Name: _______________________ Period: ______
Proofs:

Draw a picture
o
add any auxiliary lines needed for your proof (you may not need any)
o
label important points
o
label important given information (≅, ║, ⊥, º)

Begin with given(s) – the hypothesis

Use Laws of Syllogism and Detachment along with:
o
Postulates
o
Theorems & Corollaries
o
Definitions
o
Properties

To reach your Conclusion

State the reason (postulate, theorem, definition, property) for each step
Note:

Be careful – do not use the conclusion as a reason in any proof.

Be careful - do not confuse the measure of the angle or length of a segment with the angle or
the segment – keep measures and congruence in separate steps.

Do not use the theorem number from the book as a reason in your proof – use the name of
the theorem or the shorthand given in the text.
The Basics:
Term
Name
Diagram
Point
Line
Plane
Segment
Ray
Angle

Postulate 1: Ruler Postulate – the points on a line can be put in 1-to-1 correspondence with ℝ.

Postulate 2: Segment Addition Postulate

Postulate 3: Protractor Postulate

Postulate 4: Angle Addition Postulate

Postulate 5: Through any 2 points there is exactly 1 line.

Postulate 6:

Postulate 7: If 2 lines intersect, they intersect in exactly 1 point.

Postulate 8: Through any 3 non-collinear points there is exactly one plane containing them.



Postulate 9:
Postulate 10: If 2 points lie in a plane, the line containing those points lies in the plane.
Postulate 11: If 2 planes intersect, they intersect in exactly 1 line.
Reasoning and Proof- Chap 2
Conjecture:
Hypothesis:
Conclusion:
Inductive Reasoning:
Deductive Reasoning:
Laws of Detachment and Syllogism are NOT to be used as a reason in a proof:

Law of Detachment:

Law of Syllogism:

Reflexive Property of Congruence:

Symmetric Property of Congruence:

Transitive Property of Congruence:

Linear Pair Theorem:

Congruent Supplements Theorem:

Right ∠ Congruence Theorem:

Congruent Complements Theorem:

Vertical ∠s Theorem:

If 2 ≅ ∠s are supplementary, then each ∠ is a right ∠.
Angles

Corresponding Angles Postulate – 2 ∥ lines cut by transversal ⟹ corresponding ∠s are ≅.

Alternate Interior Angles Theorem – 2 ∥ lines cut by transversal ⟹ alt int ∠s are ≅.

Alternate Exterior Angles Theorem – 2 ∥ lines cut by transversal ⟹ alt ext ∠s are ≅.

Same Side Interior Angles Theorem – 2 ∥ lines cut by transversal ⟹ same side int ∠s are supplementary.

Parallel Postulate – Through a point 𝒫 not on a line ℓ there is exactly 1 line ∥ to ℓ.

Converse of the Corresponding Angles Postulate

Converse of the Alternate Interior Angles Theorem

Converse of the Alternate Exterior Angles Theorem

Converse of the Same Side Interior Angles Theorem

2 intersecting lines form a linear pair of ≅ ∠s ⇒ lines ⊥.

Perpendicular Transversal Theorem

Coplanar lines ⊥ to the same line ⇒ 2 lines ∥.

Parallel Lines Theorem

Perpendicular Lines Theorem
Triangles

Triangle Sum Theorem
o
Acute angles of a right ∆ are complementary
o
Each angle of an equiangular ∆ measures 60º

Exterior Angle Theorem

Third Angles Theorem

SSS

SAS

ASA

AAS

HL

CPCTC

Isosceles Triangle Theorem

Converse of Isosceles Triangle Theorem
o
Equilateral → Equiangular
o
Equiangular → Equilateral

Perpendicular Bisector Theorem

Converse of the Perpendicular Bisector Theorem

Angle Bisector Theorem

Converse of the Angle Bisector Theorem.

Circumcenter Theorem

Incenter Theorem

Centroid Theorem

Triangle Midsegment Theorem

In ∆, larger ∠is opp. longer side.

In ∆, longer side is opp. larger ∠.

Triangle Inequality Theorem

Hinge Theorem

Converse of the Hinge Theorem

Converse of the Pythagorean Theorem

Pythagorean Inequalities Theorem

45º - 45º - 90º Triangle Theorem

30º - 60º - 90º Triangle Theorem
Polygons

Polygon ∠ Sum Theorem

Polygon Exterior ∠ Sum Theorem
Properties of Parallelograms:

⇒ opp. sides ≅

⇒ opp. ∠s ≅

⇒ consec. ∠s suppl.

⇒ diags. bisect each other
Conditions for Parallelograms (Converses)
Def:
Both pairs
opposite sides
One pair
opposites sides
Both pairs
opposite ∠s
Thm: Quad with opp. sides ≅ ⇒
Thm: Quad with pair of opp sides ∥ and ≅ ⇒
Thm: Quad with opposite ∠𝑠 ≅ ⇒
Thm: Quad with ∠ supp. to cons. ∠𝑠 ≅ ⇒
One ∠
Thm: Quad with diags. bisecting each other ⇒
Diagonals
Properties of Rectangles:

Rect. ⇒

Rect. ⇒ diags. ≅
Conditions for Rectangles (Converses)
Condition
Def:
Thm:
Thm:
with one rt. ∠ ⇒ rect.
with diags. ≅ ⇒ rect.
Example
Properties of Rhombuses:

Rhombus ⇒

Rhombus ⇒ diags. ⊥

Rhombus ⇒ each diag. bisects opp. ∠s
Conditions for Rhombuses (Converses)
Condition
Def:
Thm:
with one pair cons. sides ≅ ⟹ Rhombus
Thm:
with diags. ┴ ⟹ Rhombus
Thm:
with diag. bisecting opp. ∠s ⟹ Rhombus
Properties of Kites:

(def) Kite –

Kite ⇒ diags. ⊥

Kite ⇒ exactly one pair opp. ∠𝑠 ≅.
Example
Trapezoids:

Isosc. trap. ⇒ base ∠𝑠 ≅.

Trap. with pair base ≅ ∠𝑠 ⇒ isosc. trap.

Isosc. trap ⇔ diags. ≅

Trapezoid Midsegment Theorem:
Similarity

Cross Product Property

Properties of Proportions

Similar Polygons
Similar Polygons
Definition
Diagram

AA Similarity

SSS Similarity

SAS Similarity

Reflexive property of similarity

Symmetric property of similarity

Transitive property of similarity

Triangle Proportionality Theorem
Statements

Converse of the Triangle Proportionality Theorem

Two Transversal Proportionality Corollary

Triangle Angle Bisector Theorem

Proportional Perimeters and Areas Theorem
Circles

Tangent to a circle

Tangent segments

Minor arcs are congruent

If one chord is a perpendicular bisector

If a diameter of a circle is perpendicular to a chord

In the same circle, or in congruent circles, two chords are congruent

Measure of an Inscribed Angle

If two inscribed angles of a circle intercept the same arc

If a right triangle is inscribed in a circle

A quadrilateral can be inscribed in a circle

If a tangent and a chord intersect at a point on the circle

Angles Inside the Circle

Angles Outside the Circle

Segments of Chords Theorem

Segments of Secants Theorems

Segments of Secants and Tangents Theorem
DEFINITIONS TO KNOW:
Angle(s)
Interior
Exterior
Vertex
Degree
Acute
Right
Obtuse
Angle Pairs:
Adjacent
Straight
Complementary
Supplementary
Linear Pair
Vertical
Corresponding
Alternate Interior
Alternate Exterior
Bisect/Bisector
Circles
Arcs
Congruent
Intercepted
Minor
Major
Center
Central Angle
Chord
Inscribed Angle
Secant
Semicircle
Standard Equation of a Circle
Tangent
Collinear
Conclusion
Condition
Conditional Statements
Biconditional
Converse
Inverse
Contrapositive
Congruent
Angles
Polygons
Segments
Conjecture
Contrapositive
Converse
Coordinate
Coplanar
Corollary
Definition
Diagonal
Distance/length
Endpoint
Hypothesis
Inverse
Kite
Line Pairs
Parallel
Intersecting
Skew
Midpoint
Polygon
Postulate
Property
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Trapezoid
Kite
Similar
Theorem
Trapezoid
Base
Legs
Base angles
Triangles
Isosceles
Scalene
Equilateral
Equiangular
Acute, Obtuse, Right