Geometry Proofs Flashcards

Angles

• Supplementary angles sum to 180 degrees.
• Complementary angles sum to 90 degrees.
• Vertical angles are opposite angles formed when two lines intersect and are congruent.
• Adjacent angles share a common vertex and side.

Basic Properties for Proofs

• Reflexive Property: a is equal to a.
• Symmetric Property: if a equals b, then b equals a.
• Transitive Property: if a equals b and b equals c, then a equals c.
• Substitution allows for replacing equal values in equations.
• Addition and subtraction properties maintain equality during operations.
• Multiplication and division can be applied to both sides of an equation as long as the divisor is not zero.
• Commutative Property indicates that order does not affect addition or multiplication.
• Associative Property highlights grouping does not affect addition or multiplication.
• Distributive Property connects multiplication with addition.

Parallel Lines and Transversals

• Corresponding angles formed are congruent when parallel lines are cut by a transversal.
• Alternate interior angles are congruent.
• Alternate exterior angles are congruent.
• Consecutive interior angles sum to 180 degrees (supplementary).

Proving Lines are Parallel

• Congruent corresponding angles imply lines are parallel.
• Congruent alternate interior angles signify parallel lines.
• Congruent alternate exterior angles indicate lines are parallel.
• Supplementary consecutive interior angles confirm parallelism.
• Lines sharing the same slope are parallel.

Proving a Right Angle

• Perpendicular lines intersect at right angles. Can be demonstrated using the slope formula or given values.

Proving Congruent Triangles

• Congruence can be established via:
  • Side-Angle-Side (SAS)
  • Side-Side-Side (SSS)
  • Angle-Side-Angle (ASA)
  • Angle-Angle-Side (AAS)
• CPCTC: Corresponding Parts of Congruent Triangles are Congruent.

Quadrilaterals

• Trapezoid:

  • One pair of parallel sides (bases).
  • Angles on the same side of bases are supplementary.

• Isosceles Trapezoid:

  • One pair of parallel sides.
  • Congruent legs and congruent base angles.
  • Diagonals are congruent.

• Kite:

  • Two pairs of consecutive congruent sides.
  • One pair of congruent opposite angles.
  • Diagonals are perpendicular.

• Parallelogram:

  • Opposite sides are congruent.
  • Opposite angles are congruent.
  • Consecutive angles are supplementary.
  • If one angle is right, all angles are right.
  • Diagonals bisect each other.
  • Each diagonal divides the parallelogram into two congruent triangles.

• Rectangle:

  • A parallelogram with all angles congruent.
  • Diagonals are congruent.

• Rhombus:

  • A parallelogram with all sides congruent.
  • Diagonals are perpendicular and bisect opposite angles.

• Square:

  • A figure that is both a rhombus and a rectangle.
  • All sides and angles are congruent.

Proving a Quadrilateral is a Parallelogram

• Achieve by demonstrating:
  • Both pairs of opposite sides are parallel.
  • Both pairs of opposite sides are congruent.
  • Both pairs of opposite angles are congruent.
  • An interior angle is supplementary to consecutive angles.
  • Diagonals bisect each other.
  • One pair of opposite sides is both parallel and congruent.

Proving a Quadrilateral is a Rectangle

• Confirm by showing:
  • All four angles are congruent.
  • It is a parallelogram and its diagonals are congruent.

Proving a Quadrilateral is a Rhombus

• Validate by proving:
  • All sides are congruent.
  • It is a parallelogram with perpendicular diagonals.
  • Each diagonal bisects a pair of opposite angles.

Proving a Quadrilateral is a Square

• Demonstrate it is both a rhombus and a rectangle.

Proving a Quadrilateral is a Kite

• Establish if it has:
  • Two pairs of congruent sides.
  • Opposite sides are not congruent.
  • Diagonals are perpendicular.
  • One pair of congruent opposite angles.

Proving a Quadrilateral is an Isosceles Trapezoid

• Show:
  • It is a trapezoid with a pair of congruent legs.
  • It has congruent base angles.
  • Its diagonals are congruent.

Proving Similar Triangles

• Establish similarity through:
  • Angle-Angle (AA) congruence.
  • Sides in proportion (SSS).
  • Two sides in proportion with an included angle (SAS).