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Prove It How do we create truth? 2:1:16:Geometric Concepts: Angles III TITLE OF LESSON Geometry Unit 1 Lesson 16 – Geometric Concepts: Angles III Prove it! What’s on the outside? What’s on the inside? of Geometry TIME ESTIMATE FOR THIS LESSON One class period ALIGNMENT WITH STANDARDS California – Geometry Introductory lesson necessary for: 4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. 6.0 Students know and are able to use the triangle inequality theorem. 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. MATERIALS Pre-cut angles Note: For this you can just cut triangles out of paper, but try to have them the approximate measure that you name them. You’ll need one per student, no two measurements the same. Keep them between 5° and 180° or 270°. Also, be sure to cut out your two demo angles—90° and 35°. LESSON OBJECTIVES • To apply understanding of angles to doing problems associated with angles • To add angles and to recognize the different types of angles. FOCUS AND MOTIVATE STUDENTS 1) Homework Check – Stamp/initial complete homework assignment. Pass back graded work and have students place in the appropriate sections of their binders. 2) Binders – Pass back graded binders. 3) Agenda – Have students copy the agenda. 4) Present Homework – (5 minutes) Have individual students present one or two observations from their homework. For instance ask one student to describe an object she has selected in her homework and explain what type of angle describes that object. Have her draw the angle on the board and write the type of angle (right, acute, obtuse, etc.) Ask her to estimate the measure of the angle. Have five or so students do this, and collect the homework from all students, or stamp and sign it and have them leave it in their binders for when you collect them. ACTIVITIES – INDIVIDUAL AND GROUP 1. Lecture: Angle Addition Postulate – (5 minutes): Introduce the angle addition postulate. Write the postulate below on the board and have the students copy it into the postulates and theorems section of their binders. → If AC is interior to the angle BAD then the measure of angle BAD = the measure of angle BAC plus the measure of angle CAD. Draw the following diagram associated with this definition and have them copy the drawing: B C D A 1 © 2001 ESubjects Inc. All rights reserved. Prove It How do we create truth? 2:1:16:Geometric Concepts: Angles III 2. Practice Problems – (1 minutes) Do a few practice problems to understand the angle addition postulate. Ask questions such as: If the measure of angle BAD is 70° and the measure of angle CAD is 40° what is the measure or angle BAC? (30°). If the measure of angle BAD is 80° and the measure of angle BAC is 35° what is the measure of angle CAD (45°). If the measure of angle BAC is 30° and the measure of angle CAD is 35° what is the measure of angle BAD? (65°). What types of angles are these? (These are adjacent angles.) 3. Pair Work – (4 minutes) Next, have two demo angles to use (one 90° and one 35°), and explain that they’re going to have 3 minutes to quickly do this next activity. Showing them with your demo angles, explain that they will be pairing up with each other student. In each pairing, one angle will be smaller than the other. Both students will write the other student’s name, then decide whose angle is smaller. The smaller angle will become one of the two adjacent angles in the larger angle. Demonstrating with the 90° and 35° angles, place the 35° angle inside the 90° angle. Explain that their jobs will be to determine the measurement of the other, missing adjacent angle in their set. In the case of the demo, the missing angle is 55°. Then, have students take out a blank piece of paper and number it with one fewer than the number of students in the class. Give each student one of the pre-made angles and tell them they have 3 minutes to pair up with another student in the class and complete the assignment. 4. Review – (5 minutes) Review the types of angles that you discussed during the previous angle lesson (acute, right, obtuse, straight, reflex, adjacent, vertical, congruent, complimentary, and supplementary). Ask for student volunteers to give the definition of each of these types of angles. They may read them from their notes. 5. Acute Angles – (5 minutes) Problems associated with acute angles. Acute angles are angles less than 90°. Ask questions such as: What is the largest acute angle? (Answer: 89°) Refer to the diagram in number 1 of today’s activities. If angle BAD is 90° what type of angle is angle BAC? (Answer: Acute). Why must this be true? (Answer: Since an acute angle is less than 90° and the measure of BAD (90°) is the sum of the measures of BAC and CAD, then BAC must be less than 90° and is therefore acute.) What about angle CAD? (Answer: It is also acute for the same reason.) 6. Right Angles – (5 minutes): Problems associated with right angles. A right angle is an angle that measures 90°. B C A D Draw this diagram on the board. Point out that the angle for a right angle is drawn with a rectangle rather than semicircular angle designator. Any time you see this designation, you can assume that the angle is 90°. Ask: If angle BAC measures 40° and angle CAD measure 45°is angle BAD a right angle? (No they do not add up to 90° but instead add up to 85° and therefore BAD is not a right angle.) If BAC is 40°, what would the measurement of CAD have to be for BAD to be a right angle assuming that? (50°). Have some students form the original set of angles forming BAC, CAD and BAD adding up to (approximately) 85°. What adjustments could be made to create a right angle BAD? (You can increase either BAC or CAD by 5°) Is there any other way you could do it? (You could increase both BAC and CAD by 2.5°) Have the students demonstrate each of these questions, forming angles by standing in the appropriate configuration. Have one student draw the appropriate diagram on the board and label the size of each angle. 7. Obtuse Angles – (5 minutes): Problems associated with obtuse angles. An obtuse angle is an angle that measures more than 90° but less than 180°. Draw the diagram below on the board and ask which angle is the obtuse angle? (BAD) How do we know that it is obtuse? (CAD is 90° since it has the rectangular angle designator and BAC appears to be acute. We need BAD to be greater than 90° but we also need for it to be less than 180° implying that BAC is acute since CAD is a right angle and is 90°) Do we know this for sure? (Not really since we are deducing this from the appearance of the diagram and a diagram is not the geometrical object but is only a representation of the 2 © 2001 ESubjects Inc. All rights reserved. Prove It How do we create truth? 2:1:16:Geometric Concepts: Angles III object. If CAD is 20° can BAC be 60° and still imply that BAD is obtuse? (No BAC must be greater than 70° for BAD to still be obtuse.) C B A D 8. Straight Angles – (5 minutes): Problems associated with straight angles. A straight angle is an angle that measures 180°. Ask a student to draw a straight angle on the board. Have three students form a straight angle by standing in the room. Ask the question: If we divide a straight angle (a line) into two adjacent angles and one of the two adjacent angles is an acute angle what type of angle is the other angle? (Obtuse). If we divide a straight angle into two adjacent angles and one of the two adjacent angles is a right angle what type of angle is the other angle? (Right angle.) Have a student draw this on the board. Be sure to have them designate the right angles with rectangular angle designators. What types of objects can be drawn using straight angles and right angles? 9. Reflex Angles – (5 minutes) Problems associated with reflex angles. An angle that measures more than 180° but less than 360°is a reflex angle. Have a student draw a reflex angle on the board. It should look similar to the angle below. Note that it is the outside angle that is the reflex angle. If an angle is 60° and we add an adjacent angle to it that is 130°, what is the measure of the two angles formed by adding these two together? (190°) Is this a reflex angle? (Yes because it is greater than 180° and less than 360°) If you add two obtuse angles together what do you get? (A reflex angle) 10. Homework – (5 minutes) Explain today’s homework and have students copy the necessary information from the board. HOMEWORK Add the following angles together and describe the result by recording its measurement (when possible) and the type of angle. For instance a 45° added to a 45° is a 90° angle, which is a right angle. A 10° angle added to a 10°angle produces an acute angle of 20°. Have the students give the type of angle, the measure of the angle (if appropriate) and create an approximate drawing. 10° added to 20° 10° added to 40° 40° added to 50° 60° added to 30° 70° added to 30° 70° added to 70° 90° added to 90° 100° added to 80° 130° added to 50° 130° added to 70° 3° added to 20° Right added to acute Right added to obtuse Obtuse added to obtuse Straight added to right Straight added to obtuse Straight added to acute Acute added to acute (There’s leeway here for the answer to be acute or not.) GROUP ROLES None DOCUMENTATION FOR PORTFOLIO None 3 © 2001 ESubjects Inc. All rights reserved.