DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Name Class 4.1 Date Angles Formed by Intersecting Lines Essential Question: How can you find the measures of angles formed by intersecting lines? Explore 1 Resource Locker Exploring Angle Pairs Formed by Intersecting Lines When two lines intersect, like the blades of a pair of scissors, a number of angle pairs are formed. You can find relationships between the measures of the angles in each pair. A Using a straightedge, draw a pair of intersecting lines like the open scissors. Label the angles formed as 1, 2, 3, and 4. Possible answer: © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©MNI/ Shutterstock 4 B 1 3 2 Use a protractor to find each measure. Module 4 GE_MNLESE385795_U2M04L1.indd 163 Angle Measure of Angle m∠1 120° m∠2 60° m∠3 120° m∠4 60° m∠1 + m∠2 180° m∠2 + m∠3 180° m∠3 + m∠4 180° m∠1 + m∠4 180° 163 Lesson 1 01/04/14 11:37 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT Correcti You have been measuring vertical angles and linear pairs of angles. When two lines intersect, the angles that are opposite each other are vertical angles. Recall that a linear pair is a pair of adjacent angles whose non-common sides are opposite rays. So, when two lines intersect, the angles that are on the same side of a line form a linear pair. Reflect 1. Name a pair of vertical angles and a linear pair of angles in your diagram in Step A. Possible answers: vertical angles: ∠1 and ∠3 or ∠2 and ∠4 ; linear pairs: ∠1 and ∠2, ∠2 and ∠3, ∠3 and ∠4, or ∠1 and ∠4 2. Make a conjecture about the measures of a pair of vertical angles. Vertical angles have the same measure and so are congruent. 3. Use the Linear Pair Theorem to tell what you know about the measures of angles that form a linear pair. The angles in a linear pair are supplementary, so the measures of the angles add up to 180°. Explore 2 Proving the Vertical Angles Theorem The conjecture from the Explore about vertical angles can be proven so it can be stated as a theorem. The Vertical Angles Theorem If two angles are vertical angles, then the angles are congruent. 4 1 3 2 ∠1 ≅ ∠3 and ∠2 ≅ ∠4 Follow the steps to write a Plan for Proof and a flow proof to prove the Vertical Angles Theorem. Given: ∠1 and ∠3 are vertical angles. Prove: ∠1 ≅ ∠3 Module 4 GE_MNLESE385795_U2M04L1.indd 164 4 1 3 2 164 © Houghton Mifflin Harcourt Publishing Company You have written proofs in two-column and paragraph proof formats. Another type of proof is called aflow proof. A flow proof uses boxes and arrows to show the structure of the proof. The steps in a flow proof move from left to right or from top to bottom, shown by the arrows connecting each box. The justification for each step is written below the box. You can use a flow proof to prove the Vertical Angles Theorem. Lesson 1 20/03/14 8:14 AM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A A Complete the final steps of a Plan for Proof: Because ∠1 and ∠2 are a linear pair and ∠2 and ∠3 are a linear pair, these pairs of angles are supplementary. This means that m∠1 + m∠2 = 180° and m∠2 + m∠3 = 180°. By the Transitive Property, m∠1 + m∠2 = m∠2 + m∠3. Next: Subtract m∠2 from both sides to conclude that m∠1 = m∠3. So, ∠1 ≅ ∠3. B Use the Plan for Proof to complete the flow proof. Begin with what you know is true from the Given or the diagram. Use arrows to show the path of the reasoning. Fill in the missing statement or reason in each step. ∠1 and ∠2 are a linear pair. Given (see diagram) ∠1 and ∠2 are supplementary . Linear Pair Theorem m∠1 + m∠2 = 180° Def. of supplementary angles ∠1 and ∠3 are vertical angles. Given ∠2 and ∠3 are a linear pair. Given (see diagram) ∠1 ≅ ∠3 © Houghton Mifflin Harcourt Publishing Company Def. of congruence ∠2 and ∠3 are supplementary. Linear Pair Theorem m∠2 + m∠3 = 180° Def. of supplementary angles m∠1 = m∠3 m∠1 + m∠2 = m∠2 + m∠3 Subtraction Property of Equality Transitive Property of Equality Reflect 4. Discussion Using the other pair of angles in the diagram, ∠2 and ∠4, would a proof that ∠2 ≅ ∠4 also show that the Vertical Angles Theorem is true? Explain why or why not. Yes, it does not matter which pair of vertical angles is used in the proof. Similar statements and reasons could be used for either pair of vertical angles. 5. Draw two intersecting lines to form vertical angles. Label your lines and tell which angles are congruent. Measure the angles to check that they are congruent. A D E C Possible answer: B By the Vertical Angles Theorem, ∠AEC ≅ ∠DEB and ∠AED ≅ ∠CEB. Checking by measuring, m∠AEC = m∠DEB = 45° and m∠AED = m∠CEB = 135°. Module 4 GE_MNLESE385795_U2M04L1.indd 165 165 Lesson 1 20/03/14 8:14 AM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT Correcti Using Vertical Angles Explain 1 You can use the Vertical Angles Theorem to find missing angle measures in situations involving intersecting lines. Example 1 Cross braces help keep the deck posts straight. Find the measure of each angle. 146° 5 7 ∠6 Because vertical angles are congruent, m 6 ∠5 and ∠7 From Part A, m ∠6 = 146°. ∠6 = 146°. Because ∠5 and ∠6 form a linear pair , they are supplementary and m ∠5 = 180° - 146° = 34° . m∠ also forms a linear pair with ∠6, or because it is a 7 = 34° because ∠ 7 vertical angle with ∠5. Your Turn 6. The measures of two vertical angles are 58° and (3x + 4) °. Find the value ofx. 58 = 3x + 4 18 = x 7. The measures of two vertical angles are given by the expressions (x + 3) ° and(2x - 7) °. Find the value of x. What is the measure of each angle? x + 3 = 2x - 7 x + 10 = 2x 10 = x The measure of each angle is (x + 3)° = (10 + 3)° = 13°. Module4 GE_MNLESE385795_U2M04L1 166 166 © Houghton Mifflin Harcourt Publishing Company 54 = 3x Lesson1 3/27/14 2:44 PM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Using Supplementary and Complementary Angles Explain 2 Recall what you know about complementary and supplementary angles. Complementary angles are two angles whose measures have a sum of 90°. Supplementary angles are two angles whose measures have a sum of 180°. You have seen that two angles that form a linear pair are supplementary. Example 2 Use the diagram below to find the missing angle measures. Explain your reasoning. C B D 50° F A Find the measures of E ∠AFC and ∠AFB. ∠AFC and ∠CFD are a linear pair formed by an intersecting line and ray, ‹ › → − ‾ , so they are supplementary and the sum of their measures is 180°. By the AD and FC diagram, m∠CFD = 90°, so m∠AFC = 180° - 90° = 90° and ∠AFC is also a right angle. Because together they form the right angle ∠AFC, ∠AFB and ∠BFC are complementary and the sum of their measures is 90°. So, m∠AFB = 90° - m∠BFC = 90° - 50° = 40°. Find the measures of ∠DFE and ∠AFE. ∠BFA and ∠DFE are formed by two so the angles are vertical intersecting lines and are opposite each other, angles. So, the angles are congruent. From Part A m∠AFB = 40°, so m∠DFE = 40° also. © Houghton Mifflin Harcourt Publishing Company Because ∠BFA and ∠AFE form a linear pair, the angles are supplementary and the sum of their measures is 180° . So, m∠AFE = 180° - m∠BFA = 180° - 40° = 140° . Reflect 8. In Part A, what do you notice about right angles ∠AFC and ∠CFD? Make a conjecture about right angles. Possible answer: Both angles have measure 90°. Conjecture: All right angles are congruent. Module4 GE_MNLESE385795_U2M04L1.indd 167 167 Lesson1 20/03/14 8:14 AM DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT Correcti Your Turn You can represent the measures of an angle and its complement as x° and (90 - x)°. Similarly, you can represent the measures of an angle and its supplement as x° and (180 - x)°. Use these expressions to find the measures of the angles described. 9. The measure of an angle is equal to the measure of its complement. 10. The measure of an angle is twice the measure of its supplement. x = 2(180 - x) x = 90 - x 2x = 90 x = 360 - 2x x = 45; so, 90 - x = 45 3x = 360 x = 120; so, 180 - x = 60 The measure of the angle is 45° the measure of its complement is 45°. The measure of the angle is 120° the measure of its supplement is 60°. Elaborate 11. Describe how proving a theorem is different than solving a problem and describe how they are the same. Possible answer: A theorem is a general statement and can be used to justify steps in solving problems, which are specific cases of algebraic and geometric relationships. Both proofs and problem solving use a logical sequence of steps to justify a conclusion or to find a solution. 12. Discussion The proof of the Vertical Angles Theorem in the lesson includes a Plan for Proof. How are a Plan for Proof and the proof itself the same and how are they different? Possible answer: A plan for a proof is less formal than a proof. Both start from the given information and reach the final conclusion, but a formal proof presents every logical step in detail, while a plan describes only the key logical steps. 13. Draw two intersecting lines. Label points on the lines and tell what angles you know are congruent and which are supplementary. Possible answer: K L J H Vertical angles are congruent: ∠GLK ≅ ∠JLH and ∠GLJ ≅ ∠KLH Angles in a linear pair are supplementary: ∠GLJ and ∠GLK; ∠GLJ and ∠JLH; ∠JLH and ∠HLK; and ∠HLK and ∠GLK © Houghton Mifflin Harcourt Publishing Company G 14. Essential Question Check-In If you know that the measure of one angle in a linear pair is 75°, how can you find the measure of the other angle? The angles in a linear pair are supplementary, so subtract the known measure from 180°: 180 - 75 = 105, so the measure of the other angle is 105°. Module 4 GE_MNLESE385795_U2M04L1.indd 168 168 Lesson 1 20/03/14 8:14 AM
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