Core Geometry Theorems & Properties – Practice Flashcards

Question-and-answer flashcards that cover fundamental geometry properties, postulates, and theorems from equality properties to circle theorems.

What does the Addition Property of Equality state?

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

What does the Subtraction Property of Equality state?

If a = b, then a – c = b – c.

According to the Multiplication Property of Equality, what happens when you multiply both sides of an equation by the same number?

If a = b, then ac = bc.

State the Division Property of Equality.

If a = b, then a⁄c = b⁄c (where c ≠ 0).

What is the Reflexive Property of Congruence?

Any geometric quantity is congruent to itself; symbolically, a ≅ a.

Describe the Symmetric Property of Congruence.

If a geometric quantity a is congruent to b, then b is congruent to a (a ≅ b ⇒ b ≅ a).

Explain the Transitive Property of Congruence.

If a ≅ b and b ≅ c, then a ≅ c.

What does the Substitution Property of Equality allow you to do in geometry?

Replace any geometric quantity with another equal quantity within an expression (if a = b, substitute b for a).

What is the intersection of two unique planes?

A line.

What is the intersection of two unique lines?

A single point.

How many points determine a unique line?

Two points.

How many non-collinear points determine a unique plane?

Three non-collinear points.

If two points lie on a plane, where is the line containing them?

The line is also on the plane.

Write the Distance Formula between points (x₁, y₁) and (x₂, y₂).

d = √[(x₂ – x₁)² + (y₂ – y₁)²].

State the Midpoint Theorem for segment AC when B is the midpoint.

AB = ½ AC (and AB = BC).

What does the Overlapping Segment Theorem state for collinear points P–K–E–C if PK = EC?

Then PE = KC.

State the Converse of the Overlapping Segment Theorem for collinear points P–K–E–C.

If PE = KC, then PK = EC.

State the Alternate Exterior Angles Theorem (AEA).

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

What is the converse of the Alternate Exterior Angles Theorem?

If a transversal forms congruent alternate exterior angles, the two lines are parallel.

State the Alternate Interior Angles Theorem (AIA).

If two parallel lines are cut by a transversal, alternate interior angles are congruent.

Give the converse of the Alternate Interior Angles Theorem.

If alternate interior angles are congruent, the lines cut by the transversal are parallel.

State the Same-Side Interior Angles Theorem (SSI).

If two parallel lines are cut by a transversal, same-side interior angles are supplementary.

Converse of the Same-Side Interior Angles Theorem?

If same-side interior angles are supplementary, the two lines are parallel.

What does the Vertical Angle Theorem tell us about vertical angles?

Vertical angles formed by two intersecting lines are congruent.

State the Pythagorean Theorem.

In a right triangle, (leg)² + (leg)² = (hypotenuse)².

What is the converse of the Pythagorean Theorem?

If a² + b² = c² for the sides of a triangle (with c the longest side), the triangle is right.

State the Isosceles Triangle Theorem.

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

What does the Exterior Angle Theorem (Remote Interior Angle Theorem) say?

An exterior angle of a triangle equals the sum of the two remote interior angles.

State the Triangle Inequality Theorem.

The sum of any two side lengths of a triangle is greater than the third side.

What is the Triangle Angle Sum Theorem?

The interior angles of a triangle sum to 180°.

Explain CPCTC.

Corresponding Parts of Congruent Triangles are Congruent—if two triangles are congruent, all matching sides and angles are equal.

If the same pair of opposite sides in a quadrilateral are both congruent and parallel, what can you conclude?

The quadrilateral is a parallelogram.

What if opposite angles of a quadrilateral are congruent and the opposite sides are parallel?

The quadrilateral is a parallelogram.

How do diagonals confirm a parallelogram?

If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

State the intersecting chords theorem for a circle.

If two chords intersect inside a circle, (segment₁ × segment₂) of one chord equals (segment₁ × segment₂) of the other chord.

What is the external secant segment theorem?

If two secants intersect outside a circle, (whole secant × external part) = (other whole secant × its external part).

State the tangent-secant theorem.

(Tangent segment)² = (whole secant × external secant segment).