Proving the Vertical Angles Theorem

Proving the Vertical Angles TheoremAdapted from Walch Education

Key Concepts1.8.1: Proving the Vertical Angles Theorem2Angles 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.

Straight Angle1.8.1: Proving the Vertical Angles Theorem3Straight angles are straight lines.Straight angleNot a straight angleBCD 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.

Adjacent Angles1.8.1: Proving the Vertical Angles Theorem4Adjacent 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 Theorem5Adjacent anglesABC is adjacent to CBD. They share vertex B and .

ABC and CBD have no common interior points.

1.8.1: Proving the Vertical Angles Theorem6Nonadjacent anglesABE is not adjacent to FCD. They do not have a common vertex.

1.8.1: Proving the Vertical Angles Theorem7Nonadjacent anglesPQS is not adjacent to PQR. They share common interior points within PQS.

Linear pair1.8.1: Proving the Vertical Angles Theorem8Linear pairs are pairs of adjacent angles whose non-shared sides form a straight angleLinear pairABC 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 Theorem9Not a linear pairABE and FCD are not a linear pair. They are not adjacent angles.

Vertical Angles1.8.1: Proving the Vertical Angles Theorem10Vertical angles are nonadjacent angles formed by two pairs of opposite rays.

TheoremVertical Angles TheoremVertical angles are congruent.1.8.1: Proving the Vertical Angles Theorem11Vertical anglesABC and EBD are vertical angles.

ABE and CBD are vertical angles.

1.8.1: Proving the Vertical Angles Theorem12Not vertical anglesABC and EBD are not vertical angles. and are not opposite rays. They do not form one straight line.

Angle Addition Postulate1.8.1: Proving the Vertical Angles Theorem13PostulateAngle Addition PostulateIf D is in the interior of ABC, then mABD + mDBC = mABC.

If mABD + mDBC = mABC, then D is in the interior of ABC.

Key Concepts1.8.1: Proving the Vertical Angles Theorem14The 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 180. Supplementary angles can form a linear pair or be nonadjacent.Supplementary Angles1.8.1: Proving the Vertical Angles Theorem15In the diagram below, the angles form a linear pair.

mABD + mDBC = 180

Supplementary Angles1.8.1: Proving the Vertical Angles Theorem16A pair of supplementary angles that are nonadjacent.

mPQR + mTUV = 180

1.8.1: Proving the Vertical Angles Theorem17TheoremSupplement TheoremIf two angles form a linear pair, then they are supplementary.TheoremAngles supplementary to the same angle or to congruent angles are congruent.If and , then .

Properties of Congruence1.8.1: Proving the Vertical Angles Theorem18TheoremCongruence of angles is reflexive, symmetric, and transitive.Reflexive Property: Symmetric Property: If , then .Transitive Property: If and , then .

Perpendicular Lines1.8.1: Proving the Vertical Angles Theorem19Perpendicular lines form four adjacent and congruent right angles.TheoremIf 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.Perpendicular Lines1.8.1: Proving the Vertical Angles Theorem20The 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.

Concepts, continued1.8.1: Proving the Vertical Angles Theorem21Perpendicular 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 Theorem22TheoremPerpendicular Bisector TheoremIf 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 Theorem23TheoremIf is the perpendicular bisector of , then DA = DC. If DA = DC, then is the perpendicular bisector of .

Complementary Angles1.8.1: Proving the Vertical Angles Theorem24Complementary angles are two angles whose sum is 90. Complementary angles can form a right angle or be nonadjacent.

Complementary Angles1.8.1: Proving the Vertical Angles Theorem25

Complementary Angles1.8.1: Proving the Vertical Angles Theorem26The diagram at right shows a pair of adjacent complementary angles labeled with numbers.

1.8.1: Proving the Vertical Angles Theorem27TheoremComplement TheoremIf 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.Practice1.8.1: Proving the Vertical Angles Theorem28Prove that vertical angles are congruent given a pair of intersecting lines, and .

Draw a Diagram1.8.1: Proving the Vertical Angles Theorem29

Supplement Theorem1.8.1: Proving the Vertical Angles Theorem30Supplementary angles add up to 180.

Use Substitution1.8.1: Proving the Vertical Angles Theorem31Both expressions are equal to 180, so they are equal to each other. Rewrite the first equation, substituting m2 + m3 in for 180.

m1 + m2 = m2 + m3Reflexive Property1.8.1: Proving the Vertical Angles Theorem32m2 = m2Subtraction Property1.8.1: Proving the Vertical Angles Theorem33Since m2 = m2, these measures can be subtracted out of the equation m1 + m2 = m2 + m3.

This leaves m1 = m3.Definition of Congruence1.8.1: Proving the Vertical Angles Theorem34Since m1 = m3, 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.

Try this one1.8.1: Proving the Vertical Angles Theorem35In 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?

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