Algebra and Geometry Proofs Flashcards

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What does the Angle Addition Postulate state?

The measure of the larger angle is the sum of the measures of the two smaller angles.

What does the Linear Angle Theorem state?

If two angles form a linear pair, then they are supplementary.

What is the Transitive Property of Equality?

If a = b and b = c, then a = c.

What is the Definition of a Segment Bisector?

A point, segment, line, or plane that divides a line segment into two equal parts. Signup and view all the answers

What does the Property of Substitution state?

If x = y, then x can be substituted for y in any equation, and y can be substituted for x in any equation. Signup and view all the answers

Match the following properties of equality with their examples:

Addition = If x - 5 = 10, then x = 15. Subtraction = If x + 5 = 10, then x = 5. Multiplication = If r/3 = 4, then r = 12. Division = If 2x = 16, then x = 8. Signup and view all the answers

What does the Vertical Angle Theorem state?

If angle A and angle B are vertical, then angle A is congruent to angle B. Signup and view all the answers

What is the Definition of Supplementary Angles?

If two angles are supplementary, then they add up to 180 degrees. Signup and view all the answers

What is the Definition of an Angle Bisector?

An angle bisector is a point, ray, line, etc. that splits an angle into two equal parts. Signup and view all the answers

What does the Definition of Congruency state?

If segment AB is congruent to segment BC, then AB = BC. Signup and view all the answers

What does the Corresponding Angle Postulate state?

If two parallel lines are cut by a transversal, then the corresponding angles are congruent. Signup and view all the answers

What does the Alternate Interior Angle Conjecture state?

If two parallel lines are intersected by a transversal, then alternate interior angles are congruent. Signup and view all the answers

What does the Alternate Exterior Angle Theorem state?

If a pair of parallel lines is cut by a transversal, then the alternate exterior angles are congruent. Signup and view all the answers

What is the Same Side Interior Angle Theorem?

If two parallel lines are cut by a transversal, then the same side interior angles are supplementary. Signup and view all the answers

What does the Converse of the Corresponding Angle Postulate state?

If corresponding angles are congruent, then two lines are parallel. Signup and view all the answers

What does the Converse of the Alternate Interior Angle Postulate state?

If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. Signup and view all the answers

What does the Converse of the Alternate Exterior Angle Postulate state?

If two lines are cut by a transversal, and alternate exterior angles are congruent, then the two lines are parallel. Signup and view all the answers

What does the Triangle Sum Theorem state?

If you have a triangle, then all the angles add up to 180 degrees. Signup and view all the answers

What is the Definition of Perpendicular Lines?

If two lines are perpendicular, then all four angles formed are right angles. Signup and view all the answers

What does the 2 Lines Perpendicular Theorem state?

If two lines are perpendicular to the same line, then they themselves are parallel. Signup and view all the answers

What does the All Right Angles are Congruent state?

If two angles are right angles, then they are congruent. Signup and view all the answers

What does the 2 Lines Parallel Theorem state?

If two lines are parallel to the same line, then those lines are parallel. Signup and view all the answers

What does the SSS Postulate state?

If three sides of two triangles are congruent, then the triangles are congruent. Signup and view all the answers

What does the SAS Postulate state?

If a side and included angle, and another side of one triangle are congruent to a second triangle, then the triangles are congruent. Signup and view all the answers

What does the Angle Side Angle Postulate (ASA) state?

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Signup and view all the answers

What does the Angle Side Side Postulate (ASS) OR Side Side Angle Postulate (SSA) state?

If two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then these two triangles are congruent. Signup and view all the answers

What does the Angle Angle Side Postulate (AAS) state?

If two congruent angles and two non-inclusive congruent sides, then the triangles are congruent. Signup and view all the answers

What is the Reflexive Property?

Segment RN is congruent to segment RN. Signup and view all the answers

What does the Overlapping Line Segment Theorem state?

If two overlapping segments have congruent non-overlapping parts, then they are congruent. Signup and view all the answers

What does the Hypotenuse Leg theorem state?

If any two right triangles have a congruent hypotenuse and a corresponding congruent leg, then they are congruent triangles. Signup and view all the answers

What is the Definition of a Right Triangle?

If a triangle has a right angle, then it is a right triangle. Signup and view all the answers

What does the Definition of a Midpoint state?

A midpoint is a point that divides a line segment into two equal parts. Signup and view all the answers

What does the Congruent Parts Congruent Triangles Congruent (CPCTC) state?

If two triangles are congruent, then their parts are congruent. Signup and view all the answers

What does the Isosceles Triangle Theorem (ITT) state?

If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Signup and view all the answers

What does the Converse of ITT state?

If two angles of a triangle are congruent, then the two sides opposite are congruent. Signup and view all the answers

Study Notes

Angle Addition Postulate

States that if point B is in the interior of angle AOC, then the measure of angle AOC equals the sum of the measures of angles AOB and BOC.

Linear Angle Theorem

Asserts that when two angles form a linear pair, they are supplementary, meaning their measures add up to 180 degrees.

Transitive Property of Equality

If two values are equal to a third value, then they are equal to each other (if a = b and b = c, then a = c).

Definition of Segment Bisector

Defined as a point, segment, line, or plane that divides a line segment into two equal halves, always including the midpoint.

Property of Substitution

If one variable is equal to another, it can be replaced in any equation, allowing interchangeable use of these variables.

Properties of Equality

Includes rules for effective algebra manipulation:
- Addition: x - 5 = 10 leads to x = 15.
- Subtraction: x + 5 = 10 results in x = 5.
- Multiplication: r/3 = 4 implies r = 12.
- Division: 2x = 16 gives x = 8.

Vertical Angle Theorem

States that if angle A and angle B are vertical angles, then angle A is congruent to angle B.

Definition of Supplementary Angles

Two angles are supplementary if their measures sum to 180 degrees.

Definition of an Angle Bisector

A ray, line, or segment that divides an angle into two equal parts, ensuring the two resulting angles are congruent.

Definition of Congruency

States that if segment AB is congruent to segment BC, then their lengths are equal (AB = BC).

Definition of Congruence

Similar to congruency, reinforces that if two segments are congruent, they are equal in length.

Corresponding Angle Postulate

When two parallel lines are crossed by a transversal, the corresponding angles are congruent.

Alternate Interior Angle Conjecture

Asserts that alternate interior angles are congruent when two parallel lines are intersected by a transversal.

Alternate Exterior Angle Theorem

States that alternate exterior angles are congruent when two parallel lines are cut by a transversal.

Same Side Interior Angle Theorem

States that if two parallel lines are intersected by a transversal, then the same side interior angles are supplementary.

Converse of the Corresponding Angle Postulate

States that if corresponding angles are congruent, then the two lines are parallel.

Converse of the Alternate Interior Angle Postulate

States that if a transversal creates congruent alternate interior angles, then the two lines must be parallel.

Converse of the Alternate Exterior Angle Postulate

If alternate exterior angles are congruent, then the two lines cut by the transversal are parallel.

Triangle Sum Theorem

States that the sum of the interior angles of a triangle is always 180 degrees.

Definition of Perpendicular Lines

When two lines intersect to form right angles, they are considered perpendicular. If one angle is right, the lines are perpendicular.

2 Lines Perpendicular Theorem

If two lines are both perpendicular to the same line, then they are parallel to each other.

All Right Angles are Congruent

All right angles (90 degrees) measure the same and are therefore congruent.

2 Lines Parallel Theorem

If two lines are parallel to a third line, they are parallel to each other.

SSS Postulate (Side Side Side)

If all three sides of two triangles are congruent, then the triangles themselves are congruent.

SAS Postulate (Side Angle Side)

If two sides and the included angle of one triangle are congruent to those of another triangle, then the triangles are congruent.

Angle Side Angle Postulate (ASA)

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Angle Side Side Postulate (ASS) or Side Side Angle Postulate (SSA)

States that if two angles and a non-included side of one triangle are congruent to those of another triangle, the triangles are congruent.

Angle Angle Side Postulate (AAS)

If two angles and a corresponding non-inclusive side of one triangle are congruent to those of another triangle, then the triangles are congruent.

Reflexive Property

A segment is congruent to itself (e.g., segment RN is congruent to segment RN).

Overlapping Line Segment Theorem

If two overlapping segments share congruent non-overlapping parts, the entire segments are congruent.

Hypotenuse Leg

If two right triangles have a congruent hypotenuse and one corresponding leg, then the triangles are congruent.

Definition of Right Triangle

A triangle that contains a right angle (90 degrees).

Definition of a Midpoint

The exact middle point of a line segment, splitting it into two equal parts.

Congruent Parts Congruent Triangles Congruent (CPCTC)

If two triangles are congruent, then each corresponding part (angles and sides) is also congruent.

Isosceles Triangle Theorem (ITT)

If two sides of a triangle are congruent, then the angles opposite those sides are also congruent.

Converse of ITT

If two angles of a triangle are congruent, then the sides opposite those angles must also be congruent.

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Description

This quiz focuses on key concepts in algebraic and geometric proofs, including the Angle Addition Postulate, Linear Angle Theorem, and Transitive Property of Equality. Ideal for students looking to reinforce their understanding of foundational mathematical principles.