Bellringer 6 2 1 Parallel Lines MAFS 912
Bellringer #6
2 -1: Parallel Lines MAFS. 912. G-CO. 1. 1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MAFS. 912. G-CO. 3. 9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems.
Identify the pairs of angles of each angle type made by the motorcycle tracks • Corresponding Angles • Alternate Exterior Angles • Alternate Interior Angles • Same-Side Interior Angles
Which angle pairs include… ∠ 4? ∠ 7?
Same-Side Interior Angles Postulate (Explore in Realize) • If a transversal intersects two parallel lines, then same-side interior angles are supplementary.
How can you express each of the numbered angles in terms of x°? • m∠ 2 = • m∠ 4 = • m∠ 6 = • m∠ 8 =
If m∠ 4 = 118°, what is the measure if each of the other angles? • m∠ 1 = • m∠ 5 = • m∠ 2 = • m∠ 6 = • m∠ 3 = • m∠ 7 = • m∠ 4 = 118° • m∠ 8 =
Alternate Interior Angles Theorem (Explore in Realize) • If a transversal intersects two parallel lines, then alternate interior angles are congruent.
Corresponding Angles Theorem • If a transversal intersects two parallel lines, then corresponding angles are congruent.
Alternate Exterior Angles Theorem • If a transversal intersects two parallel lines, then alternate exterior angles are congruent.
Statements • • 1. 2. 3. 4. 5. 6. 7. Reasons • • 1. 2. 3. 4. 5. 6. 7.
• What should be the m∠ 1 and m∠ 3 be? If m∠EJF = 56, find m∠FHK
Bellringer #7
2 -2: Proving Lines Parallel MAFS. 912. G-CO. 3. 9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems.
Suppose two lines are not parallel. • Can corresponding angles still be congruent? • Could ∠ 3 be supplementary to 120° angle?
State theorem for which the converse can be used to show that m ∥ n if m∠ 3 = m∠ 6
Converse of the Same-Side Interior Angles Postulate (Explore in Realize) • If two lines and a transversal form same-side interior angles that are supplementary, then the lines are parallel.
Converse of the Alternate Exterior Angles Theorem (Explore in Realize) • If two lines and a transversal form alternate exterior angles that are congruent, then the lines are parallel.
Determine Whether Lines Are Parallel • The edges of a new sidewalk must be parallel in order to meet accessibility requirements. Concrete is poured between straight strings. How does an inspector know that the edges of the sidewalk are parallel? • What is ��∠�� ? What should ∠�� measure in order to guarantee that the sidewalk is parallel to Main Street? Explain.
When building a gate, how does Bailey know that the vertical boards v and w are parallel? • What should ∠ 1 measure to ensure board b is parallel to board a? • Bailey also needs board c to be parallel to board a. What should ∠�� measure? Explain. • Is ��∥�� ? Explain.
Bellringer #8
2 -3: Parallel Lines and Triangle Angle Sums MAFS. 912. G-CO. 3. 10 Prove theorems about triangles; use theorems about triangles to solve problems.
Triangle Angle-Sum Theorem • The sum of the measures of all the angles of a triangle is 180.
Prove the Triangle Angle-Sum Theorem •
What are the values of x and y?
Triangle Exterior Angle Theorem • The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.
What is the missing angle measure in the figure?
What is the missing angle measure in the figure?
Applying the Triangle Theorems • Cheyenne built this display for her ornament collection. Each shelf is parallel to the base. She recalls only the angle measures shown in the diagram. Now she wants to build another just like it. What are the measures of ∠�� , and ∠�� ? • Can you find the measures of ∠ 4, and ∠ 5?
2 -4: Slopes of Parallel and Perpendicular Lines MAFS. 912. G-GPE. 2. 5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e. g. , find the equation of a line parallel or perpendicular to a given line that passes through a given point).
Relationships of lines •
Checking Parallelism • Are lines k and n parallel? • Are lines m and q parallel?
Determine if line m is parallel to line n.
Checking Perpendicularity • Are lines j and k perpendicular? • Are lines h and ℓ perpendicular? • Are lines k and m perpendicular?
Writing Equations of Parallel and Perpendicular Lines • What is an equation of the line through P that is parallel to ℓ? • What is the equation of the line through P that is perpendicular to ℓ?
What are equations of lines parallel and perpendicular to the given line k passing through point T? • y = -3 x + 2; T(3, 1) • 3 x – 4 y = 20; T(12, -2) • y + 2 = 4(x – 1); T(1, -3)
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