Geometry Theorems Reference Guide
Unit 1 Reference Sheet
Constructions & Rigid Transformations
Kendall Hunt Illustrative Mathematics • High School Geometry
Printable Unit Unit 1 Complete
Unit 1 Core Principle: Symmetry and Rigid Motions In Unit 1, students explore compass-and-straightedge constructions and use reflections, rotations, and translations to establish the fundamental properties of perpendicular bisectors and angle bisectors.
Conditional
Perpendicular Bisector Theorem & Converse
Theorem: If a point lies on the perpendicular bisector of a segment, then it is equidistant from the segment's endpoints.
Converse: If a point is equidistant from the endpoints of a segment (\(PA = PB\)), then it lies on the perpendicular bisector of that segment.
P A B
Conditional
Angle Bisector Equidistance Theorem & Converse
Theorem: If a point lies on the interior bisector of an angle, then it is equidistant from the two rays forming the angle (\(PX = PY\)).
Converse: If a point in the interior of an angle is equidistant from the angle's rays, then it lies on the angle bisector.
P
Foundational Axiom
Preservation of Distance & Angle Measure
If a geometric figure undergoes a rigid motion (translation, rotation, or reflection), then every segment length is preserved (\(A'B' = AB\)) and every angle measure is preserved (\(\text{m}\angle A'B'C' = \text{m}\angle ABC\)).
Length Invariance \(A'B' = AB\)
Angle Invariance \(\text{m}\angle A'B'C' = \text{m}\angle ABC\)
Collinearity Lines map to lines
Key Construction Theorems
Perpendicular Bisector Construction If two circles of equal radius centered at \(A\) and \(B\) intersect, then the line passing through their intersection points is the perpendicular bisector of segment \(AB\).
Angle Bisector Construction If a rhombus is constructed using the rays of an angle from its vertex, then the diagonal passing through the vertex bisects the angle.
Kendall Hunt Illustrative Mathematics • High School Geometry Unit 1 Complete • Page 1 of 7
Unit 2 Reference Sheet • Part 1
Angles, Lines & Triangle Congruence
Kendall Hunt Illustrative Mathematics • High School Geometry
Printable Unit Unit 2 (1 of 2)
Relationships
Angle, Line & Isosceles Triangle Theorems
Vertical Angles Theorem
If two lines intersect to form vertical angles, then the vertical angles are congruent (\(\angle 1 \cong \angle 3\) and \(\angle 2 \cong \angle 4\)).
1 3
Triangle Angle Sum Theorem
If a figure is a triangle, then the sum of its interior angle measures is \(180^\circ\) (\(\angle 1 + \angle 2 + \angle 3 = 180^\circ\)).
Alternate Interior Angles Theorem & Converse
Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Converse: If alternate interior angles are congruent, then the lines are parallel.
Corresponding Angles Theorem & Converse
Theorem: If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Converse: If corresponding angles are congruent, then the lines are parallel.
Isosceles
Isosceles Triangle Base Angles Theorem & Converse
Theorem: If two sides of a triangle are congruent (\(AB \cong AC\)), then the angles opposite those sides are congruent (\(\angle B \cong \angle C\)).
Converse: If two angles of a triangle are congruent (\(\angle B \cong \angle C\)), then the sides opposite those angles are congruent (\(AB \cong AC\)).
Congruence Criteria
Triangle Congruence Theorems & SSA Analysis
SAS
Side-Angle-Side Congruence Theorem
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
ASA
Angle-Side-Angle Congruence Theorem
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.
SSS
Side-Side-Side Congruence Theorem
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
AAS
Angle-Angle-Side Congruence Theorem
If 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.
HL
Hypotenuse-Leg Congruence Theorem (Right Triangles)
If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
Non-Criterion
Why SSA is Not Valid & The Long-Side Exception
Failure Condition: If two sides and a non-included angle are given (\(\angle A\), side \(b\), and opposite side \(a\)) where the opposite side is shorter than the adjacent side (\(a < b\)), then swinging side \(a\) creates two possible non-congruent triangles (\(\triangle AB_1C\) and \(\triangle AB_2C\)).
Valid Exception: If the side opposite the known angle is longer than or equal to the adjacent side (\(a \ge b\)), then the side can only intersect the base ray at one unique point, guaranteeing a unique congruent triangle.
A C B&sub2; B&sub1; b a
Kendall Hunt Illustrative Mathematics • High School Geometry Unit 2 (Part 1) • Page 2 of 7
Unit 2 Reference Sheet • Part 2
Quadrilateral Theorems & Midsegments
Kendall Hunt Illustrative Mathematics • High School Geometry
Printable Unit Unit 2 (2 of 2)
Deductive Progression: Parallelograms to Special Quadrilaterals In IM Geometry, students use triangle congruence criteria along diagonals to establish properties of parallelograms, then specialize these conditionals to rectangles, rhombuses, and midsegments.
Parallelogram: Opposite Sides Theorem
If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent.
Parallelogram: Opposite Angles Theorem
If a quadrilateral is a parallelogram, then both pairs of opposite angles are congruent and consecutive angles are supplementary.
Parallelogram: Diagonals Bisect Theorem
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Parallelogram Converse Conditions
If a quadrilateral has both pairs of opposite sides congruent, OR one pair of opposite sides parallel and congruent, OR diagonals that bisect each other, then it is a parallelogram.
Rectangle Diagonals Theorem
If a parallelogram is a rectangle, then its diagonals are congruent (\(AC \cong BD\)).
Rhombus Diagonals Theorem
If a parallelogram is a rhombus, then its diagonals are perpendicular bisectors of each other and bisect the opposite interior angles.
Triangle Midsegment Theorem & Converse
Midsegment
Theorem: If a segment connects the midpoints of two sides of a triangle, then it is parallel to the third side and half as long (\(DE \parallel BC\) and \(DE = \frac{1}{2}BC\)).
Converse: If a line passes through the midpoint of one side parallel to a second side, then it bisects the third side.
Kendall Hunt Illustrative Mathematics • High School Geometry Unit 2 (Part 2) • Page 3 of 7
Unit 3 Reference Sheet
Similarity Theorems & Proofs
Kendall Hunt Illustrative Mathematics • High School Geometry
Printable Unit Unit 3 Complete
Unit 3 Foundation: Dilations and Proportional Reasoning In IM Geometry, similarity is defined through dilations: two figures are similar if there exists a sequence of dilations and rigid motions mapping one onto the other.
Side-Splitter
Triangle Proportionality Theorem & Converse
Theorem: If a line is parallel to one side of a triangle intersecting the other two sides, then it divides those two sides proportionally (\(\frac{AD}{DB} = \frac{AE}{EC}\)).
Converse: If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
A D E B C
AA, SAS, SSS
Triangle Similarity Criteria
AA: If two angles of one triangle are congruent to two angles of another, then the triangles are similar.
SAS: If two sides are proportional and their included angles are congruent, then the triangles are similar.
SSS: If all three corresponding sides are proportional, then the triangles are similar.
Geometric Mean
Right Triangle Altitude Theorem
If an altitude is drawn to the hypotenuse of a right triangle, then it divides the triangle into two triangles that are similar to each other and to the original triangle (\(\triangle ACD \sim \triangle CBD \sim \triangle ABC\)), yielding \(h^2 = xy\), \(a^2 = xc\), and \(b^2 = yc\).
x y h
Pythagoras
The Pythagorean Theorem & Converse
Theorem: If a triangle is a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), then \(a^2 + b^2 = c^2\).
Converse: If the side lengths of a triangle satisfy \(a^2 + b^2 = c^2\), then the triangle is a right triangle.
a b c
Proportions
Triangle Angle Bisector Theorem
If a ray bisects an interior angle of a triangle, then it divides the opposite side into segments proportional to the adjacent sides: \(\frac{BD}{DC} = \frac{AB}{AC}\).
BD DC AB AC
Kendall Hunt Illustrative Mathematics • High School Geometry Unit 3 Complete • Page 4 of 7
Units 4 & 5 Reference Sheet
Trigonometry & Solid Geometry
Kendall Hunt Illustrative Mathematics • High School Geometry
Printable Unit Units 4 & 5 Complete
Unit 4
Right Triangle Trigonometry Conditionals
Complementary Angles Theorem
If two acute angles are complementary (\(\angle A + \angle B = 90^\circ\)), then \(\sin(A) = \cos(B)\) and \(\cos(A) = \sin(B)\).
θ 90-θ
Pythagorean Trig Identity
If \(\theta\) is an acute angle in a right triangle, then \(\sin^2(\theta) + \cos^2(\theta) = 1\).
cos θ sin θ 1
Ratio Invariance Theorem
If two right triangles share an acute angle of measure \(\theta\), then their corresponding side ratios (\(\sin\theta\), \(\cos\theta\), \(\tan\theta\)) are equal.
Tangent & Slope Relation
If a line has an angle of inclination \(\theta\) relative to the horizontal, then its slope equals \(\tan(\theta) = \frac{\Delta y}{\Delta x}\).
Unit 5
Solid Geometry & Volume Conditionals
Cavalieri's Principle
If two solids have equal heights and equal cross-sectional areas at every parallel plane, then they have equal volumes (\(V_1 = V_2\)).
Pyramid & Cone Volume Theorem
If a pyramid or cone has base area \(B\) and height \(h\), then its volume is \(V = \frac{1}{3}Bh\).
h
Dimensional Scaling Theorems
If all linear dimensions of a solid are multiplied by scale factor \(k\), then its perimeter/length scales by \(k^1\), its surface area scales by \(k^2\), and its volume scales by \(k^3\).
Kendall Hunt Illustrative Mathematics • High School Geometry Units 4 & 5 Complete • Page 5 of 7
Unit 6 Reference Sheet
Coordinate Geometry Theorems
Kendall Hunt Illustrative Mathematics • High School Geometry
Printable Unit Unit 6 Complete
Analytic Proof: Transforming Geometry with Coordinates In Unit 6, students translate synthetic geometric properties into algebraic equations using slopes, the midpoint formula, and the distance formula.
Parallel Lines Slope Theorem & Converse
Theorem: If two non-vertical lines are parallel, then their slopes are equal (\(m_1 = m_2\)).
Converse: If two non-vertical lines have equal slopes, then they are parallel.
Perpendicular Lines Slope Theorem & Converse
Theorem: If two non-vertical lines are perpendicular, then their slopes are negative reciprocals (\(m_1 \cdot m_2 = -1\)).
Converse: If the slopes of two lines satisfy \(m_1 \cdot m_2 = -1\), then the lines are perpendicular.
Equation of a Circle Theorem
Theorem: If a circle has center \((h,k)\) and radius \(r\), then a point \((x,y)\) lies on the circle if and only if \((x - h)^2 + (y - k)^2 = r^2\).
(h,k) (x,y)
Coordinate Verification Conditionals
Parallelogram If both pairs of opposite sides have equal slopes, then the figure is a parallelogram.
Rectangle If adjacent sides have slopes satisfying \(m_1 \cdot m_2 = -1\), then the figure is a rectangle.
Rhombus If all four side lengths calculated by distance are equal, then the figure is a rhombus.
Kendall Hunt Illustrative Mathematics • High School Geometry Unit 6 Complete • Page 6 of 7
Unit 7 Reference Sheet
Circle Theorems & Proofs
Kendall Hunt Illustrative Mathematics • High School Geometry
Printable Unit Unit 7 Complete
Radial Symmetry, Inscribed Angles & Tangents In Unit 7, students combine isosceles triangles formed by radii with the Exterior Angle Theorem to establish angle, arc, and tangent relationships.
Tangent \(\perp\) Radius Theorem & Converse
Theorem: If a line is tangent to a circle at point \(T\), then it is perpendicular to the radius drawn to \(T\).
Converse: If a line is perpendicular to a radius at its circle endpoint, then it is tangent.
Tangent Segments Theorem
If two tangent segments are drawn to a circle from the same external point \(P\), then the tangent segments are congruent (\(PA \cong PB\)).
Inscribed Angle Theorem
Core Circle Theorem
If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc (\(\text{m}\angle ABC = \frac{1}{2}\text{m}\widehat{AC}\)) and half the measure of the central angle subtending the same arc.
Thales's Theorem (Semicircle)
If an angle is inscribed in a semicircle, then it is a right angle (\(\text{m}\angle = 90^\circ\)).
Angles on Same Arc Theorem
If two inscribed angles intercept the same arc, then the angles are congruent.
Cyclic Quadrilateral Theorem & Circumscribed Angles
Theorem: If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary (\(\angle A + \angle C = 180^\circ\) and \(\angle B + \angle D = 180^\circ\)).
Circumscribed: If an angle is circumscribed about a circle, then it is supplementary to the central angle that intercepts the same arc.
A C
Kendall Hunt Illustrative Mathematics • High School Geometry Unit 7 Complete • Page 7 of 7
