PROVING THE VERTICAL ANGLES THEOREM. Adapted from Walch Education

Transcription

1 PROVING THE VERTICAL ANGLES THEOREM Adapted from Walch Education

2 Key Concepts 2 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.

3 Straight Angle 3 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.

4 Adjacent Angles 4 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.

5 5 ABC is adjacent to CBD. They share vertex B and. ABC and CBD have no common interior points. Adjacent angles

6 6 Nonadjacent angles ABE is not adjacent to FCD. They do not have a common vertex.

7 7 Nonadjacent angles PQS is not adjacent to PQR. They share common interior points within PQS.

8 Linear pair 8 Linear pairs are pairs of adjacent angles whose nonshared 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.

9 9 Not a linear pair ABE and FCD are not a linear pair. They are not adjacent angles.

10 Vertical Angles 10 Vertical angles are nonadjacent angles formed by two pairs of opposite rays. Theorem Vertical Angles Theorem Vertical angles are congruent.

11 11 ABC and EBD are vertical angles. ABE and CBD are vertical angles. Vertical angles ÐEBD ÐCBD

12 12 Not vertical angles ABC and EBD are not vertical angles. and are not opposite rays. They do not form one straight line.

13 Angle Addition Postulate 13 Postulate 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.

14 Key Concepts 14 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.

15 Supplementary Angles 15 In the diagram below, the angles form a linear pair. m ABD + m DBC = 180

16 Supplementary Angles 16 A pair of supplementary angles that are nonadjacent. m PQR + m TUV = 180

17 17 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.

18 Properties of Congruence 18 Theorem Congruence of angles is reflexive, symmetric, and transitive. Reflexive Property: Ð1 Symmetric Property: If Ð2, then Ð1. Transitive Property: If Ð1@ Ð2 and Ð3, then Ð1@ Ð3.

19 Perpendicular Lines 19 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.

20 Perpendicular Lines 20 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.

21 Concepts, continued 21 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.

22 22 Theorem 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.

23 23 Theorem If is the perpendicular bisector of AC, then DA = DC. If DA = DC, then is the perpendicular bisector of AC.

24 Complementary Angles 24 Complementary angles are two angles whose sum is 90º. Complementary angles can form a right angle or be nonadjacent.

25 Complementary Angles 25 mðb + mðe = 90

26 Complementary Angles 26 The diagram at right shows a pair of adjacent complementary angles labeled with numbers. mð1+ mð2 = 90

27 27 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.

28 Practice 28 Prove that vertical angles are congruent given a pair of intersecting lines, and.

29 Draw a Diagram 29

30 Supplement Theorem 30 Supplementary angles add up to 180º. mð1+ mð2 = 180 mð2 + mð3 = 180

31 Use Substitution 31 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

32 Reflexive Property 32 m 2 = m 2

33 Subtraction Property 33 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.

34 Definition of Congruence 34 Since m 1 = m 3, by the definition of congruence, Ð1@ Ð3. 1 and 3 are vertical angles and they are congruent. This proof also shows that angles supplementary to the same angle are congruent.

35 Try this one 35 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?

36 Thanks For Watching! Dr. Dambreville