Geometry Guide Basic Terms and Definitions Term Point Line Plane Space Collinear Coplanar Segment Ray Opposite rays Intersection Congruent Congruent segments Segment midpoint Segment bisector Angle Acute angle Obtuse angle Right angle Straight angle Congruent angles Angle bisector Adjacent angles

Vertical angles

Definition A location or position. A point has no dimension. A point has no length, width, or height. An infinite set of points that extend in two opposite directions. A line has length (infinite), but no width, or thickness. An infinite set of points that form a flat surface extending in all directions. A plane has length and width (infinite), but no thickness. The set of all points. Collinear points lie on the same line. Noncollinear points do not lie on the same line. Coplanar points lie on the same plane. Noncoplanar points do not lie on the same plane. A section of a line designated by two endpoints and the set of all points between them. A section of a line with one endpoint and extending in one direction. Two rays with the same endpoint that form a straight line. The set of points in both objects. Two objects are congruent if they have the same size and shape. Two segments are congruent if they have the same length. A midpoint divides a segment into two ≅ segments. A line, ray, or segment that intersects a segment at its midpoint. A figure formed by two rays with the same endpoint. The endpoint is the vertex. The rays form the sides of the angle. ∠A is acute if 0 < m∠A < 90 ∠A is obtuse if 90 < m∠A < 180 ∠A is a right angle if m∠A = 90 ∠A is a straight angle if m∠A = 180 Two angles are congruent if they have equal measure. A ray that divides an angle into two congruent angles. Two angles are adjacent if they have: 1. Common vertex 2. Common side 3. No points in common The two angles “across” from each other at the intersection of lines.

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Basic Postulates and Theorems about Points, Lines, and Planes: • • •

A line contains at least 2 points. (P.5) A plane contains at least 3 noncollinear points. (P.5) Space contains at least 4 points not all in one plane. (P.5)

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Through any 2 points there is exactly one line. (P.6) Through any 3 noncollinear points there is exactly one plane. (P.7)

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If 2 points are in a plane, then the line formed by the points is in the plane. (P.8)

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If 2 planes intersect, then their intersection is a line. (P.9) The intersection of two planes is a line.

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If 2 lines intersect, then they intersect in one point. (1.1) The intersection of two lines is a point.

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If 2 lines intersect, then one plane contains the lines. (1.3)

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Through a line and a point not on the line there is exactly one plane. (1.2)

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Segment Concepts: • •

A

C

SAP: Segment Addition Postulate: (P.2) AC + CB = AB “The sum of the parts equals the whole.” Midpoint Theorem: (2.1)/Definition of Midpoint: If M is midpoint of AB, then it divides AB into 2 ≅ segments. If M is midpoint of AB, AM ≅ MB. If AM ≅ MB, then M is midpoint of AB.

M

B

If M is midpoint of AB, then, AM = ½ AB and MB = ½ AB. If AM = ½ AB or MB = ½ AB, then M is midpoint of AB.

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Angle Concepts: • •

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AAP: Angle Addition Postulate: (P.4) m∠ABX + m∠XBC = m∠ABC. B “The sum of the parts equals the whole.” C Bisector Theorem: (2.2)/Definition of Bisector: If BX bisects ∠ABC, then ∠ABX ≅ ∠XBC. If ∠ABX ≅ ∠XBC, then BX bisects ∠ABC. If BX bisects ∠ABC, then m∠ABX = ½ m∠ABC and m∠XBC = ½ m∠ABC. If m∠ABX = ½ m∠ABC or m∠XBC = ½ m∠ABC, then BX bisects ∠ABC. Definition of Supplementary ∠s: If ∠1 and ∠2 supplementary, then m∠1 + m∠2 = 180. If m∠1 + m∠2 = 180, then ∠1 and ∠2 supplementary. Definition of Complementary ∠s: If ∠3 and ∠4 complementary, then m∠3 + m∠4 = 90. If m∠3 + m∠4 = 90, then ∠3 and ∠4 complementary.

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Vertical angles are ≅. (2.3)

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If 2 ∠s are supplements to ≅ angles (or to the same angle), then the 2 ∠s are ≅. (2.7) If 2 ∠s are complements to ≅ angles (or to the same angle), then the 2 ∠s are ≅. (2.8)

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Properties from Algebra: •

Addition/subtraction property of equality. If a = b , then a + c = b + c . (add the same thing to both sides). If a = b and c = d , then a + c = b + d . (add equals to both sides).

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Multiplication/division property of equality. If a = b , then ac = bc . (multiply both sides by the same/equal thing.) a b If a = b , then = (divide both sides by the same/equal thing.) c c

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Distributive property. a(b + c) = ab + ac

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Substitution If a + b = c and a = d , then d + b = c . (replace d for a) If a + b = z and x + y = z , then a + b = x + y . (two expressions equal to same thing)

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Transitive If a = b and b = c , then a = c . (for equality) If a ≅ b and b ≅ c , then a ≅ c . (for congruence)

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Reflexive a=a

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Symmetric a = b and b = a .

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Perpendicular Concepts: •

Definition of ⊥ lines: If 2 lines are ⊥, then they form right ∠s (90 degree ∠s). If 2 lines form right ∠s (90 degree ∠s), then they are ⊥.

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If two lines ⊥, then they form ≅ adjacent ∠s. (2.4) If 2 lines form ≅ adjacent ∠s, then they are ⊥. (2.5)

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If the ext. sides of 2 adjacent acute ∠s are ⊥, then the ∠s are complementary. (2.6) Two adjacent angles are complementary if their exterior sides are ⊥.

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Parallel Concepts: •

Definition of parallel lines: 2 coplanar lines that do not intersect (railroad tracks).

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Properties of parallel lines: If 2 lines ||, then … 1. Corresponding ∠s ≅. (P.10) 2. Alternate interior ∠s ≅. (3.2) 3. Same-side interior ∠s are supplementary. (3.3) 4. If one || line is ⊥ to the transversal, then other line is also ⊥ to the transversal. (3.4)

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Proving lines parallel: 1. If corresponding ∠s ≅, then lines ||. (P.11) 2. If alternate interior ∠s ≅, then lines ||. (3.5) 3. If same side interior ∠s supplementary, then lines ||. (3.6) 4. If 2 lines ⊥ to the same line, then lines ||. (3.7) 5. If 2 lines || to the same line, then lines ||. (3.10)

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Through a point outside a line, there is exactly one line || to the line. (3.8)

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Through a point outside a line, there is exactly one line ⊥ to the line. (3.9)

Basic Terms and Definitions Term Parallel lines Transversal Corresponding angles Alternate interior angles Same side interior angles Skew lines

Definition 2 coplanar lines that do not intersect (railroad tracks). A line that intersects 2 parallel lines. 2 angles that have the same relative position to || lines. 2 angles on alternate sides of the transversal and interior to the || lines. 2 angles on the same side of the transversal and interior to the || lines. 2 non-coplanar lines that do not intersect.

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Triangle Concepts: Basic Terms and Definitions Term Triangle Vertex Sides Exterior ∠ of a Δ

Definition Figure formed by 3 segments joining 3 noncollinear points. 3 points of a triangle. 3 line segments of a triangle. Angle formed when side of a triangle is extended.

Scalene Δ Isosceles Δ Equilateral Δ

Triangle with 3 different length sides. Triangle with at least 2 sides congruent. Triangle with 3 sides congruent.

Acute Δ Obtuse Δ Equiangular Δ

Triangle with 3 acute angles (all < 90). Triangle with one obtuse angle (>90 and