Sec 2.1 Geometry Parallel Lines and Angles Name:

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2 PARALLEL LINES A D Sec 2.1 Geometry Parallel Lines and Angles Name: EXTERIOR INTERIOR 5 6 E 7 8 G 1 2 B 3 4 C F These symbols imply the two lines are parallel. EXTERIOR H WORD BANK: Alternate Interior Angles DEH HEF Corresponding Angles Supplementary GBC Consecutive Interior Angles 5 Transversal Congruent 4 Vertical Angles Alternate Exterior Angles Consecutive Exterior Angles ABG 1. Give an alternate name for angle 2 using 3 points: 2. Angles ABE and CBG can best be described as: 3. Angles 6 and 3 can best be described as: 4. The line GH can best be described as a: 5. Which angle corresponds to DEB : 6. Angles FEB and CBE can best be described as: 7. Angles 1 and 8 can best be described as: 8. Which angle is an alternate interior angle with CBE : 9. Angles GBC and BEF can best be described as: 10. Angles 2 and 8 can best be described as: 11. Which angle is an alternate exterior angle with ABG : 12. Which angle is a vertical angle to ABG : 13. Which angle can be described as consecutive exterior angle with 1 : 14. Any two angles that sum to 180 can be described as angles. M. Winking Unit 2-1 page 19

3 TRIANGLE s INTERIOR ANGLE SUM 1. a. First, Create a random triangle on a piece of patty papers. b. Using your pencil, write a number inside each interior angle a label c. Next, cut out the triangle d. Finally, tear off or cut each of the angles from the triangle e. Using tape, carefully put all 3 angles next to one another so that they all have the same vertex and the edges are touching but they aren t overlapping Paste or Tape your 3 vertices here: Common Vertex 2. What is the measure of a straight angle or the angle that creates a line by using two opposite rays from a common vertex? 3. Collectively does the sum of your 3 interior angles of a triangle form a straight angle? What about others in your class? 4. Make a conjecture about the sum of the interior angles of a triangle. Do you think your conjecture will always be true? (please explain using complete sentences) M. Winking Unit 2-1 page 20

4 5. More formally, why do the 3 interior angles of any triangle sum to 180? Consider ABC. The segment AB is extended into a line and a parallel line is constructed through the opposite vertex. So, AB CD. a. Why is 1 2? b. Why is 5 4? c. Why is m 2 + m 3 + m 4 = 180? d. Using substitution we can replace m 2 with m 1 and m 4 with m 5 to show that the interior angles of a triangle must always sum to 180. ( ) + m 3 + ( ) = 180 Write the angle number in the and then write the letter that corresponds with the number based on the code at the bottom in the box. 7. Angle 2 and Angle are alternate exterior angles. 8. Angle 7 and Angle are alternate exterior angles. 9. Angle 4 and Angle are corresponding angles. 10. Angle 5 and Angle are consecutive interior angles. 11. Angle 3 and Angle are alternate interior angles. 12. Angle 7 and Angle are consecutive exterior angles. 13. Angle 6 and Angle are vertical angles. 14. Angle 2 and Angle are a linear pair and on the same side of the transversal. 15. Angle 1 and Angle are corresponding angles. 1=D 2=U 3= L 4 = A 5 = N 6 = I 7 = E 8 = C What type of Geometry is this? M. Winking Unit 2-1 page 21

5 16. Given lines p and q are parallel, find the value of x that makes each diagram true. a. b. (x)º q (x)º q 140º p 155º p x = x = 17. Given lines p and q are parallel, find the value of x that makes each diagram true. a. b. (6x+5)º q (6x+5)º q (2x+45)º p (2x+15)º p x = x = 18. Given lines m and n are parallel, find the value y of that makes each diagram true. a. b. m 130º n 40º m 40º yº 130º yº n y = y = M. Winking Unit 2-1 page 22

6 19. ANGLE PUZZLE. Find m AEF Given: m DEF = 85 m ABG = 50 BAE is a right angle CGE and DEG are supplementary B G C D m AEF = A E F 20. Converse of AIA, AEA, CIA, CEA. Which sets of lines are parallel and explain why? q a. b. 54º 75º m r 105º n 28º t p c. d. 70º q j 120º g 120º p h r M. Winking Unit 2-1 page 23

7 Sec 2.2 Geometry - Constructions Name: 1. [COPY SEGMENT] Construct a segment with an endpoint of C and congruent to the segment AB. A B C **Using a ruler measure the two lengths to make sure they have the same measure. 2. [COPY ANGLE] Construct an angle with ray HI and congruent to the angle DEF Step I: Open the compass to any length and create an arc on the original angle with the needle at the vertex Step 2: Leave the compass set to the exact same length and create a new arc on the new ray with the needle at the end point of the ray. Step 3: Open the compass to the length of the intercepted arc in the original angle and draw a small arc to be sure the created arcs intersect on the ray of the angle. Step 4: Leave the compass open to the exact same length as it was from step 3. Then with the needle at the intersection of the second arc and the copied ray, create an arc that intersects the first arc Step 5: Finally, use a straight edge to draw a ray from the endpoint of the original bottom ray through the intersection of the two created arcs. D E F H I **Using a protractor measure the two angles to make sure they have the same measure. M. Winking Unit 2-2 page 24

8 3. [Perpendicular Bisector] Construct a perpendicular bisector to the segment. AB Step I: Open the compass to a length that is certainly larger than half of the segment length. Draw an arc with the needle at the left endpoint of the segment. Step 2: Leave the compass open to the same length from step 1 and create another arc only this time starting with the needle at the right endpoint. Step 3: Using a straight edge and a pencil, draw a line that passes through the intersection point of the two arcs. A B **Using a ruler measure the two halves of the segment to make sure they have the same measure. M. Winking Unit 2-2 page 25

9 4. [Angle Bisector] Construct an angle bisector of the angle DEF Step I: Open the compass to any length and create an arc with the needle of the compass at the vertex of the angle. Step 2: Open the compass so that the needle is on one intersection of the arc and the angle and the pencil is at the other intersection of the arc and the angle. Step 3: Create an arc with the opened length determined in step 2 towards the interior of the angle. Step 4: Move the compass so that the needle and pencil are on the opposite intersection points that they were on in step 2 and create another arc to the exterior of the angle. Step 5: Using a straight edge, draw ray from the vertex of the angle through were the two outer arcs intersect. D E F **Using a ruler measure the two halves of the segment to make sure they have the same measure. M. Winking Unit 2-2 page 26

10 5. [Perpendicular to a Line Through a Point] Construct a perpendicular line to AB through point C. Step I: Start by placing the needle on the point not on the line. Then, open your compass to a length such that when you draw an arc it will pass through the line twice and draw the arc. Step 2: Open the compass so that the needle is on one intersection of the arc and the pencil is on the other intersection of the arc. Then, create a arc above and below the line. Step 3: Opening the compass to the same length as the previous step and switching the needle and pencil to the opposite intersections of arcs that they were on in step 2, create another arc above and below the line. Step 3: Using a straight edge, draw a line passing through the intersection of the two most recent arc intersections. The line should incidentally also pass through the point not on the line. C A B M. Winking Unit 2-2 page 27

11 6. [Hexagon inscribed in a Circle] Construct a circle with radius AB and an inscribed regular hexagon. Step I: Start by placing the needle on the point that is to be the center of the circle and the pencil on the other endpoint of the radius. Create the entire circle with the compass. Step 2: With the compass still open to the exact length of the radius place the needle on the endpoint of the radius that is on the circle and with the pencil mark an intersection on the circle with a new arc. Step 3: Leaving the compass still open to the same length move the needle to the new intersection point and mark another intersection point on the circle again. Continue repeating this process until you have made it all the way around the circle. Step 4: The last mark should end right were you started with the needle. Then, connect each consecutive arc intersection with a segment. A B M. Winking Unit 2-2 page 28

12 7. [Triangle inscribed in a Circle] Construct a circle with radius AB and an inscribed regular triangle. Step I: Start by placing the needle on the point that is to be the center of the circle and the pencil on the other endpoint of the radius. Create the entire circle with the compass. Step 2: With the compass still open to the exact length of the radius place the needle on the endpoint of the radius that is on the circle and with the pencil mark an intersection on the circle with a new arc. Step 3: Leaving the compass still open to the same length move the needle to the new intersection point and mark another intersection point on the circle again. Continue repeating this process until you have made it all the way around the circle. Step 4: The last mark should end right were you started with the needle. Then, connect every other consecutive arc intersection with a segment. A B M. Winking Unit 2-2 page 29

13 8. [Square inscribed in a Circle] Construct a circle with radius AB and an inscribed square. Step I: Start by placing the needle on the point that is to be the center of the circle and the pencil on the other endpoint of the radius. Create the entire circle with the compass. Step 2: Line your straight edge up with the radius and extend the radius segment to create a diameter. Step 3: Create a perpendicular bisector of the newly created diameter (see previous construction #3 if needed) Step 4: Connect the each endpoint of the diameter with each endpoint of where the perpendicular bisector intersects the circle. A B M. Winking Unit 2-2 page 30

14 9. [Construct a Parallel Line given a point and a line] Construct a parallel line to AB through point C. Step I: Start drawing a general line that passes through point O and intersects line MN. Step 2: Open the compass so that the needle is on the intersection of the new line just created and line MN. Then, open the compass a little (it needs to be less than the distance to reach point O ). Create an arc as shown below. Step 3: Leave the compass set to the same opening and recreate another arc of the same radius but with the needle at point O. Step 4: Put the compass needle on the intersection of the transversal line and first arc that you created as shown below and open the pen to the intersection of the first arc and line MN. Create a small arc to verify the intersection of the arcs mark the correct intersection. Step 5: Leave the compass open to the same length as the previous step and put the compass needle on the intersection of the transversal line and the second arc that you created. Then, create an arc of the same radius to intersect the second arc created as shown below. Step 6: D C A B M. Winking Unit 2-2 page 31

15 Sec 2.3 Geometry Dilations Name: [Example Dilation]: Dilate the ABCD by a factor of 2. 0 from point E. Step I: Measure the distance from the point of dilation to a point to be dilated (preferably using centimeters). Step 2: Multiply the measured distance by the scale factor. 2. 5cm 2. 0 = 5 cm Step I: With the ruler in the same place as it was in step #1, mark a point at the measured distance determined in step #2 as the image of the original point. Repeat the process for all points that are to be dilated. Usually, denoted with the same letter but a single quote after it (referred to as a prime). 1. Dilate the ABC by a factor of 3 2 from point D. B D A C a. Measure the length of AB in centimeters to the nearest tenth. AB = b. Measure the length of A B in centimeters to the nearest tenth. A B = c. Determine the value of A B divided by AB. A B AB = d. What might you conclude about the scale factor and the ratio of dilated segment measure to its pre-image? e. Measure angle BAC and the angle B A C using a protractor. m BAC = m B A C = f. What might you conclude about each pair of corresponding angles? M. Winking Unit 2-3 page 32

16 2. Consider the following picture in which BCDE has been dilated from point A. a. What is the scale factor of the dilation based on the sides? b. What is the area of BCDE? c. What is the area of B C D E? d. What is the value of the area of B C D E divided by area of BCDE? e. What might you conclude about the ratio of two dilated shapes sides compared to the ratio of their areas? 3. Dilate the line AB by a factor of 0. 5 from point C. A B C How could you characterize the lines AB and A B? M. Winking Unit 2-3 page 33

17 4. Dilate the line AB by a factor of 2. 2 from point C. A C B How could you characterize the lines AB and A B? 5. Consider the following picture in which rectangular prism A has been dilated from point G. a. What is the scale factor of the dilation based on the sides? b. What is the volume of rectangular prism A? c. What is the volume of rectangular prism A? d. What is the value of the volume of prism A divided by volume of prism A? e. What might you conclude about the ratio of two dilated solids sides compared to the ratio of their volumes? M. Winking Unit 2-3 page 34

18 1. Plot the following points and connect the consecutive points. Coordinate Dilations Name: A(5,1) B(8,2) C(9,3) D(9,7) E(7,8) F(8,7) G(7,5) H(6,4) I(5,6) J(6,5) K(5,7) L(4,6) M(4,4) N(3,5) O(2,7) P(3,8) Q(1,7) R(1,3) S(2,2) A(5,1) 2. First use the dilation rule D:(x,y) (0.5x, 0.5y) using the points from the previous problem, plot the newly created points, and connect the consecutive points. A ( ) B ( ) C ( ) D ( ) E ( ) F ( ) G ( ) H ( ) I ( ) J ( ) K ( ) L ( ) M ( ) N ( ) O ( ) P ( ) Q ( ) R ( ) S ( ) A ( ) 3. What would happen with the rule: D:(x,y) (3x, 3y)? 4. What would happen with the rule: D:(x,y) (0.5x, 1y)? 5. What would happen with the rule: R:(x,y) ( 1x, 1y)? 6. What would happen with the rule: R:(x,y) (1x, 1y)? M. Winking Unit 2-3 page 35

19 6 in. 8 in. 7. Figure FCDE has been dilated to create to create F C D E. a. What is the dilation scale factor? b. What is the location of the center of dilation? c. What is the ratio of the areas? 8. Which point would be the center of dilation? A B C D 9. The different sizes of soft drink cups at a movie theater are created by using dilations. If the large is 8 inches tall and the medium is 6 inches tall, answer the following. a. What is the scale factor of the dilation from a large to a medium drink? b. What is the ratio of the volumes of the two drinks? c. If the large holds 30 ounces, how much does the medium hold? M. Winking Unit 2-3 page 36

20 Sec 2.4 Geometry Similar Figures Name: Two figures are considered to be SIMILAR if the two figures have the same shape but may differ in size. To be similar by definition, all corresponding sides have the same ratio OR all corresponding angles are congruent. Alternately, if one figure can be considered a transformation (rotating, reflection, translation, or dilation) of the other then they are also similar. Two triangles are similar if one of the following is true: 1) (AA) Two corresponding pairs of angles are congruent. ABC ~ FDE Notice that in the similarity statement above that corresponding angles must match up. 2) (SSS) Each pair of corresponding sides has the same ratio. ABC ~ DEF EF = DE = FD = 3 = 1. 5 BC AB CB 2 3) (SAS) Two pairs of corresponding sides have the same ratio and the angle between the two corresponding pairs the angle is congruent. FE = ED = 2 AB BC ABC FED ABC ~ FED Determine whether the following figures are similar. If so, write the similarity ratio and a similarity statement. If not, explain why not M. Winking Unit 2-4 page 37

21 Assuming the following figures are similar use the properties of similar figures to find the unknown y = 6. n = x = 2x 4 x x = 8. t = 3 x 9 8 t Given the similarity statement ABC ~ DEF and the following measures, find the requested measures. It may help to draw a picture. AB = 8 DE = 20 m ABC = 40 AC = 10 EF = 30 m EFD = 30 a. Find the measure of DF = b. Find the measure of BC = c. Find the measure of m DEF = d. Find the measure of m BCA = e. Find the measure of m CAB = f. Which angles are ACUTE? g. Which angles are OBTUSE? M. Winking Unit 2-4 page 38

22 16. Explain why the reason the triangles are similar and find the measure of the requested side. A) B) Triangle mid-segment 10 What is the measure of TU? What is the measure of AU? 17. Using (SSS, AA, SAS) which triangles can you determine must be similar? (explain why) A) B) C) E circle one Similar? YES NO SSS AA SAS circle one Similar? YES NO SSS AA SAS circle one Similar? YES NO SSS AA SAS D) E) circle one Similar? YES NO SSS AA SAS circle one Similar? YES NO SSS AA SAS M. Winking Unit 2-4 page 39

23 18. Using some type of similar figure find the unknown lengths. O Z A. B. C. 3 cm x N P c Y V E 5 cm 6 cm M Q X T 22a. x = 22b. x = 22c. y = C. D. 6 cm HINT: 22c. x = 22d. x = M. Winking Unit 2-4 page 40

24 Prove the Pythagorean Theorem (a 2 + b 2 = c 2 ) using similar right triangles: M. Winking Unit 2-4 page 41

25 62 in. 35 f t 10. Thales was one of the first to see the power of the property of ratios and similar figures. He realized that he could use this property to measure heights and distances over immeasurable surfaces. Once, he was asked by a great Egyptian Pharaoh if he knew of a way to measure the height of the Great Pyramids. He looked at the Sun, the shadow that the pyramid cast, and his 6 foot staff. By the drawing below can you figure out how he found the height of the pyramid? Height of Pyramid: 11. Using similar devices he was able to measure ships distances off shore. This proved to be a great advantage in war at the time. How far from the shore is the ship in the diagram? 3 ft 5 ft Distance from Shore: 12. Using a mirror you can also create similar triangles (Thanks to the properties of reflection similar triangles are created). Can you find the height of the flag pole? Height of the Flag Pole: 21 in. mirror 72 in. 13. Use your knowledge of special right triangles to measure something that would otherwise be immeasurable. M. Winking Unit 2-4 page 42

26 14. Find the unknown area based on the pictures below.?? in 2 36 in 2 Scale Length Area Factors 5 in. 15 in. Area of the small square: 15. If the small can holds 20 gallons how much will the big trashcan hold (assuming they are similar shapes) Scale Similarity Length Area Volume Factors Ratios 4 ft 2 ft Volume of the Large Trashcan: 16. If the smaller spray bottle holds 37 fl. oz., then how much does the larger one hold assuming they are similar shapes? Scale Similarity Length Area Volume Factors Ratios Volume of the Large Spray Bottle: 2 in. 3 in The smaller of the two cars is a Matchbox car set at the usual th 64 scale (the length) and it takes fluid ounces to paint the car. If the smaller is a perfect scale of the actual car and the ratios of the paint remains the same then how many gallons of paint will be needed for the real car? (128 fl. oz = 1 gallon) Scale Similarity Length Area Volume Factors Ratios 2 in. 128 in. Amount of Paint Needed: M. Winking Unit 2-4 page 43

27 18. If the following are similar determine the length of the unknown sides. A. 18a. x = B. 18 b. V = C. Volume = 4 cm 3 18c. x = Volume = 20 cm 3 D. Volume = 90 cm 3 Volume =??cm 3 18d. V = M. Winking Unit 2-4 page 44

28 Sec 2.5 Geometry Congruence Name: Any two congruent figures can be mapped onto one another using a series of rigid or isometric transformation (reflections, rotations, and translations). See GSP Lab (Transformations) Each of the following pairs of figures shown below are congruent. Write a congruence statement for each and tell whether or not a reflection would be needed to map the pre-image onto the image Given the following congruencies find the requested unknown angle. 5. ABC ONM 6. GEM TOR (3x+20) 50 (3x) m MNO = m GEM = M. Winking Unit 2-5 page 45

29 Given the following congruencies find the requested unknown side. 7. TRI ANG 8. MTH FUN 17 8 GN = UF = The following pairs of triangles are congruent. Provide a suggested transformation or series of transformations that can map one triangle onto the other congruent triangle. (In each diagram ABC DEF ) Circle which transformation(s) could be used to map ABC onto DEF. Translation Reflection Rotation Dilation Circle which transformation(s) could be used to map ABC onto DEF. Translation Reflection Rotation Dilation Circle which transformation(s) could be used to map ABC onto DEF. Translation Reflection Rotation Dilation M. Winking Unit 2-5 page 46 Circle which transformation(s) could be used to map ABC onto DEF. Translation Reflection Rotation Dilation

30 Angle Puzzles. (angles are not drawn to scale) 9. Find mm BBBBBB 10. Find mm GGGGGG Given: AAAA bisects DDDDDD mm DDDDDD = 50 AAAAAA is a right angle Given: GGGG bisects FFFFFF mm FFFFFF = 60 mm GGGGGG = 75 FFFFFF is a right angle B E C F G A D H I mm BBBBBB = mm GGGGGG = M. Winking Unit 2-5 page 47

31 Sec 2.6 Geometry Triangle Proofs Name: COMMON POTENTIAL REASONS FOR PROOFS Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Definition of Midpoint: The point that divides a segment into two congruent segments. Definition of Angle Bisector: The ray that divides an angle into two congruent angles. Definition of Perpendicular Lines: Lines that intersect to form right angles or 90 Definition of Supplementary Angles: Any two angles that have a sum of 180 Definition of a Straight Line: An undefined term in geometry, a line is a straight path that has no thickness and extends forever. It also forms a straight angle which measures 180 Reflexive Property of Equality: any measure is equal to itself (a = a) Reflexive Property of Congruence: any figure is congruent to itself (FFFFFFFFFFFF AA FFFFFFFFFFFF AA) Addition Property of Equality: if a = b, then a + c = b + c Subtraction Property of Equality: if a = b, then a c = b c Multiplication Property of Equality: if a = b, then ac = bc Division Property of Equality: if a = b, then a = b c c Transitive Property: if a = b & b =c then a = c OR if a b & b c then a c. Segment Addition Postulate: If point B is between Point A and C then AB + BC = AC Angle Addition Postulate: If point S is in the interior of PQR, then m PQS + m SQR = m PQR Side Side Side Postulate (SSS) : If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Side Angle Side Postulate (SAS): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Angle Side Angle Postulate (ASA): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Angle Angle Side Postulate (AAS) : If two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the triangles are congruent Hypotenuse Leg Postulate (HL): If a hypotenuse and a leg of one right triangle are congruent to a hypotenuse and a leg of another right triangle, then the triangles are congruent Right Angle Theorem (R.A.T.): All right angles are congruent. Vertical Angle Theorem (V.A.T.): Vertical angles are congruent. Triangle Sum Theorem: The three angles of a triangle sum to 180 Linear Pair Theorem: If two angles form a linear pair then they are adjacent and are supplementary. Third Angle Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. Alternate Interior Angle Theorem (and converse): Alternate interior angles are congruent if and only if the transversal that passes through two lines that are parallel. Alternate Exterior Angle Theorem (and converse): Alternate exterior angles are congruent if and only if the transversal that passes through two lines that are parallel. Corresponding Angle Theorem (and converse) : Corresponding angles are congruent if and only if the transversal that passes through two lines that are parallel. Same-Side Interior Angles Theorem (and converse) : Same Side Interior Angles are supplementary if and only if the transversal that passes through two lines that are parallel. Pythagorean Theorem (and converse): A triangle is right triangle if and only if the given the length of the legs a and b and hypotenuse c have the relationship a 2 +b 2 = c 2 Isosceles Triangle Theorem (and converse): A triangle is isosceles if and only if its base angles are congruent. Triangle Mid-segment Theorem: A mid-segment of a triangle is parallel to a side of the triangle, and its length is half the length of that side. CPCTC: Corresponding Parts of Congruent Triangles are Congruent by definition of congruence.

32 1. Tell which of the following triangle provide enough information to show that they must be congruent. If they are congruent, state which theorem suggests they are congruent (SAS, ASA, SSS, AAS, HL) and write a congruence statement. Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information Circle one of the following: SSS SAS ASA AAS HL Congruence Statement if necessary: Not Enough Information M. Winking Unit 2-6 page 49

33 2. Prove which of the following triangles congruent if possible by filling in the missing blanks: a. Given CCCC AAAA and CCCC AAAA Statements 1. CCCC AAAA Reasons 2. CCCC AAAA 3. CCCCCC AAAAAA 4. BBBB BBBB 5. BBBBBB DDDDDD b. Given PPPP YYYY and Point A is the midpoint of PPPP Statements 1. Given Reasons 2. Given 3. Definition of Midpoint 4. Reflexive property of congruence 5. By steps 1,3,4 and SSS c. Given VVVV RRRR and PPPP VVVV Statements 1. Given Reasons 2. Given PPPPPP EEEEEE M. Winking Unit 2-6 page 50

34 Prove the Isosceles Triangle Theorem and the rest of the suggested proofs. d. Given WWWWWW is isosceles and point R is the midpoint of WWWW Statements 1. WWWWWW is isosceles Reasons 2. WWWW KKKK 3. R is the midpoint of WWWW 4. WWWW KKKK 5. OOOO OOOO 6. WWWWWW KKKKKK 7. OOOOOO OOOOOO e. Given point I is the midpoint of XXXX and point I is the midpoint of AAAA Statements 1. I is the midpoint of XXXX Reasons 2. Definition of Midpoint 3. I is the midpoint of AAAA AAAAAA OOOOOO 6. AAAAAA OOOOOO f. Given MMMMMM and HHHHHH are right angles and MMMM TTTT Statements 1. MMMMMM & HHHHHH are right angles 2. MMMM TTTT 3. MMMM MMMM Reasons 4. MMMMMM HHHHHH M. Winking Unit 2-6 page 51

35 Prove the suggested proofs by filling in the missing blanks. g. Given GGGG PPPP and GGGG CCCC Statements 1. GGGG PPPP Reasons GGGG CCCC GGGGGG PPPPPP h. Given RRRR HHHH, TTTTTT MMMMMM and TTTT MMMM Statements 1. HHHH OOOO Reasons 2. RRRR = HHHH 3. RRRR + OOOO = HHHH + EEEE 4. RRRR + OOOO = RRRR and HHHH + EEEE = HHHH 5. RRRR = HHHH Substitution Property 6. RRRR OOOO 7. TTTT MMMM 8. MMMMMM TTTTTT 9. TTTTTT MMMMMM 10. TTTTTT MMMMMM M. Winking Unit 2-6 page 52

36 Prove the suggested proofs by filling in the missing blanks. i. Given HHHHHH TTTTTT, TTTTTT HHHHHH, and Point A is the midpoint of EEEE ii. Statements 1. TTTTTT HHHHHH Reasons 2. HHHHHH is an isosceles triangle 3. HHHH TTTT 4. A is the midpoint of EEEE 5. EEEE AAAA 6. TTTTTT HHHHHH 7. TTTTTT HHHHHH j. Given that AAAA bisects NNNNNN. Also, AAAA and AAAA are radii of the same circle with center A. Statements 1. AAAA bisects NNNNNN Reasons 2. Definition of Angle Bisector Radii of the same circle are congruent. Reflexive property of congruence 5. AAAAAA AAAAAA M. Winking Unit 2-6 page 53

37 Prove the suggested proofs by filling in the missing blanks. i. Given: AAAA GGGG ΔΔΔΔΔΔΔΔ forms an isosceles triangle with base CCCC Prove: AAAAAA GGGGGG Statements AAAA GGGG Reasons 4. CCCCCC is an isosceles triangle 5. Isosceles Triangle Theorem Prove the suggested proofs by filling in the missing blanks. k. Given: The circle has a center at point C ΔΔΔΔΔΔΔΔ forms an isosceles triangle with base AAAA Prove: AAAAAA DDDDDD Statements 1. AAAAAA is an isosceles triangle w/ base AAAA 2. AAAA DDDD 3. The circle is centered at point C Reasons 4. AAAA DDDD 5. Reflexive Property of Congruence 6. AAAAAA DDDDDD M. Winking Unit 2-6 page 54

38 Sec 2.7 Geometry Locus of Points & Triangle Centers Name: 1. Anna is stuck out on a swimming dock in the lake. If the picture below is a scale drawing, how far must she swim to get to shore (convert using the scale provided)? SCALE 10 ft 2. Measure the distance (cm) from the point to the line in each of the following problems a. b. 3. Describe how you attempted to measure the distance from a point to a line?? Locus of Points. 1. Mrs. and Mr. Scitamehtam were looking to purchase a new house. The map below shows where both of the two work. They both wish to be completely fair to each other in selecting a new house that is equal in distance from each of their respective places of work. Using a ruler or other construction utility can you show all such possible locations. Mrs. Scitamehtam s place of employment Mr. Scitamehtam s place of employment M. Winking Unit 2-7 page 55

39 DEFINITIONS: 1. MEDIAN of a triangle: All triangle s have 3 medians. Each median is a segment with one endpoint on the midpoint of a side of a triangle and the other endpoint at the opposite vertex of the triangle. 2. ALTITUDE of a triangle: All triangle s have 3 altitudes. Each altitude is a line that passes through a vertex of the triangle and is perpendicular to the opposite side. CONSTRUCTIONS: 1. Construct all 3 medians of the triangle below using either a straight edge & compass or a MIRA 2. Construct all 3 altitudes of the triangle below using either a compass compass and straight edge or a mira M. Winking Unit 2-7 page 56

40 3. Construct all 3 perpendicular bisectors of each side of the triangle below using either a compass compass and straight edge or a mira 4. Construct all 3 angle bisectors of the triangle below using either a compass and straight edge or a mira Each one of these constructions should create a common center of a triangle. 1. Circumcenter: Is the center of a circle that perfectly passes through each vertex. 2. Incenter: Is the center of largest possible circle that still completely fits inside the circle. 3. Centroid: Is the center that is the center of gravity of the triangle (balancing point) and there is a constant ratio between the distance from the centroid to the midpoint and centroid to the vertex. 4. Orthocenter: Is the fourth common center but has no unique properties other than it is on the EULER line. CAN YOU GUESS WHICH CENTER IS WHICH? (Hint: LOOK at the BOLD letters in each prompt. M. Winking Unit 2-7 page 57

41 Additional information about the Common Triangle Centers. It is the CENTROID created by the MEDIANS. It is the INCENTER created by the ANGLE BISECTORS p.34 If you wanted to balance a cut out of this triangle on one finger, you would hold the triangle at the centroid. It represents the center of gravity. It is the point that all sides are equidistant from and the center of a circle that is circumscribed by the triangle. It is the ORTHOCENTER and it is created by the ALTITUDES of the triangle. The only additional interesting fact about the orthocenter is that is a point on Euler s Line (the line also includes the Centroid and Circumcenter. M. Winking Unit 2-7 page 58 It is the CIRCUMCENTER and it is created by the PERPENDICULAR BISECTORS It is the point that is equidistant from all of the triangle s vertices and the center of a circle that circumscribes the triangle.

42 There are some interesting uses of some of the common triangle centers. Identify which center would be best for each situation. 1. A person wishes to make a triangular table with a single post holding up the table. Which triangle center should the post connect to that would help keep the table the most balanced and stable? 2. A person has a triangular piece of scrap ornate fabric. The person wants to create a circular table cloth. Which center might help her cut out the biggest possible circle from the fabric? 3. The space shuttle is monitoring 3 GPS satellites and has positioned itself so that it is equidistant from each of the satellites. Which triangle center might have helped the pilot determine this particular location? M. Winking Unit 2-7 page 59

43 Sec 2.8 Geometry Polygons & Quadrilaterals Name: Polygon: A closed plane figure formed by three or more segments such that each segment intersects or connects end to end to form a closed shape. Simple Polygon: A polygon in which sides only share each endpoint with one other side. Regular Polygon: A polygon that is both equilateral and equiangular Determine whether each figure below is a polygon or not a polygon Circle one of the following: It is a Not a Polygon Polygon Name if Polygon: Circle one of the following: It is a Not a Polygon Polygon Name if Polygon: Circle one of the following: It is a Not a Polygon Polygon Name if Polygon: Circle one of the following: It is a Not a Polygon Polygon Name if Polygon: Circle one of the following: It is a Not a Polygon Polygon Name if Polygon: Circle one of the following: It is a Not a Polygon Polygon Name if Polygon: Concave Polygon: A polygon in which a diagonal can be drawn such that part of one of the diagonals contains point in the exterior of the polygon. Convex Polygon: A polygon in which no diagonal contains points in the exterior of the polygon. Determine whether each figure below is a Convex or Concave Circle one of the following: CONVEX CONCAVE Circle one of the following: CONVEX CONCAVE Circle one of the following: CONVEX CONCAVE M. Winking Unit 2-8 page 60

44 Quadrilateral: A four-sided polygon. Parallelogram: A quadrilateral with two pairs of parallel sides. Trapezoid: A quadrilateral with exactly one pair of parallel sides. Rectangle: A quadrilateral with four right angles. Rhombus: A quadrilateral with four congruent sides. Square: A quadrilateral with four congruent sides and four right angles. Kite: A quadrilateral with exactly two pairs of congruent consecutive sides but opposite sides are not congruent. The Hierarchy of Quadrilaterals Convex Quadrilateral Parallelograms Trapezoids. Concave Quadrilateral Rhombi. Squares. Rectangles. Kites. Answer each of the following with ALWAYS, SOMETIMES, or NEVER A square is a rhombus. A rhombus is a square. A trapezoid is a parallelogram. A rectangle is a rhombus. A kite is a concave quadrilateral. A parallelogram is a rectangle. A rhombus is a trapezoid. A convex quadrilateral is a trapezoid. A rectangle is a parallelogram. M. Winking Unit 2-8 page 61

45 Using your knowledge of congruent triangles and parallel triangles determine the highest level and most appropriate definition of a quadrilateral for each ABCD quadrilateral below. Each definition in the word wall should be used exactly once. Briefly explain why for each quadrilateral. Square Rhombus Kite Convex-Quadrilateral Rectangle Parallelogram Concave-Quadrilateral Trapezoid Please assume that different congruent marks represent incongruent angles Both dashed figures can be assumed to be circles. 5. AAAAAA and CCCCCC 6. are complimentary M. Winking Unit 2-8 page 62

46 Sec 2.9 Geometry Polygon Angles Name: The Asimov Museum has contracted with a company that provides Robotic Security Squads to guard the exhibits during the hours the museum is closed. The robots are designed to patrol the hallways around the exhibits and are equipped with cameras and sensors that detect motion. Each robot is assigned to patrol the area around a specific exhibit. They are designed to maintain a consistent distance from the wall of the exhibits. Since the shape of the exhibits change over time, the museum staff must program the robots to turn the corners of the exhibit. Below, you will find a map of the museum's current exhibits and the path to be followed by the robot. One robot is assigned to patrol each exhibit. When a robot reaches a corner, it will stop, turn through a programmed angle, and then continue its patrol. Your job is to determine the angles that R1, R2, R3, and R4 will need to turn as they patrol their area. Keep in mind the direction in which the robot is traveling and make sure it always faces forward as it moves around the exhibits. Robot 4 Robot 1 Robot 2 Robot 3 Try measuring the angles on the following pages using a protractor or using the Geometer s Sketchpad animation found at Original Task developed by Janet Davis at Georgia Department of Education 2016 Modified, Updated, and Edited by M. Winking mmwinking@gmail.com p.63

47 1. What angles will Robot 1 need to turn? What is the total of these turns? Exterior Angle at Vertex A : Exterior Angle at Vertex B : Exterior Angle at Vertex C : Exterior Angle at Vertex D : + TOTAL: 2. What angles will Robot 2 need to turn? What is the total of these turns? Exterior Angle at Vertex E : Exterior Angle at Vertex F : Exterior Angle at Vertex G : Exterior Angle at Vertex H : Exterior Angle at Vertex I : + Original Task developed by Janet Davis at Georgia Department of Education 2016 Modified, Updated, and Edited by M. Winking mmwinking@gmail.com p.64 TOTAL:

48 3. What angles will Robot 3 need to turn? What is the total of these turns? Exterior Angle at Vertex J : Exterior Angle at Vertex K : Exterior Angle at Vertex L : Exterior Angle at Vertex M : Exterior Angle at Vertex N : Exterior Angle at Vertex O : + TOTAL: 4. What angles will Robot 4 need to turn? What is the total of these turns? Exterior Angle at Vertex P : Exterior Angle at Vertex Q : Exterior Angle at Vertex R : Exterior Angle at Vertex S : Exterior Angle at Vertex T : Exterior Angle at Vertex U : Exterior Angle at Vertex V : + TOTAL: Or Original Task developed by Janet Davis at Georgia Department of Education 2016 Modified, Updated, and Edited by M. Winking mmwinking@gmail.com p.65

49 5. What do you notice about the sum of the angles? Do you think this will always be true? (please explain using complete sentences) 6. The museum also requested a captain robot, CR, to patrol the entire exhibit area. What angles will CR need to turn? What is the total of these turns? Exterior Angle at Vertex A : Exterior Angle at Vertex B : Exterior Angle at Vertex C : Exterior Angle at Vertex D : Exterior Angle at Vertex E : Exterior Angle at Vertex F : Exterior Angle at Vertex G : Exterior Angle at Vertex H : + TOTAL: Original Task developed by Janet Davis at Georgia Department of Education 2016 Modified, Updated, and Edited by M. Winking mmwinking@gmail.com p.66

50 7. What makes Captain Robot s path different from the other robot s paths (other than the number of sides being different)? (please explain using complete sentences) (RECALL FROM EARLIER IN THIS UNIT) 8. a. Create a random triangle on a piece of patty papers. b. Write a number inside each interior angle c. Cut out the triangle d. Tear off or cut each of the angles from the triangle e. Paste all 3 angles next to one another so that they all have the same vertex and the edges are touching but they aren t overlapping Paste or Tape your 3 vertices here: Common Vertex 9. What is the measure of a straight angle or the angle that creates a line by using two opposite rays from a common vertex? 10. Collectively does the sum of your 3 interior angles of a triangle form a straight angle? What about others in your class? 11. Make a conjecture about the sum of the interior angles of a triangle. Do you think your conjecture will always be true? (please explain using complete sentences) Original Task developed by Janet Davis at Georgia Department of Education 2016 Modified, Updated, and Edited by M. Winking mmwinking@gmail.com p.67

51 12. Determine the measure of the interior angles of the polygons of the polygons formed by Exhibits A through D. Exhibit A (Quadrilateral) Exhibit B (Pentagon) Exhibit C (Hexagon) Exhibit D (Heptagon) Interior Angle at Vertex A : Interior Angle at Vertex B : Interior Angle at Vertex C : Interior Angle at Vertex D : TOTAL: + Interior Angle at Vertex E : Interior Angle at Vertex F : Interior Angle at Vertex G : Interior Angle at Vertex H : Interior Angle at Vertex I : TOTAL: + Interior Angle at Vertex J : Interior Angle at Vertex K : Interior Angle at Vertex L : Interior Angle at Vertex M : Interior Angle at Vertex N Interior Angle at Vertex O : TOTAL: + Interior Angle at Vertex P : Interior Angle at Vertex Q : Interior Angle at Vertex R : Interior Angle at Vertex S : Interior Angle at Vertex T : Interior Angle at Vertex U Interior Angle at Vertex V : Look at the sums of the angles in each of the polygons, do you notice a pattern? Can you briefly describe the pattern? TOTAL: Original Task developed by Janet Davis at Georgia Department of Education 2016 Modified, Updated, and Edited by M. Winking mmwinking@gmail.com p.68

52 14. Using your hypothesis from the previous problem can you determine the sum of the interior angles of the following: A. Octagon (8-Sides) B. Dodecagon (12-sides) C. Icosagon (20 sides) D. A polygon with n-sides 15. Using the information from above what would be the measure of a single interior angle each were regular polygons? A. Regular Octagon B. Regular Dodecagon C. Regular Icosagon D. A regular polygon with n-sides 16. The museum intends to create regular polygons for its 6 th exhibition. For regular polygons the robot will make the same turn at each vertex. Can you determine the angle of each turn for the robot for a regular Pentagon? a. A regular Pentagon? b. A regular Hexagon? c. A regular Nonagon? 17. A sixth exhibit was added to the museum. The robot patrolling this exhibit will make 15º turns at each vertex. How many sides must the exhibit have and what is the name of the polygon? What makes it possible for the robot to make the same turn each time (what type of polygon must the exhibit be)? 18. Find the value of x in each diagram below. a. b. c. Original Task developed by Janet Davis at Georgia Department of Education 2016 Modified, Updated, and Edited by M. Winking mmwinking@gmail.com p.69

53 Sec 2.10 Geometry Quadrilateral Properties Name: Name all of the properties of a parallelogram and its diagonals. 1. Opposite Sides are parallel 2. Opposite Sides are congruent 3. Opposite Angles are congruent 4. Consecutive Angles are supplementary 5. Diagonals bisect each other The properties of a rectangle and its diagonals: 1. All angles are right 2. Opposite Sides are parallel 3. Opposite Sides are congruent 4. Diagonals bisect each other 5. Diagonals are congruent The properties of a rhombus and its diagonals: 1. Opposite angles are congruent 2. Opposite Sides are parallel 3. All Sides are congruent 4. Consecutive angles are supplementary 5. Diagonals are perpendicular 6. Diagonals bisect interior angles. The properties of a square and its diagonals: 1. All angles are right 2. Opposite Sides are parallel 3. All Sides are congruent 4. Diagonals bisect each other 5. Diagonals are perpendicular 6. Diagonals bisect angles 7. Diagonals are congruent M. Winking Unit 2-10 page 70

54 Find the value of x in each diagram below using properties of quadrilaterals (8x +5 ) (3x + 10) 1. x = 2. x = x 50 AC = 9x + 4 BD = 6x x = 4. x = x x 2x 8 5. x = 6. x = M. Winking Unit 2-10 page 71

55 Plot points A(-3, -1), B(-1, 2), C(4, 2), and D(2, -1). 1. What specialized geometric figure is quadrilateral ABCD? Support your answer mathematically. 2. Draw the diagonals of ABCD. Find the coordinates of the midpoint of each diagonal. What do you notice? 3. Find the slopes of the diagonals of ABCD. What do you notice? 4. The diagonals of ABCD create four small triangles. Are any of these triangles congruent to any of the others? Why or why not? Plot points E(1, 2), F(2, 5), G(4, 3) and H(5, 6). 5. What specialized geometric figure is quadrilateral EFHG? Support your answer mathematically using two different methods. 6. Draw the diagonals of EFHG. Find the coordinates of the midpoint of each diagonal. What do you notice? 7. Find the slopes of the diagonals of EFHG. What do you notice? 8. The diagonals of EFHG create four small triangles. Are any of these triangles congruent to any of the others? Why or why not? M. Winking Unit 2-10 page 72

56 Plot points P(4, 1), W(-2, 3), M(2,-5), and K(-6, -4). 9. What specialized geometric figure is quadrilateral PWKM? Support your answer mathematically. 10. Draw the diagonals of PWKM. Find the coordinates of the midpoint of each diagonal. What do you notice? 11. Find the lengths of the diagonals of PWKM. What do you notice? 12. Find the slopes of the diagonals of PWKM. What do you notice? 13. The diagonals of ABCD create four small triangles. Are any of these triangles congruent to any of the others? Why or why not? Plot points A(1, 0), B(-1, 2), and C(2, 5). 14. Find the coordinates of a fourth point D that would make ABCD a rectangle. Justify that ABCD is a rectangle. 15. Find the coordinates of a fourth point D that would make ABCD a parallelogram that is not also a rectangle. Justify that ABCD is a parallelogram but is not a rectangle. M. Winking Unit 2-10 page 73