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**Proving the Vertical Angles Theorem**

Adapted from Walch Education

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Key Concepts Angles can be labeled with one point at the vertex, three points with the vertex point in the middle, or with numbers. If the vertex point serves as the vertex for more than one angle, three points or a number must be used to name the angle. 1.8.1: Proving the Vertical Angles Theorem

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**Straight Angle Straight angles are straight lines. Straight angle**

Not a straight angle ∠BCD is a straight angle. Points B, C, and D lie on the same line. ∠PQR is not a straight angle. Points P, Q, and R do not lie on the same line. 1.8.1: Proving the Vertical Angles Theorem

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Adjacent Angles Adjacent angles are angles that lie in the same plane and share a vertex and a common side. They have no common interior points. Nonadjacent angles have no common vertex or common side, or have shared interior points. 1.8.1: Proving the Vertical Angles Theorem

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**∠ABC is adjacent to ∠CBD. They share vertex B and .**

Adjacent angles ∠ABC is adjacent to ∠CBD. They share vertex B and ∠ABC and ∠CBD have no common interior points. 1.8.1: Proving the Vertical Angles Theorem

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**∠ABE is not adjacent to ∠FCD. They do not have a common vertex.**

Nonadjacent angles ∠ABE is not adjacent to ∠FCD. They do not have a common vertex. 1.8.1: Proving the Vertical Angles Theorem

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Nonadjacent angles ∠PQS is not adjacent to ∠PQR. They share common interior points within ∠PQS. 1.8.1: Proving the Vertical Angles Theorem

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Linear pair Linear pairs are pairs of adjacent angles whose non- shared sides form a straight angle Linear pair ∠ABC and ∠CBD are a linear pair. They are adjacent angles with non-shared sides, creating a straight angle. 1.8.1: Proving the Vertical Angles Theorem

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**a linear pair. They are not adjacent angles.**

Not a linear pair ∠ABE and ∠FCD are not a linear pair. They are not adjacent angles. 1.8.1: Proving the Vertical Angles Theorem

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Vertical Angles Vertical angles are nonadjacent angles formed by two pairs of opposite rays. Theorem Vertical Angles Theorem Vertical angles are congruent. 1.8.1: Proving the Vertical Angles Theorem

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**∠ABC and ∠EBD are vertical angles.**

∠ABE and ∠CBD are vertical angles. 1.8.1: Proving the Vertical Angles Theorem

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Not vertical angles ∠ABC and ∠EBD are not vertical angles and are not opposite rays. They do not form one straight line. 1.8.1: Proving the Vertical Angles Theorem

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**Angle Addition Postulate**

If D is in the interior of ∠ABC, then m∠ABD + m∠DBC = m∠ABC. If m∠ABD + m∠DBC = m∠ABC, then D is in the interior of ∠ABC. 1.8.1: Proving the Vertical Angles Theorem

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Key Concepts The Angle Addition Postulate means that the measure of the larger angle is made up of the sum of the two smaller angles inside it. Supplementary angles are two angles whose sum is 180º. Supplementary angles can form a linear pair or be nonadjacent. 1.8.1: Proving the Vertical Angles Theorem

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Supplementary Angles In the diagram below, the angles form a linear pair. m∠ABD + m∠DBC = 180 1.8.1: Proving the Vertical Angles Theorem

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Supplementary Angles A pair of supplementary angles that are nonadjacent. m∠PQR + m∠TUV = 180 1.8.1: Proving the Vertical Angles Theorem

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

Theorem Supplement Theorem If two angles form a linear pair, then they are supplementary. Theorem Angles supplementary to the same angle or to congruent angles are congruent. If and , then 1.8.1: Proving the Vertical Angles Theorem

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**Properties of Congruence**

Theorem Congruence of angles is reflexive, symmetric, and transitive. Reflexive Property: Symmetric Property: If , then Transitive Property: If and , then 1.8.1: Proving the Vertical Angles Theorem

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Perpendicular Lines Perpendicular lines form four adjacent and congruent right angles. Theorem If two congruent angles form a linear pair, then they are right angles. If two angles are congruent and supplementary, then each angle is a right angle. 1.8.1: Proving the Vertical Angles Theorem

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Perpendicular Lines The symbol for writing perpendicular lines is , and is read as “is perpendicular to.” Rays and segments can also be perpendicular. In a pair of perpendicular lines, rays, or segments, only one right angle box is needed to indicate perpendicular lines. 1.8.1: Proving the Vertical Angles Theorem

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Concepts, continued Perpendicular bisectors are lines that intersect a segment at its midpoint at a right angle; they are perpendicular to the segment. Any point along the perpendicular bisector is equidistant from the endpoints of the segment that it bisects. 1.8.1: Proving the Vertical Angles Theorem

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**Perpendicular Bisector Theorem **

If a point lies on the perpendicular bisector of a segment, then that point is equidistant from the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment. 1.8.1: Proving the Vertical Angles Theorem

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**If is the perpendicular bisector of , then DA = DC. **

Theorem If is the perpendicular bisector of , then DA = DC. If DA = DC, then is the perpendicular bisector of 1.8.1: Proving the Vertical Angles Theorem

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Complementary Angles Complementary angles are two angles whose sum is 90º. Complementary angles can form a right angle or be nonadjacent. 1.8.1: Proving the Vertical Angles Theorem

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Complementary Angles 1.8.1: Proving the Vertical Angles Theorem

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Complementary Angles The diagram at right shows a pair of adjacent complementary angles labeled with numbers. 1.8.1: Proving the Vertical Angles Theorem

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**Theorem Complement Theorem**

If the non-shared sides of two adjacent angles form a right angle, then the angles are complementary. Angles complementary to the same angle or to congruent angles are congruent. 1.8.1: Proving the Vertical Angles Theorem

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Practice Prove that vertical angles are congruent given a pair of intersecting lines, and 1.8.1: Proving the Vertical Angles Theorem

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Draw a Diagram 1.8.1: Proving the Vertical Angles Theorem

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**Supplement Theorem Supplementary angles add up to 180º.**

1.8.1: Proving the Vertical Angles Theorem

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Use Substitution Both expressions are equal to 180, so they are equal to each other. Rewrite the first equation, substituting m∠2 + m∠3 in for 180. m∠1 + m∠2 = m∠2 + m∠3 1.8.1: Proving the Vertical Angles Theorem

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**Reflexive Property m∠2 = m∠2**

1.8.1: Proving the Vertical Angles Theorem

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Subtraction Property Since m∠2 = m∠2, these measures can be subtracted out of the equation m∠1 + m∠2 = m∠2 + m∠3. This leaves m∠1 = m∠3. 1.8.1: Proving the Vertical Angles Theorem

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**Definition of Congruence**

Since m∠1 = m∠3, by the definition of congruence, ∠1 and ∠3 are vertical angles and they are congruent. This proof also shows that angles supplementary to the same angle are congruent. 1.8.1: Proving the Vertical Angles Theorem

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Try this one… In the diagram at right, is the perpendicular bisector of If AD = 4x – 1 and DC = x + 11, what are the values of AD and DC ? 1.8.1: Proving the Vertical Angles Theorem

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Thanks For Watching! Ms. Dambreville

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