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Reflexive Property Equality

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Cards (129)

Section 1

(50 cards)

Reflexive Property Equality

Front

a=a (Reflexive)

Back

(The Vertical Angle Theorem)

Front

Vertical angles are congruent.

Back

A median of a triangle

Front

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.

Back