Introduction Think about crossing a pair of chopsticks and the angles that are created when they are opened at various positions. How many angles are formed? What are the relationships among those angles? This lesson explores angle relationships. We will be examining the relationships of angles that lie in the same plane. A plane is a two-dimensional figure, meaning it is a flat surface, and it extends infinitely in all directions. Planes require at least three non-collinear points. Planes are named using those points or a capital script letter. Since they are flat, planes have no depth : Proving the Vertical Angles Theorem

Key Concepts Angles can be labeled with one point at the vertex, three points with the vertex point in the middle, or with numbers. See the examples that follow : Proving the Vertical Angles Theorem

Key Concepts, continued Straight angles are angles with rays in opposite directions—in other words, straight angles are straight lines : Proving the Vertical Angles Theorem Straight angleNot 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.

Key Concepts, continued Adjacent angles are angles that lie in the same plane and share a vertex and a common side. They have no common interior points : Proving the Vertical Angles Theorem Adjacent angles ∠ ABC is adjacent to ∠ CBD. They share vertex B and. ∠ ABC and ∠ CBD have no common interior points.

Key Concepts, continued Nonadjacent angles have no common vertex or common side, or have shared interior points. (continued) : Proving the Vertical Angles Theorem Nonadjacent angles ∠ ABE is not adjacent to ∠ FCD. They do not have a common vertex.

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

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

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

Key Concepts, continued Vertical angles are nonadjacent angles formed by two pairs of opposite rays : Proving the Vertical Angles Theorem Theorem Vertical Angles Theorem Vertical angles are congruent.

Key Concepts, continued : Proving the Vertical Angles Theorem Vertical angles ∠ ABC and ∠ EBD are vertical angles. ∠ ABE and ∠ CBD are vertical angles.

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

Key Concepts, continued : Proving the Vertical Angles Theorem 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.

Key Concepts, continued Informally, 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 : Proving the Vertical Angles Theorem

Key Concepts, continued In the diagram below, the angles form a linear pair. m ∠ ABD + m ∠ DBC = : Proving the Vertical Angles Theorem

Key Concepts, continued The next diagram shows a pair of supplementary angles that are nonadjacent. m ∠ PQR + m ∠ TUV = : Proving the Vertical Angles Theorem

Key Concepts, continued : Proving the Vertical Angles Theorem Theorem Supplement Theorem If two angles form a linear pair, then they are supplementary.

Key Concepts, continued Angles have the same congruence properties that segments do : Proving the Vertical Angles Theorem Theorem Congruence of angles is reflexive, symmetric, and transitive. Reflexive Property: Symmetric Property: If, then. Transitive Property: If and, then.

Key Concepts, continued : Proving the Vertical Angles Theorem Theorem Angles supplementary to the same angle or to congruent angles are congruent. If and, then.

Key Concepts, continued Perpendicular lines form four adjacent and congruent right angles : Proving the Vertical Angles Theorem 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.

Key Concepts, continued The symbol for indicating perpendicular lines in a diagram is a box at one of the right angles, as shown below : Proving the Vertical Angles Theorem

Key Concepts, continued The symbol for writing perpendicular lines is, and is read as “is perpendicular to.” In the diagram,. 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 : Proving the Vertical Angles Theorem

Key Concepts, continued (continued) : Proving the Vertical Angles Theorem 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.

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

Key Concepts, continued Complementary angles are two angles whose sum is 90º. Complementary angles can form a right angle or be nonadjacent. The following diagram shows a pair of nonadjacent complementary angles : Proving the Vertical Angles Theorem

Key Concepts, continued : Proving the Vertical Angles Theorem

Key Concepts, continued The diagram at right shows a pair of adjacent complementary angles labeled with numbers : Proving the Vertical Angles Theorem

Key Concepts, continued : Proving the Vertical Angles Theorem 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.

Guided Practice Example 1 Prove that vertical angles are congruent given a pair of intersecting lines, and : Proving the Vertical Angles Theorem

Supplementary angles add up to 180º : Proving the Vertical Angles Theorem m ∠ 1 + m ∠ 2 = m ∠ 2 + m ∠ 3 m ∠ 1 = m ∠ 3.

Guided Practice Example 2 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 ? : Proving the Vertical Angles Theorem

WORK: : Proving the Vertical Angles Theorem

Introduction Think about all the angles formed by parallel lines intersected by a transversal. What are the relationships among those angles? In this lesson, we will prove those angle relationships. First, look at the diagram on the next slide of a pair of parallel lines and notice the interior angles versus the exterior angles : Proving Theorems About Angles in Parallel Lines Cut by a Transversal

Introduction, continued The interior angles lie between the parallel lines and the exterior angles lie outside the pair of parallel lines. In the following diagram, line k is the transversal. A transversal is a line that intersects a system of two or more lines. Lines l and m are parallel. The exterior angles are ∠ 1, ∠ 2, ∠ 7, and ∠ 8. The interior angles are ∠ 3, ∠ 4, ∠ 5, and ∠ : Proving Theorems About Angles in Parallel Lines Cut by a Transversal

Key Concepts, continued : Proving Theorems About Angles in Parallel Lines Cut by a Transversal Postulate Corresponding Angles Postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent. Corresponding angles: The converse is also true. If corresponding angles of lines that are intersected by a transversal are congruent, then the lines are parallel.

Key Concepts, continued : Proving Theorems About Angles in Parallel Lines Cut by a Transversal Theorem Alternate Interior Angles Theorem If two parallel lines are intersected by a transversal, then alternate interior angles are congruent. Alternate interior angles: The converse is also true. If alternate interior angles of lines that are intersected by a transversal are congruent, then the lines are parallel.

Key Concepts, continued : Proving Theorems About Angles in Parallel Lines Cut by a Transversal Theorem Same-Side Interior Angles Theorem If two parallel lines are intersected by a transversal, then same-side interior angles are supplementary. Same-side interior angles: The converse is also true. If same-side interior angles of lines that are intersected by a transversal are supplementary, then the lines are parallel.

Key Concepts, continued : Proving Theorems About Angles in Parallel Lines Cut by a Transversal Theorem Alternate Exterior Angles Theorem If parallel lines are intersected by a transversal, then alternate exterior angles are congruent. Alternate exterior angles: The converse is also true. If alternate exterior angles of lines that are intersected by a transversal are congruent, then the lines are parallel.

Key Concepts, continued : Proving Theorems About Angles in Parallel Lines Cut by a Transversal Theorem Same-Side Exterior Angles Theorem If two parallel lines are intersected by a transversal, then same-side exterior angles are supplementary. Same-side exterior angles: The converse is also true. If same-side exterior angles of lines that are intersected by a transversal are supplementary, then the lines are parallel.

Key Concepts, continued : Proving Theorems About Angles in Parallel Lines Cut by a Transversal Theorem Perpendicular Transversal Theorem If a line is perpendicular to one line that is parallel to another, then the line is perpendicular to the second parallel line. The converse is also true. If a line intersects two lines and is perpendicular to both lines, then the two lines are parallel.

Guided Practice Example 3 In the diagram, and. If,, and, find the measures of the unknown angles and the values of x and y : Proving Theorems About Angles in Parallel Lines Cut by a Transversal

WORK: : Proving the Vertical Angles Theorem

Guided Practice Example 4 In the diagram,. If m ∠ 1 = 35 and m ∠ 2 = 65, find m ∠ EQF : Proving Theorems About Angles in Parallel Lines Cut by a Transversal

Guided Practice: Example 4, continued 1.Draw a third parallel line that passes through point Q. Label a second point on the line as P : Proving Theorems About Angles in Parallel Lines Cut by a Transversal

WORK: : Proving Theorems About Angles in Parallel Lines Cut by a Transversal