(3.1) Properties of Parallel Lines

What are we learning? Students will… 1. Identify angles formed by two lines and a transversal. 2. Proving and using properties of parallel lines. Evidence Outcome: Prove geometric theorems (lines, angles, triangles, parallelograms). Purpose (Relevancy): Do you think it is important for architects and builders to know if lines are parallel or perpendicular?

Identifying Angles Transversal: A line that intersects two coplanar lines at two distinct points. l a b m k c How many angles are formed by a transversal?

Identifying Angles Alternate Interior Angles: Nonadjacent interior angles that lie on opposite sides of the transversal. Same-Side Interior Angles: Angles that lie on the same side of the transversal between the two lines it intersects Corresponding Angles: Angles that lie on the same side of the transversal in corresponding positions relative to the two lines it intersects

Identifying Angles Alternate Interior Angles: and are alternate interior angles 5 1 6 3 4 2 7 8 Also: Same-Side Interior Angles: and are same-side interior angles (AKA co-interior angles) Also: Corresponding Angles: and are corresponding angles Also:

Properties of Parallel Lines 1 Note: Notation for parallel lines 2 m Postulate 3-1: Corresponding Angles Postulate: If a transversal intersects two parallel lines, then corresponding angles are congruent.

Properties of Parallel Lines Let’s say this angle is 72°… 72° 72° 108° 72° Alternate Interior Angles are congruent!!!

Properties of Parallel Lines b t 1 3 2 Theorem 3-1: Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then alternate interior angles are congruent.

Two-Column Proof

Proof of Alternate Interior Angles Theorem b t 4 1 3 2 Statements Reasons 1. 2. 3. 4.

Properties of Parallel Lines 72° 72° 108° 72° Same-Side Interior Angles are supplementary!!!

Properties of Parallel Lines 1 3 2 a b t Theorem 3-2: Same-Side Interior Angles Theorem If a transversal intersects two parallel lines, then same-side interior angles are supplementary.

Identifying Angles Alternate Exterior Angles: Nonadjacent exterior angles that lie on opposite sides of the transversal. Same-Side Exterior Angles: Angles that lie on the same side of the transversal outside of the two lines it intersects

Identifying Angles Alternate Exterior Angles: and are alternate exterior angles 6 5 Also: 1 3 4 2 Same-Side Exterior Angles: and are same-side exterior angles (AKA co-exterior angles) 7 8 Also:

Properties of Parallel Lines 108° 72° 72° 108° 72° 108° Alternate Exterior Angles are congruent!!!

Properties of Parallel Lines b 1 3 2 Theorem 3-3: Alternate Exterior Angles Theorem If a transversal intersects two parallel lines, then alternate exterior angles are congruent.

Proof of Alternate Exterior Angles Theorem 1 2 4 b 3 Statements Reasons 1. 2. 3. 4.

Properties of Parallel Lines 108° 72° 72° 108° 72° 108° Same-Side Exterior Angles are supplementary!!!

Properties of Parallel Lines b 1 3 2 Theorem 3-4: Same-Side Exterior Angles Theorem If a transversal intersects two parallel lines, then same-side exterior angles are supplementary.

Let’s Apply What We Have Learned? Find the values of x and y in the diagram below. Use a two-column proof to show your work. Under statements, write each step and under reasons, write the definition, property, postulate, or theorem that supports your ideas. The first statement should be the lines are parallel and the first reason should be given. x° y° 50° 70°

Let’s Apply What We Have Learned, K? Find the values of x and y in the diagram below. Use a two-column proof to show your work. Under statements, write each step and under reasons, write the definition, property, postulate, or theorem that supports your ideas. The first statement should be the lines are parallel and the first reason should be given. x° y° 66° 52°

HOMEWORK: (3.1) Pg. 131, #5-9 all, 11-16 all, 19-25 all TERMS: transversal, alternate interior (exterior) angles, same-side interior (exterior) angles, corresponding angles Thinking Page: Using a ruler, draw two parallel lines, cut by a transversal. Use the symbol for parallel and label two parallel lines c and d. Label the transversal as line t. 1. Put a star to show one set of corresponding angles. 2. Put a checkmark to show one set of same-side exterior angles. 3. Put a dot to show one set of vertical angles. 4. Put an arch to show alternate interior angles.