6: The median drawn from the vertex angle of an isosceles triangle is
Front
the angle bisector of the vertex angle.
Back
Theorem 3.7: P → CAC
Front
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Back
Theorem 3.9: If two parallel lines are cut by a transversal, then same-side interior angles are
Front
supplementary.
Back
Definition: perpendicular bisector
Front
In a plane, the perpendicular bisector of a segment is the line that is perpendicular to the segment at its midpoint.
Back
A diagonal of a polygon is
Front
a segment that joins nonconsecutive vertices of the polygon.
Back
Theorem 3.14: The sum of the measures of the interior angles of a convex polygon with n sides is
Front
(n−2)⋅180
Back
Theorem 3.6: Given two coplanar lines and a transversal, if a pair of same-side interior angles are supplementary,
Front
then the lines are parallel.
Back
Theorem 2.9: The altitudes drawn to the legs of an isosceles triangle are
Front
congruent.
Back
Theorem 2.10: Halves of congruent angles are
Front
congruent.
Back
Theorem 3.8: P → AIC
Front
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Back
Section 3
(29 cards)
Definition: A trapezoid is a
Front
quadrilateral with exactly one pair of opposite sides parallel.
Back
geometric mean
Front
If the means in a proportion are equal 3/x=x/12 — then we call x the geometric mean or the mean proportional of 3 and 12.
Back
Theorem 4.12: If a quadrilateral is a rhombus, then
Front
the diagonals bisect the angles of the rhombus and the diagonals are perpendicular
Back
Theorem 4.15: The base angles of an isosceles trapezoid are
Front
congruent.
Back
the midpoint of the hypotenuse
Front
For any right triangle ABC, the midpoint of the hypotenuse BC is equidistant from the 3 vertices A, B, C. (Comment: This is one direction of the Carpenter Locus Theorem. The other direction says that if BC is a segment with midpoint O and if A is a point with OA = OB = OC, then angle BAC is a right angle.)
Back
Theorem 4.14: If M is the midpoint of hypotenuse AB of right triangle ABC,
Front
then MA=MB=MC.
Back
Definition: A square is
Front
a parallelogram with four right angles and all four sides congruent.
Back
If a:b=c:d,
what are the extremes
what are the means
Front
This is derived from the colon notation, where there are numbers written on the far outside, and numbers written in between. If a:b=c:d, then we call a and d the extremes of the proportion, while b and c are the means.
Back
Theorem 4.10: If a quadrilateral is a rectangle, then
Front
the diagonals are congruent.
Back
Theorem 4.9: If ℓ and m are two parallel lines and points A and B are on ℓ, then the distance from A to line m is equal to the distance from
Front
B to line m.
Back
Theorem 5.3: If three parallel lines intersect two transversals,
Front
then those transversals are cut proportionally.
Back
Theorem 4.16: The median of a trapezoid is
Front
parallel to the bases of the trapezoid and the length of the median is equal to half the sum of the lengths of the bases.
Back
Theorem 5.5: SAS Similarity Theorem (OPTIONAL)
Front
If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are proportional, then the triangles are similar.
Back
Theorem 5.2: The Side-Splitter Theorem
Front
If a line is parallel to a side of a triangle and intersects the other two sides in two points, then it divides those two sides proportionally.
Back
AA Postulate:
Front
If two angles of one triangle are congruent to two angles of a second triangle, then the triangles are similar.
Back
Corollary:
Front
If a line goes through the midpoint of one side of a triangle and is parallel to a second side, then the line goes through the midpoint of the third side of the triangle.
Back
there is a ratio of proportionality or a scale factor between
Front
similar polygons
Back
Ratio
Front
the quantitative relation between two amounts showing the number of times one value contains or is contained within the other.
Back
Definition: A rhombus is a
Front
parallelogram with all four sides congruent.
Back
Theorem 5.1: If a line is parallel to one side of a triangle and intersects the other two sides in two points, then the line cuts off a triangle that is
Front
similar to the given triangle.
Back
Theorem 5.4: The Triangle Angle-Bisector Theorem
Front
If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the adjacent two sides of the triangle.
Back
Theorem 4.13: The Midline Theorem
Front
The segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length of the third side.
Back
Theorem 4.11: If a quadrilateral is a rhombus, then
Front
the diagonals are perpendicular to each other.
Back
A proportion is an
Front
equality between two ratios
Back
A trapezoid is an isosceles trapezoid if its
Front
legs are congruent.
Back
Definition: The median of a trapezoid is
Front
the segment joining the midpoints of the legs of the trapezoid.
Back
Definition: A midline of a triangle is
Front
a segment whose endpoints are the midpoints of two sides of the triangle.
Back
Definition: A rectangle is a
Front
parallelogram with four right angles.
Back
Theorem 5.6: SSS Similarity Theorem (OPTIONAL)
Front
If three sides of one triangle are proportional to the three corresponding sides of a second triangle, then the two triangles are similar.