Summer Geometry 2026

Point

0 dimensions

Represented by a dot

Line

1 dimension

Represented by ↔

Extends infinitely

Plane

2 dimensions

Represented by 3 points on it

Extends infinitely

Segment

2 endpoints

Ray

1 endpoint

Collinear Points

Lie on the same line

Coplanar Points

Lie on the same plane

Postulate or Axiom

A rule that is accepted without proof

Coordinate of a Point

Real number that corresponds to a point

Example: (5, 3)

Segment Addition Postulate

If B is between A and C, AB + BC = AC

Congruent Segments

Same length

Midpoint

Point that divides a segment into two congruent segments

Segment Bisector

A point, ray, line, line segment, or plane that intersects a segment at its midpoint

Midpoint Formula

Distance Formula

Angles

Consist of two different rays with the same endpoint that are the sides of the angle

Angle Addition Postulate

If P is the interior of angle RST, then mRST = mRSP + mPST

Angle Bisector

A ray that divides an angle into 2 congruent angles

Complementary Angles

2 angles whose measures sum to 90 degrees

Supplementary Angles

2 angles whose measures sum to 180 degrees

Adjacent Angles

2 angles that share a common vertex and side but no common interior points

IMPORTANT: vertical angles CANNOT be adjacent angles

Linear Pair

2 angles’ uncommon sides are opposite rays and they are supplementary

Vertical Angles

Angles whose sides are opposite rays and they are congruent

Polygon

Closed plane figure formed by 3+ line segments and each side intersects exactly 2 sides

Vertex of a Polygon

Endpoint of a side

Convex Polygon

No line that contains the sides of a polygon contains a point inside the polygon

Concave Polygon

A line that contains the sides of a polygon also contains a point inside

Equilateral Polygon

All congruent sides

Equiangular Polygon

All congruent angles

Regular Polygon

Convex, equilateral, and equiangular

Polygon Number of Sides Names Chart

Conjecture

Unproven statement based on observations

Inductive Reasoning

Find a pattern in specific cases then use a conjecture to apply for the general case

Prove a Conjecture True

Prove for every single case

Disprove a Conjecture

Find 1 counterexample

Counterexample

Used to disprove a conjecture

A special case where the conjecture is false

Conditional Statement

A logical statement with 2 parts (eg. If the weather is nice, then I will play outside)

• Hypothesis (eg. If the weather is nice)

• Conclusion (eg. then I will play outside)

If p, then q.

Converse

Switch the hypothesis and conclusion

If q, then p

Inverse

Negate the hypothesis and conclusion

If not p, then not q

Contrapositive

Write the converse then inverse

If not q, then not p

Equivalent Statements

When two statements are both true or both false

Examples:

• The conditional and contrapositive are either both true or both false

• The inverse and converse are either both true or both false

Biconditional Statement

When a conditional statement and its converse are both true and can rewrite as one statement

Us if and only if

Deductive Reasoning

Uses facts, definitions, accepted properties, and laws of logic to form a logical argument (NO PATTERNS)

Postulates So Far

• Ruler Postulates (not needed)

• Segment Addition Postulate

• Protractor Postulate (not needed)

• Angle Addition Postulate

• Through any 2 points there exists exactly 1 line

• A line contains at least 2 points

• If 2 lines intersect, then their intersection is exactly 1 point

• Through any 3 noncollinear points, there exists exactly 1 plane

• A plane contains at least 3 noncollinear points

• If 2 points lie in a plane, then the line containing them lies in the plane

• If two planes intersect, then their intersection is a line

Proof

Logical argument using deductive reasoning that shows a statement is true and is created by making 1 fact-based statement at a time

2 Column Proof

Contains numbers statements and reasons

2 Column Proof

Contains numbers statements and reasons

Segment Congruence Theorem

Segment congruence is reflexive

• For any segment AB, AB = AB

Segment congruence is symmetric

• If AB = CD, then CD = AB

Segment congruence is transitive

• If AB = CD and CD = EF, then AB = EF

Angle Congruence Theorem

Angle congruence is reflexive

• For any angle A, A = A

Angle congruence is symmetric

• If A = B, then B = A

Angle congruence is transitive

• If A = B and B = C, then A = C

Angle Theorems

• All right angles are congruent

• If 2 angles are supplementary to the same angle, then they are congruent

• If two angles are complementary to the same angle, then they are congruent

Angle Postulates

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

• Vertical angles are congruent

Parallel Lines

Lines that are coplanar and never intersect

IMPORTANT: Lines that do not intersect are SOMETIMES parallel

Skew Lines

Lines that do not intersect but are not coplanar

Parallel Planes

Planes that do not intersect

Angles Formed by Transversals

• Corresponding angles if they have corresponding positions

• Alternate interior angles if they lie between the two lines on opposite sides

• Alternate exterior angles if they lie outside the two lines on opposite sides

• Consecutive interior angles if they lie between the two lines on the same side

Parallel Postulate

If there is a line and a point not on the lines, then there is exactly one line through the point parallel to the line

Perpendicular Postulate

If there is a line and a point not on the lines, then there is exactly one line through the point perpendicular to the line

Corresponding Angles Postulate

If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent

Alternate Interior Angles Converse

If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel

Corresponding Angles Converse

If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel

Alternate Interior Angles Theorem

If two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel

Alternate Exterior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent

Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, the the pairs of consecutive interior angles are supplementary

Alternate Exterior Angles Converse

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

Consecutive Interior Angles Converse

If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel

Transitive Property of Parallel Lines

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

Slope

Ration of the vertical change to the horizontal change of a nonvertical lines

• Negative slope: falls left to right

• Positive slope: rises left to right

• Zero slope: horizontal

• Undefined: vertical

Slopes of Parallel Lines Postulate

In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope

Two vertical lines are always parallel

Slopes of Perpendicular Lines Postulate

In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1

Horizontal lines are perpendicular to vertical lines

Slopes with Different Lines (Parallel and Perpendicular)

Parallel lines have the same slope

Perpendicular lines have opposite reciprocal slops

Slope Intercept Form

y = mx + b

m = slope

b = y intercept

Point Slope Form

y - y_1 = m(x - x_1)

m = slope

Standard Form

Ax + By = C

If two lines intersect to form a linear pair of congruent angles…

then the lines are perpendicular

If two lines are perpendicular…

then they intersect to form 4 right angles

If two sides of two adjacent acute angles are perpendicular…

then the angles are complementary

If a transversal is perpendicular to one of two parallel lines…

then it is perpendicular to the other

In a plane, if two lines are perpendicular to the same line…

then they are parallel to each other

The distance from a point to a lines is…

the length of the perpendicular segment from the point to the lines

Scalene Triangles

No congruent sides

Isosceles Triangle

At least 2 congruent sides

Equilateral Triangles

3 congruent sides

Acute Triangles

3 acute angles

Right Triangle

1 right angle

Obtuse Triangles

1 obtuse angle

Equiangular Triangles

3 congruent angles

Triangle Sum Theorem

The sum of the measures of the interior angles of a triangle is 180 degrees

Corollary to the Triangle Sum Theorem

The acute angles of a right triangle are complementary

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles

Congruent Figures

Exactly the same size and shape and all parts of one figure are exactly the same to the corresponding parts of the other

Third Angles Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent

Properties of Congruent Triangles

Reflexive property of congruent triangles:

• For any triangle ABC, ABC is congruent to ABC

Symmetric property of congruent triangles:

• If ABC is congruent to DEF, then DEF is congruent to ABC

Transitive property of congruent triangles:

• If ABC is congruent to DEF and DEF is congruent to JKL, then ABC is congruent to JKL