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Geometry Mrs. Franks 10th Grade Unit Plan Information Properties of Triangles Jessica Franks 10th Grade Mathematics Geometry Learning Objectives: See daily lessons Essential Questions: What are the properties of a triangle? How can I identify congruent triangles? What is an altitude? What are the different types of triangles? What are corresponding parts? What happens when the medians of a triangle meet? What happens when the altitudes of a triangle meet? When you connect the midpoints of a triangle what do you get? Enduring Understandings: Students are able to geometric relationships are evident in real-life situations. Students will be able to recognize math processes in the future and be able to locate appropriate resource materials to assist them. Students will be able to recognize reasoning and proof as fundamental aspects of mathematics. Students will be able to see relationships that exist between the angles and sides of geometric figures can be proven. At the conclusion of this unit the students should be able to use properties, theorems and postulates to prove the congruency of triangles to one another. Instructional Procedures: See Daily Lesson Plans Geometry Mrs. Franks 10th Grade Standards: NY State NY State Geometry Standards Technology Standards G.G.28 Determine the congruence of 1.a Students demonstrate a sound two triangles by using one of the five understanding of the nature and congruence techniques (SSS, SAS, operation of technology systems ASA, AAS, HL), given sufficient 2.b Students practice responsible use of information about the sides and/or technology systems, information, and angles of two congruent triangles software. G.G.29 Identify corresponding parts of 2.c Students develop positive attitudes congruent triangles toward technology uses that support G.G.30 Investigate, justify, and apply lifelong learning, collaboration, theorems about the sum of the personal pursuits, and productivity. measures of the angles of a triangle 3.a Students use technology tools to G.G.31 Investigate, justify, and apply enhance learning, increase productivity, the isosceles triangle theorem and its and promote creativity. converse 5.b Students use technology tools to G.G.32 Investigate, justify, and apply process data and report results. theorems about geometric inequalities, 6.a Students use technology resources using the exterior angle theorem for solving problems and making G.G.33 Investigate, justify, and apply informed decisions. the triangle inequality theorem 6.b Students employ technology in the G.G.34 Determine either the longest development of strategies for solving side of a triangle given the three angle problems in the real world. measures or the largest angle given the lengths of three sides of a triangle G.G.43 Investigate, justify, and apply theorems about the centroid of a triangle, dividing each median into segments whose lengths are in the ratio 2:1 G.G.44 Establish similarity of triangles, using the following theorems: ASA, SAS, and SSS G.G.45 Investigate, justify, and apply theorems about similar triangles Geometry Mrs. Franks 10th Grade DATE OBJECTIVE: DATE OBJECTIVE: 12/05 - Understand the key properties of 12/05 - Identify congruent figures and triangles using geometer’s corresponding parts. sketchpad (GS) - Prove that two triangles are - Classify triangles by their sides congruent. and angles - Prove that triangles are congruent using the SSS and SAS Congruence Postulates. CONTENT: CONTENT: - Triangles - 4.2: Congruence and Triangles - 4.1: Triangles and Angles - 4.3: Proving Triangles are Congruent: SSS and SAS ACTIVITIES: ACTIVITIES: - Class discussion on Triangles - Go to computer lab - What make a triangle a triangle? - Bell Ringer - Go to the computer lab - Go over homework - Introduce geometer’s sketchpad - Using GS have class take notes, to the students practice, explore and discuss - Allow students to get the used to congruency of triangles the new program by letting them explore - Use geometer’s sketchpad to explore basic properties of triangles using geometer’s sketchpad MATERIALS NEEDED: MATERIALS NEEDED: Computers, Geometer’s Sketchpad, Computers, Geometer’s Sketchpad, Calculators, Worksheet for GS from Calculator, Compass http://sierra.nmsu.edu/morandi/Cour seMaterials/IntroToSketchpad.html (In Notes) ASSESSMENT: ASSESSMENT: Student responses – verbal and Student responses – verbal and written. Class participation – written. Class participation – Sketch. Homework assignment. Sketch. Homework assignment. PRACTICE: PRACTICE: In class - sketches In class - see written examples from Homework – Triangle Worksheet notes Homework – Explore applet at http://illuminations.nctm.org/tools/t ool_detail.aspx?id=4 . Geometry Mrs. Franks 10th Grade DATE OBJECTIVE: DATE OBJECTIVE: 12/05 - Prove that triangles are congruent 12/05 - Identify the mid-segments of a using the ASA Congruence triangle. Postulate and the AAS Congruence - Use properties of mid-segments of a Theorem. triangle. - Use properties of medians of a - Use triangle measurement to decide which side is longest or which angle is triangle. largest. - Use properties of altitudes of a - Use the triangle Inequality. triangle. CONTENT: CONTENT: - 4.4: Proving Triangles are - 5.4: Mid-segment Theorem Congruent: ASA and AAS - 5.5: Inequalities in One Triangle - 5.3: Medians and Altitudes of a Triangle ACTIVITIES: ACTIVITIES: - Go to computer lab - Go to computer lab - Bell Ringer - Bell Ringer - Go over homework - Go over homework - Using Geometer’s sketchpad have - Using Geometer’s sketchpad the class take notes, practice, have the class take notes, practice explore and discuss congruency, and discuss Mid-segment and medians and altitudes of triangles inequalities in one triangle. MATERIALS NEEDED: MATERIALS NEEDED: Computers, Geometer’s Sketchpad, Computers, Geometer’s Sketchpad, Calculator, Worksheet for GS from Calculator, Sketch from Key http://sierra.nmsu.edu/morandi/Cour Curriculum Press on web page seMaterials/sketchpadFiles.html (in Notes) ASSESSMENT: ASSESSMENT: Student responses – verbal and Student responses – verbal and written. Class participation – written. Class participation – Sketch. Homework assignment. Sketch. Homework assignment. PRACTICE: PRACTICE: In class - sketches In class - see written examples from Homework – Textbook Problems notes Homework - Textbook problems. Geometry Mrs. Franks 10th Grade DATE OBJECTIVE: DATE OBJECTIVE: 12/05 - Write the equation of a line given 12/05 - Assess Knowledge of Students a point on the line and the slope of the line - Write the equation of a line given two points on the line CONTENT: CONTENT: - 5.3 & 5.5: Writing equations of - Chapters 4 & 5 lines with two points and Point- Slope Form ACTIVITIES: ACTIVITIES: - Bell Ringer - Go to computer lab - Go over homework - Bell Ringer - Using the graphing calculator - Go over homework have class take notes, practice and - Using Geometer’s sketchpad discuss the equations of lines. assess triangles - Complete Worksheet on the writing equations of lines MATERIALS NEEDED: MATERIALS NEEDED: Computers, Calculators, notes Computers, Geometer’s Sketchpad, Calculators, Teacher created exam ASSESSMENT: ASSESSMENT: Student responses – verbal and Student responses – verbal and written. Class participation – written. Class participation – Sketch. Homework assignment. Sketch. Teacher created exam PRACTICE: PRACTICE: In class - see written examples from In class - Teacher created exam notes Individually - Teacher created Homework – Textbook problems exam Geometry Mrs. Franks 10th Grade Chapters 4&5 Course 2R Mrs. Franks Geometry Mrs. Franks 10th Grade Geometry Mrs. Franks 10th Grade Triangles What do you remember about triangles? Geometry Mrs. Franks 10th Grade Introduction to Geometer's Sketchpad In this assignment we will learn how to use the program Geometer's Sketchpad. This program is very useful for learning about geometry. We will discover several geometric facts this semester through its use. Here are several tasks to perform in Geometer's Sketchpad. You should use the program enough to be able to do these tasks with ease. When you open the program, you will see six icons on the left side of the screen. They are, from top to bottom, the arrow tool, the point tool, the compass (or circle) tool, the straightedge tool, the text tool, and the custom tool. The arrow tool is used to select objects. The next three are used to draw points, circles, and lines. One important thing to know about is how to highlight objects. By clicking on an object it will be highlighted, and then can be used in further constructions. The order in which you highlight objects can affect the resulting construction. Draw a point: Click on the point tool, then click where you want a point. Draw a line segment: Click on the line tool. The icon should show two points and a segment connecting them. To draw a line segment click the mouse where you want the segment to begin, and holding the mouse, drag it until you get to where you want the line to end, then release the mouse. Draw a ray and line: Click and hold the mouse on the line tool until you see three icons. These, from left to right, are the line segment, ray, and line tools. Click on the appropriate one, then click and hold the mouse somewhere on the screen, then drag to get the ray or line. Draw a circle: Click on the circle tool, then click and hold the mouse, move to size the circle. Alternatively, if you want a circle centered at a given point, with the circle tool, place the cursor over the point and then draw the circle. If you want the circle centered at a certain point and passing through another point, click on the center and then click on the second point. Finally, click on construct, then circle by center and point. See what happens if you highlight the points in reverse order and construct the circle by center and point. Circles are determined by two points, one being the center and the other being a point on the circle. Resize the circle: Click on the arrow tool, then on the point on the circle. Drag this point to resize the circle. Alternatively, click and drag the center. Move the circle: Click on the arrow tool, then on the circle away from the point on the circle. Drag to move the circle. Draw a triangle: using the line segment tool, draw a line segment. Then draw a second segment starting where the first segment ended. Finally, draw a third segment starting where the second segment ended and ending where the first segment started. Resize the triangle: Click the mouse on the arrow tool. Then click on one of the vertices of the triangle (i.e., one of the endpoints), then drag the mouse to resize. Alternatively, click and drag one of the sides. Geometry Mrs. Franks 10th Grade Move the triangle: Click the arrow tool. Then click on two of the sides (or the three vertices). Then drag one of the sides. Measure the angles of the triangle: Click the arrow tool. Then click three of the vertices in order. Then go to Measure, Angle. Select more than one object: Click on the arrow tool. Click on the objects you wish to select. You should see which objects are selected. Draw the interior of a triangle: Click on the arrow tool. Then click on all three vertices of the triangle. You should see large dots over each of them. Click the mouse on the menu item construct, then on polygon interior. Draw a four-sided figure: Once you have drawn it, resize it by moving one of the vertices. Notice that you can make many different shapes. Draw the four-sided figure's interior. Draw an angle bisector. Geometer's Sketchpad views an angle as three points selected in order. The middle point is the vertex, or corner, of the angle. You can then draw the angle by drawing rays from the vertex through the other two points. Once you have drawn and selected three points, click on construct, and then angle bisector. This line should cut the angle into two equal pieces. If it does not appear to do so, look carefully at the order in which you selected your three points, since there are three different angles that can be made from the three points (the three angles of the triangle formed by the three points). Find the intersection of two lines, segments, or circles: Draw two line segments (or rays or lines or circles) that cross. With the point tool, put the mouse over the intersection and click. Move one of the line segments and watch what happens to the intersection point. Alternatively, select both line segments, then click on construct, then on intersection. Draw perpendicular and parallel lines: Draw a line. Select the line and a point on the line. Then click construct, then perpendicular line. This constructs a line perpendicular to the given line and passing through the given point. Next, plot a point off of a given line. Select the line and the point. Click construct, then parallel line. This produces a line through the given point and parallel to the given line. Label points or sides: Click on the label tool (the one that looks like a hand), then click on whatever you want to label. If you want to change the label, double click on the label (after selecting either the label tool or the arrow tool). Open documents: Open the file Square.gsp. It is on my web page http://www.bataviacsd.org/webpages/JFranks/course__3r.cfm?subpage=6660 . Read the instructions once you open it and play around with them accordingly. Print documents: Click on file, then on print preview. Click on fit to page if it shows your sketch printing on two pages . Finally, click print. If you click print directly, your document may print on two pages. Resource: http://sierra.nmsu.edu/morandi/CourseMaterials/IntroToSketchpad.html Geometry Mrs. Franks 10th Grade 4.2 Congruence and Triangles Two Geometric Figures are __________________________ if they have exactly the same _____________________ and ___________________________. When two figures are ________________________, there is a correspondence between their angles and sides such that, corresponding ____________________ are congruent and corresponding ________________________ are congruent. For the triangles below you can write ABC PQR A Corresponding Angles Corresponding Sides B C P Q R Using Geometer’s Sketchpad: Create Two Congruent Triangles. Show that Corresponding Angles are Congruent and Corresponding Sides are Congruent (Using the Measure Tool). Example 1: Congruent Figures In the diagram NPLM EFGH P E F a. Find the value of x 8m L (7y + 9)º 72º b. Find the value of y N 110º 10m H G (2x – 3) m 87º M Geometry Mrs. Franks 10th Grade Theorem 4.3 Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. If A D and B E then C F E B D F A C Use Geometer’s Sketchpad: Create two triangles (NOT Congruent). Measure all of the angles in each triangle. Now move your points around so that you have two sets of angles congruent. Is the third set of angles congruent? Example 2: Find the value of x if MNL TRS : M T (2x + 30)º N 55º R 65º L S Geometry Mrs. Franks 10th Grade 4.3 Proving Triangles are Congruent: SSS and SAS Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of the sides of the two triangles. Now move the triangles such that the sides of the first triangle are congruent to the sides of the second triangle. Now without moving the triangles measure all the angles of both triangles. What do you notice? Postulate 19 Side – Side – Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are ___________________________. N If Side MN QR Side NP RS Side PM SQ Then MNP QRS R M P Q S Using a Compass Construct a triangle that is congruent to the given triangle ABC. A C B Now that we’ve used the compass try using Geometer’s Sketchpad to construct congruent triangles. Remember, you must show your arcs to have a valid construction. Hint use construct a circle. Geometry Mrs. Franks 10th Grade Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of two sides and the angle between the two sides of the two triangles. Now move the triangles such that these three measurements are congruent to each other. Now without moving the triangles measure the rest of the sides and angles of both triangles. What do you notice? Postulate 20 Side – Angle – Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are __________________. X If Side PQ WX Q Angle Q X Side QS XY P W Y Then PQS WXY S Example 3: Use the SSS Congruence Postulate to Prove the two triangles congruent. 8 6 B: (-7.00, 5.00) A: (-4.00, 5.00) F: (6.00, 5.00) B A F 4 2 E: (6.00, 2.00) D E D: (1.00, 2.00) -10 C -5 5 10 C: (-7.01, 0.00) -2 -4 -6 -8 Homework: Go to http://illuminations.nctm.org/tools/tool_detail.aspx?id=4 and play around with the applet. Answer the questions at the bottom of the page and print out your explorations. Geometry Mrs. Franks 10th Grade 4.4 Proving Triangles are Congruent: ASA and AAS Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of two angles and the side between the two them in both triangles. Now move the triangles such that these three measurements are congruent to each other. Now without moving the triangles measure the rest of the sides and angles of both triangles. What do you notice? Postulate 21 Angle – Side – Angle (ASA) Congruence Postulate If two ______________ and the included _______________ of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are _________________. If Angle A D B E Side AC DF Angle C F Then ABC DEF C F A D Use Geometer’s Sketchpad: Construct two triangles (NOT Congruent). Measure the length of two angles and a side NOT between the two angles in both triangles. Now move the triangles such that these three measurements are congruent to each other. Now without moving the triangles measure the rest of the sides and angles of both triangles. What do you notice? Theorem 4.5 Angle – Angle – Side (AAS) Congruence Theorem If two _________________ and a non-included ______________________ of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are ________________________. A F If Angle A D E Angle C F Side BC EF Then ABC DEF C B D Geometry Mrs. Franks 10th Grade This is Wonderful that Geometer’s Sketchpad is working to show us these postulates and theorems are true, but can anyone tell us why, or show us another way using Geometer’s Sketchpad to prove these postulates to us? - With a partner try to find another way to use geometer’s sketchpad to prove these to the class. Example 1: Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. M N D A F G 2 3 C 1 4 E P O B I H D B MN PO and PM ON Example 2: You want to describe the boundary lines of a triangular piece of property to a friend. You fax the note and the sketch below to your friend. Have you provided enough information to determine the boundary lines of the property? Use Geometer’s Sketchpad to explain. N The southern border is a line running cherry tree east from the apple tree, and the western border is the north – south line running from the cherry tree to 250ft the apple tree. The bearing from the easternmost point to the northernmost point is W 53.1º N. The distance between these points is 250 ft. 53.1º Geometry Mrs. Franks 10th Grade 5.3 Medians and Altitudes of a Triangle Median of a Triangle – a segment whose endpoints are a ____________ of the triangle and the ____________________ of the opposite side. A C M B Use Geometer’s Sketchpad: Construct a Triangle. Find the midpoint of Each Side. Now connect the vertex of each angle to the midpoint on the opposite side. What do you notice? Drag one vertex of the triangle to see an acute, obtuse and right triangle. What do you notice now? The medians of a triangle are __________________________. Concurrent Lines – Lines that intersect at ____________________________________. The point of concurrency is called the _________________________ of the triangle. Use Geometer’s Sketchpad: Construct a point at the centroid. Now use the Measure Tool to measure the distance from each vertex to the centroid. Use the Calculate Tool to find the ratio of each Median. What do you notice? Geometry Mrs. Franks 10th Grade Theorem Theorem 5.7 Concurrency of Medians of a Triangle The medians of a triangle intersect at a point that is two thirds of the distance from each B vertex to the midpoint of the opposite side. D If P is the centroid of ΔABC, then P E 2 2 2 AP AD, BP BF, andCP CE. C 3 3 3 F A Example 1: P is the centroid of ΔQRS shown below and PT = 5, find RT and RP. Q P S R Example 2: Find the coordinates of the centroid of ΔJKL. 12 L 10 8 6 K 4 2 J 5 10 Geometry Mrs. Franks 10th Grade Altitude of a Triangle – the ______________________ segment from a vertex to the opposite side or to the line that contains the opposite side. Use Geometer’s Sketchpad: Construct a triangle. Construct perpendicular segments from a vertex to the opposite side of the triangle. Repeat for all three sides. Do these lines intersect? If they do construct a point at the intersection. Drag one of the vertices of the triangle, What do you notice about the point of intersection? Think about the following questions. The altitude of a triangle can be where? How many altitudes does a triangle have? Are the lines concurrent? The point where they intersect is called the _______________________________________. Example 3: Where is the orthocenter located in each type of triangle? Use Geometer’s Sketchpad to see the sketch. Try to draw it. a. Acute Triangle b. Right Triangle c. Obtuse Triangle Theorem Theorem 5.6 Concurrency of Altitudes of a Triangle F A B The lines containing the altitudes of a triangle are concurrent. H If AE , BF and CD are the altitudes of D E ΔABC, then the lines AE , BF and CD intersect at some point H. C Geometry Mrs. Franks 10th Grade 5.4 Mid-segment Theorem A Mid-segment of a triangle is a segment that __________________________________ of two sides of a triangle. Example 4: Using Geometer’s Sketchpad Show that the mid-segment MN is parallel to side JK and is half as long. Hint: How do we know lines are parallel? 8 6 K 4 J 2 -5 5 L -2 Draw in the missing pieces (segments and measurements) from Sketchpad. Midsegment Theorem C Theorem 5.9 Mid-segment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the D E third side and is half as long. 1 DE AB and DE AB A 2 B Geometry Mrs. Franks 10th Grade Example 5: UV and VW are mid-segments of ΔRST. Find UW and RT. If RS 12 and VW 8 R U V T W S Geometry Mrs. Franks 10th Grade 5.5 Inequalities in One Triangle Theorems B Theorem 5.10 If one side of a triangle is longer than 5 another side, then the angle opposite 3 the longer side is larger than the angle opposite the shorter side. A C Theorem 5.11 D E If one angle of a triangle is larger than 60º 40º another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. F Largest Angle Shortest Side Longest Side Smallest Angle Example 1: Write the measurements of the triangles in order from least to greatest. a. R b. F 100 8 7 35 45 H 5 Q P G Geometry Mrs. Franks 10th Grade Use Geometer’s Sketchpad: Construct a ray. Construct a point above the ray and a point on the ray. Construct a triangle using the endpoint of the ray and the two new points that you have created. Measure the exterior and interior angles. Play around with the calculations. Do you notice anything? B A C D Theorem A Theorem 5.12 Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. m1 > mA and m1 > mB 1 D C B Use Geometer’s Sketchpad: Go to my webpage http://www.bataviacsd.org/webpages/JFranks/course__3r.cfm?subpage=6660 and open Inequalities in One Triangle. Keep clicking random break and see if you can make a triangle. What do you notice about the lengths of the sides when you can and cannot make a triangle? Theorem Theorem 5.13 Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC AC + BC > AB AB + AC > BC Geometry Mrs. Franks 10th Grade Example 3: A triangle has one side of 10 centimeters and another of 14 centimeters. Describe the possible lengths of the third side. Geometry Mrs. Franks 10th Grade 5.3 & 5.5 Writing equations of lines with two points and Point-Slope Form What two pieces of information do you need to write the equation of a line? The __________________________ and the _______________________________. What is the Slope – Intercept Form of a line? ________________________________. What is the Slope Formula? __________________________________. Example 1: Write the equation of the line that passes through the points (1, 6) and (3, -4). What do we need to write the equation? _____________________ and ____________________ What can we find with two points? ____________________________________ Writing an equation of a line given two points Step 1 Find the Slope. Substitute the coordinates of the two given points into the formula for y y1 slope, m 2 . x 2 x1 Step 2 Find the y-intercept. Substitute the slope m and the coordinates of one of the points into the slope-intercept form, y = mx + b. Step 3 Write an equation of the line. Substitute the slope m and the y-intercept b into the slope-intercept form, y = mx + b. Geometry Mrs. Franks 10th Grade Another strategy for writing the equation of a line is __________________________________. Point – Slope Form of the equation of a line The point – slope form of the equation of the nonvertical line that passes through a given point ( x1 , y1 ) with a slope of m is y y1 m(x x 1 ) 2 Example 2: Write an equation of the line given the point (2, 5) and a slope m of . 3 Example 3: Write an equation of a line given the points (-2, 3) and (-1, 1). Geometry Mrs. Franks 10th Grade Another way to write the equation of line when given two points is to use your graphing Calculator. Let’s use the last example: Write an equation of a line given the points (-2, 3) and (-1, 1). 1. First hit the STAT button and then Edit. 5. Then GRAPH it. 2. Enter your x-values into L1 and your 6. Now we want the equation of the line. y-values into L2 Go back to STAT, CALC 7. We want the equation of a line #4. Then nd 3. Now hit 2 , Y = to get into STAT PLOT type L1, L2 , Y1 (Under VARS). and then ENTER 4. Turn on your STAT PLOT 8. Hit ENTER and your coefficients will appear and you can look at the graph to see your line.