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Ch. 3 – Parallel and Perpendicular Lines Section 3.1 – Lines and Angles 1. I CAN identify relationships between figures in space. 2. I CAN identify angles formed by two lines and a transversal. Key Vocabulary: Parallel Lines Skew Lines Parallel Planes Transversal Alternate Interior Angles Same-Side Interior angles Corresponding Angles Alternate Exterior Angles Section 3.1 – Lines and Angles Section 3.1 – Lines and Angles In the solve it, you used relationships among planes in space to write the instructions. In Chapter 1, you learned about intersecting lines and planes. In this section, we will explore relationships of nonintersecting lines and planes. Section 3.1 – Lines and Angles Not all lines and not all planes intersect. Segments and rays can also be parallel or skew. They are parallel if they lie in parallel lines and skew if they lie in skew lines. Section 3.1 – Lines and Angles Problem 1 In the figure, assume that lines and planes that appear to be parallel are parallel. A) Which segments are parallel to ? B) Which segments are skew to ? C) What are two pairs of parallel lines? D) What are two segments parallel to plane BCGF? Section 3.1 – Lines and Angles A Transversal is a line that intersects two or more coplanar lines at distinct points. The diagram below shows the 8 angles formed by a transversal t and two lines l and m. NOTICE: Angles 3, 4, 5, and 6 lie between l and m. They are interior angles. Angles 1, 2, 7, and 8 lie outside of l and m. They are exterior angles. Section 3.1 – Lines and Angles • Pairs of the 8 angles have special names. Section 3.1 – Lines and Angles Problem 2 Name the pairs of alternate interior angles. Name the pairs of corresponding angles. Section 3.1 – Lines and Angles Problem 3 – Classifying an Angle Pair The photo below shows the Royal Ontario Museum in Toronto, Canada. Are angles 2 and 4 alternate interior angles, same-side interior angles, corresponding angles, or alternate exterior angles. Section 3.1 – Lines and Angles Section 3.2 – Properties of Parallel Lines 3. I CAN prove theorems about parallel lines. 4. I CAN use properties of parallel lines to find angle measures. Key Vocabulary: Same-Side Interior Angles Postulate Alternate Interior Angles Theorem Corresponding Angles Theorem Alternate Exterior Angles Theorem Section 3.2 – Properties of Parallel Lines Section 3.2 – Properties of Parallel Lines In the solve it, you identified several pairs of angles that appear congruent. You already know the relationship between vertical angles. In this lesson, you will explore the relationship between the angles you learned previously when they are formed by parallel lines and a transversal. Section 3.2 – Properties of Parallel Lines The special angle pairs formed by parallel lines and a transversal are congruent, supplementary, or both. Section 3.2 – Properties of Parallel Lines Problem 1 The measure of <3 is 55. Which angles are supplementary to <3? How do you know? Section 3.2 – Properties of Parallel Lines You can use Same-Side Interior Angles Theorem to prove other angle relationships. Section 3.2 – Properties of Parallel Lines • Proof of Alternate Interior Angles Theorem Section 3.2 – Properties of Parallel Lines Problem 2 Given: a||b Prove <1 and <8 are supplementary Prove that <1 is congruent to <7. Section 3.2 – Properties of Parallel Lines In the previous problem, you proved <1 congruent to <7. These angles are Alternate Exterior Angles. Section 3.2 – Properties of Parallel Lines • If you know the measure of one of the angles formed by two parallel lines and a transversal, you can use theorems and postulates to find the measures of the other angles. Problem 3 What are the measures of <3 and <4? Justify your answer. What is the measure of: Justify> a. <1 b. <2 c. <5 d. <6 e. <7 f. <8 Section 3.2 – Properties of Parallel Lines Problem 4 What is the value of y? Section 3.2 – Properties of Parallel Lines a. In the figure below, what are the values of x and y? b. What are the measures of the four angles in the figure Section 3.2 – Properties of Parallel Lines Section 3.3 – Proving Lines Parallel 5. I CAN determine whether two lines are parallel. Section 3.3 – Proving Lines Parallel In the Solve It, you used parallel lines to find congruent and supplementary relationships of special angle pairs. In this section, you will do the inverse. You will use the congruent and supplement relationships of the special angle pairs to prove lines parallel. Section 3.3 – Proving Lines Parallel You can use certain angle pairs to decide whether two lines are parallel. Section 3.3 – Proving Lines Parallel Problem 1 Which lines are parallel if <1 is congruent <2? Justify. Which lines are parallel if <6 is congruent to <7? Section 3.3 – Proving Lines Parallel We can use the Converse of Corresponding Angles Theorem to prove converse of the theorems and postulates we learned from section 3.2. Section 3.3 – Proving Lines Parallel Proving Converse of the Alternate Interior Angles Theorem Given: <4 is congruent <6 Prove: L||m Section 3.3 – Proving Lines Parallel Prove Converse of the Same-Side Interior Angles Postulate Given: m<3 + m<6 = 180 Prove: L||m Section 3.3 – Proving Lines Parallel Proving Converse of the Alternate Exterior Angle Theorem Given: <4 is congruent <6 Prove: L||m Section 3.3 – Proving Lines Parallel Problem 3 The fence gate at the right is made up of pieces of wood arranged in various directions. Suppose <1 is congruent to <2. Are lines r and s parallel? Explain. What is another way to explain why r||s. Justify. Section 3.3 – Proving Lines Parallel Problem 4 What is the value of x for which a||b? Section 3.3 – Proving Lines Parallel What is the value of w for which c||d? Section 3.3 – Proving Lines Parallel Section 3.3 – Proving Lines Parallel Section 3.3 – Proving Lines Parallel Section 3.4 – Parallel and Perpendicular Lines 5. I CAN relate parallel and perpendicular lines. Section 3.4 – Parallel and Perpendicular Lines In the Solve It, you likely made your conjecture about Oak Street and Court Road based on their relationships to Schoolhouse Road. In this section, you will use similar reasoning to prove that lines are parallel and perpendicular. Section 3.4 – Parallel and Perpendicular Lines You can use the relationships of two lines to a third line to decide whether the two lines are parallel or perpendicular to each other. Section 3.4 – Parallel and Perpendicular Lines There is also a relationship with perpendicular lines. Section 3.4 – Parallel and Perpendicular Lines Problem 1 A carpenter plans to install molding on the sides and the top of the doorway. The carpenter cuts the ends of the top piece and one end of each of the sides pieces 45 degrees as shown. Will the side pieces of molding be parallel? Explain. Can you assemble the pieces at the right to form a picture frame with opposite sides parallel? Section 3.4 – Parallel and Perpendicular Lines The previous theorems proved lines parallel. The perpendicular transversal theorem allows us to conclude that lines are perpendicular. Section 3.4 – Parallel and Perpendicular Lines Proving a relationship between two lines. Given: In a plane, c | b, b | d, and d |a. Prove: c | a Section 3.4 – Parallel and Perpendicular Lines Section 3.4 – Parallel and Perpendicular Lines Section 3.4 – Parallel and Perpendicular Lines Section 3.5 – Parallel Lines and Triangles 6. I CAN use parallel lines to prove a theorem about triangles. 7. I CAN find measures of triangles. Key Vocabulary Auxiliary Line Exterior Angle of a Polygon Remote Interior Angles Section 3.5 – Parallel Lines and Triangles Section 3.5 – Parallel Lines and Triangles In the Solve It, you may have discovered that you can rearrange the corners of the triangles to form a straight angle. You can do this for any triangle. Section 3.5 – Parallel Lines and Triangles The sum of the angle measures of a triangle is always the same. We will use parallel lines to prove this theorem. Section 3.5 – Parallel Lines and Triangles To prove the Triangle Angle-Sum Theorem, we must use an auxiliary line. An auxiliary line is a line that you add to the diagram to help explain relationships in proofs. The red line is the diagram is an auxiliary line. Section 3.5 – Parallel Lines and Triangles Proof of Triangle Angle-Sum Theorem Section 3.5 – Parallel Lines and Triangles Problem 1 What are the values of x and y in the diagram. Section 3.5 – Parallel Lines and Triangles An exterior angle of a polygon is an angle formed by a side and an extension of an adjacent side. For each exterior angle of a triangle, the two nonadjacent angles are its remote interior angles. In the triangle below <1 is an exterior angle and <2 and <3 are its remote interior angles. Section 3.5 – Parallel Lines and Triangles The theorem below states the relationship between an exterior angle and its two remote interior angles. Section 3.5 – Parallel Lines and Triangles Problem 2 What is the measure of <1? What is the measure of <2? Section 3.5 – Parallel Lines and Triangles Problem 3 When radar tracks an object, the reflection of signals off the ground can result in clutter. Clutter causes the receiver to confuse the real object with its reflection, called a ghost. At the right, there is a radar receiver at A, an airplane at B, and the airplane’s ghost at D. What is the value of x? Section 3.5 – Parallel Lines and Triangles Section 3.5 – Parallel Lines and Triangles Section 3.5 – Parallel Lines and Triangles Section 3.5 – Parallel Lines and Triangles