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Geometry Unit 11 Quadrilaterals Notes Packet The Quadrilateral Family Tree Quadrilaterals - Four angles - All four-sided shapes - Sum of the angles is 360° Trapezoid Kite Parallelogram - Only 2 parallel sides - Opposite sides parallel - Opposite sides congruent - Opposite angles congruent - Diagonals bisect each other - Consecutive angles supp. - 2 pairs adjacent sides congruent - Diagonals perpendicular - Angles between non-congruent sides are congruent Rhombus Rectangle - A parallelogram - Four right angles - Diagonals are congruent - A parallelogram - Equilateral sides - Diagonals bisect angles - Diagonals perpendicular Isosceles Trapezoid - Non-parallel sides (legs) are congruent - Diagonals are congruent - Base angles are congruent - Legs are congruent - All properties of a rectangle - All properties of a rhombus 3/9/2015 8:22 PM Square 1 Coordinate Geometry Reference Sheet: Important Points: 1. IT IS NOT ENOUGH to just draw a graph!! You MUST use slope, midpoint, or distance formulas to receive full (any!) credit for the problem! 2. Always write the formulas before plugging in numbers. 3. Be extremely neat and organized when showing your work. 4. Write a concluding statement (sentence) at the end of the proof. Graphing Instructions: 1. 2. 3. 4. Always use graph paper. Always label your axes, scale, equations (if any), and the coordinates of the points plotted. Always use a straightedge. Always use pencil. Formulas: Name: Formula: Slope m y2 y1 x2 x1 What it finds: How its used in proofs: The slope of a line 1. To prove two lines parallel (Show 2 equal slopes) 2. To prove two lines perpendicular (Show 2 slopes that are negative reciprocals) Midpoint d x2 x1 y2 y 1 2 x x y y midpt 1 2 , 1 2 2 2 2 The length of a line segment The midpoint of a line segment 2. To prove lines are not congruent (Show 2 unequal distances) 1. To prove two line segments bisect each other (Show they have the same midpoint, 2 equal midpoints) 3/9/2015 8:22 PM Distance 1. To prove two lines congruent (Show 2 equal distances) 2 Coordinate Geometry Explanations HOW to prove a… 1. is isosceles a.) prove only 2 sides (distance 3x) 2. is a right a.) Find the lengths of all three sides (distance 3x) and show a2+b2=c2 … OR b.) Find the slopes of the 2 sides that look and show they are negative reciprocals forming a right angle* (slopes 2x – easiest of these!) 3. quad is a a.) opposite sides (distance 4x) b.) opposite sides (slopes 4x) c.) one pair of opposite sides and (distance 2x & slopes 2x ON THE SAME SIDES) d.) diagonals bisect each other* (midpoint 2x – for diags – easiest of these!) e.) opposite angles congruent (not used in a coordinate proof) 4. quad is a rectangle a.) it has 4 right angles* (slopes 4x – easiest of these!) b.) if it is a (do ONE of the above for ) AND has 1 right angle (slopes 2x) *also easy – do 4 slopes, show opposite sides are || and 2 are negative recips c.) if it is a (do ONE of the above for ) AND diagonals (distances 2x) 5. quad is a rhombus a.) it has 4 sides* (distances 4x – easiest of these!) b.) if it is a (do ONE of the above for ) AND diagonals are (slopes 2x) c.) if it is a (do ONE of the above for ) AND 2 adjacent sides (dist 2x) 6. quad is a square a.) if it has 4 sides and 1 right angle (distance 4x & slopes 2x) b.) if it has 4 right angles and 2 adjacent sides (slopes 4x, distance 2x) c.) if it has diagonals that bisect each other (makes it a ) and are (makes it a rectangle) and are (makes it a rhombus) (midpoint 2x, distance 2x, slopes 2x,) (diags bisect each other, are congruent, and are ) d.) if it has four right angles and diagonals (slopes 6x) e.) if it has four congruent sides and congruent diagonals* (distances 6x – easiest) 7. quad is a trapezoid a.) if only one pair of opposite sides are 8. quad is isos. trap. a.) if only one pair of opposite sides are AND the 2 non-parallel sides are (slopes 4x (sides) & distance 2x (non-|| sides)) b.) if only one pair of opposite sides are AND the 2 diagonals are (slopes 4x (sides) & distance 2x (diagonals)) (slopes 4x, 2 will be =, 2 will be ) ***Include phrases like: “ = slopes → || lines ” “ negative reciprocal slopes → lines ” “ slopes → non-|| lines ” “ diags have same midpt → bisect each other ” 3/9/2015 8:22 PM ** For all proofs above, give a sentence to “explain” your work (from the list above!) 3 Geometry Unit 11 Quadrilaterals Interior and Exterior Angles of Polygons Warm-up: CW 11.1 HW 11.1 on pages: 8 & 9 The angle measures of a quadrilateral are x 5 , 2 x 10 , 2 x 4 , and 3x 1 . Solve for x. Explore: Complete the chart – draw a conclusion # of Number Sum of Interior Polygon of Sides Triangles Angles 1 Interior Angle 1 Exterior Angle Sum of Exterior Angles Triangle Quadrilateral Pentagon Hexagon Conclusions: # s = ______________ ___________ ____________ ____________ 3/9/2015 8:22 PM Octagon 4 Definitions: 1. Convex: polygon where every interior angle is _________ Diagram: 2. Concave: polygon that has at least 3. Regular: polygon with equal _____________ and equal _____________ interior angle _________ Diagram: Theorems: (where n is the number of sides) FORMULA: WHAT IT FINDS / DIAGRAM: Sum of the interior angles of a polygon v w x y z The measure of a specific interior angle of a regular polygon (finds one of these) Sum of the exterior angles of ANY polygon Exterior Angles add to __________ 1 Exterior Angle:_______________ v w x y z 360 u v w x y z 360 ... because they are a linear pair! An interior + its exterior = ________ Example: Find the sum of the measures of the interior angles of a convex octagon. 3/9/2015 8:22 PM x y 180 5 Practice: Find the sum of the measures of the interior angles of the indicated convex polygon. __________1. 13-gon __________2. 18-gon __________3. 25-gon __________4. 34-gon The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon by the number of sides. _________5. 3060 __________6. 1260° __________7. 3240° __________8. 7560° Find the measure of ONE exterior angle of each regular polygon. __________9. Decagon __________10. 20-gon __________11. 72-gon __________12. 15-gon __________13. 168 _________14. 174 __________ _________ __________15. 135 _________16. 140 __________ _________ 3/9/2015 8:22 PM If an interior angle of a polygon is given, find the exterior angle and find the number of sides: 6 17. What is the value of the variables in the diagram shown? z: ______ x: ______ v: ______ w: ______ 18. Error Analysis/Reasoning: Your friend says she has another way to find the sum of the interior angle measures of a polygon. She picks a point inside the polygon, draws a segment to each vertex, and counts the number of triangles. She multiplies the total by 180, and then subtracts 360 from the product. Does her method work? Explain. Regents Multiple Choice Practice: __________19.What is the measure of an exterior angle of a regular octagon? (1) 1080° (2) 180° (3) 135° (4) 45° __________20.What is the measure of an interior angle of a regular hexagon? (1) 540° (2) 720° (3) 120° (4) 6° __________21.For any regular polygon, what is the sum of one of its interior angles and one of its exterior angles? (2) 180° (3) 90° (4) 540° __________22.What is the measure of an exterior angle of a regular nonagon? (1) 180° (2) 40° (3) 1260° (4) 140 3/9/2015 8:22 PM (1) 360° 7 Geometry Unit 11 Quadrilaterals **TWO PAGES!! Interior and Exterior Angles of Polygons HW 11.1 Find the sum of the measures of the interior angles of the indicated convex polygon. __________1. 14-gon __________2. 23-gon The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon by the number of sides. _________3. 3060 __________4. 1260° Find the measure of ONE exterior angle of each regular polygon. Round to nearest tenth if needed. __________5. Nonagon __________6. 22-gon If an interior angle of a polygon is given, find the exterior angle and find the number of sides: __________7. 162 _________8. 171 __________ _________ What is the value of x in the diagram shown? __________a.) __________b.) 3/9/2015 8:22 PM 9. 8 Interior and Exterior Angles Practice: __________10. Find the measure of each interior angle of a regular decagon. (1) 144 (2) 135 (3) 120 (4) Cannot be determined __________11. How many degrees are there in each interior angle of a regular hexagon? (1) 108 (2) 120 (3) 144 (4) Cannot be determined __________12. If a polygon has six sides, how many degree are there in each of its exterior angles? (1) 60 (2) 30 (3) 120 (4) Cannot be determined __________13. If each interior angle of a regular polygon measures 162°, how many sides does the polygon have? (1) 20 (2) 18 (3) 16 (4) Cannot be determined __________14. How many sides does a polygon have if each of its interior angles measures 174°? (1) 20 (2) 40 (3) 60 (4) Cannot be determined __________15. How many degrees are there in the sum of the exterior angles of a dodecagon ? (1) 4320 (2) 2160 (3) 1800 (4) 360 __________16. Find the number of degrees in each exterior angle of a regular pentagon? (1) 36 (2) 72 (3) 108 (4) 360 __________17. If each exterior angle of a regular polygon contains 40°, how many sides does the polygon have? (2) 10 (3) 11 (4) 12 __________18. If the sum of the interior angles of a regular polygon is 900°, how many sides does the polygon have? (1) 7 (2) 9 (3) 10 (4) 11 3/9/2015 8:22 PM (1) 9 9 Geometry Unit 11 Quadrilaterals Properties of Parallelograms Warm-up: CW 11.2 HW 11.2 on pgs: 13 & 14 Complete the following statements about two lines cut by a transversal: 1) Alternate Interior Angles are congruent _______________________________ 2) Alternate Exterior Angles are congruent _______________________________ 3) Corresponding Angles are congruent _______________________________ 4) Co-Interior Angles are supplementary _______________________________ Notes: *Go over Quadrilateral Family Tree The 5 Properties of a Parallelogram: 1) Opposite sides ________ Diagram of the 5 Properties: 2) Opposite sides ________ 3) Opposite angles _______ 4) Diagonals bisect __________ _________ 5) Consecutive Angles are _____________________ Practice Examples: (**Draw a picture) __________1. Given parallelogram RSTW with RS 2 x 7 , ST 3 y 5 , TW 25 , and WR 16 , __________ solve for the values of x and y. __________2. Given parallelogram RSTW such that diagonals SW and RT intersect at Z, if WZ 4 x 3 , __________ ZS 13 , RZ 17 , and ZT 7 y 3 , solve for the values of x and y. __________3. Given parallelogram RSTW, with mWRS 24x , mRST 14 y 4 , mSTW 15x 27 , and mTWR 11y 20 , solve for the values of x and y. 3/9/2015 8:22 PM __________ 10 __________4. Given parallelogram RSTW, with mWRS 75 , mRST 7 x , and mSTW 11y 9 , __________ solve for the values of x and y. __________5. Given parallelogram ABCD, if mDAB 4 x 4 and mBCD 74 x , find mCDA . __________6. Given parallelogram ABCD, if mDCB 2x 9 and mABC 5x 3 , find mDAB . __________7. Given parallelogram ABCD where diagonals AC and BD intersect at E, if AC 4 y 6 and EC 3 y 1 , find AC . __________8. Given parallelogram ABCD where diagonals AC and BD intersect at E, if DE y 4 and DB 5 y 10 , find DB . To Complete a Coordiante Proof of a Parallelogram: 1. ↔ Diagonals bisect each other – Show the midpoints of diagonals are same 2. ↔ 2 pairs opp sides || - Show 4 slopes and opposite sides parallel 3. ↔ 2 pairs opp sides - Show 4 distances and opposite sides = in length 4. ↔ 1 pair opp sides and || - Show 2 distances and 2 slopes of SAME OPP. SIDES 3/9/2015 8:22 PM - MUST SHOW ONE OF THESE: 11 Coordinate Geometry Proofs: 9. The vertices of quadrilateral ABCD are given: A(1,2), B(2,5), C(5,7) D(4,4) Graph it and use slopes to show that it is a parallelogram. 3/9/2015 8:22 PM 10. Quadrilateral LMNO has vertices L 2, 4 , M (5, 2) , N 2, 1 , O 5,1 . Use distances to prove that LMNO is a parallelogram. 12 Geometry Unit 11 Quadrilaterals **TWO PAGES!! Interior and Exterior Angles of Polygons HW 11.2 __________1. Given parallelogram JKLM where diagonals MK and JL __________ intersect at N, name four pairs of congruent segments. __________ __________ __________2. Given parallelogram JKLM with mMJK 65 , find mJKL , mKLM , and mLMJ . __________ __________ __________3. Given parallelogram JKLM where diagonals MK and JL intersect at N, if NJ 7 , find JL . __________4. Given parallelogram JKLM where diagonals MK and JL intersect at N, if MK 10 , find NK . __________5. Given parallelogram JKLM, if mMJL 37 and mLJK 27 , find mJKL . __________7. Given parallelogram ABCD, if DA 2 y 5 and CB 14 y , find DA. 3/9/2015 8:22 PM __________6. Given parallelogram JKLM, if mJMK 71 and mKML 42 , find mJKL and mMKL . __________ 13 __________8. Given parallelogram ABCD , if mDAC 4 x 7 , mCAB 5x 8 and, mDCB 7 x 13 , find mDAC . 9. Quadrilateral LMNO has vertices L 2, 4 , M (5, 2) , N 2, 1 , O 5,1 . Use midpoints to prove that LMNO is a parallelogram. (**Look up what you should be doing midpoints of) Identify the error(s) in planning the solution or solving the problem. Then write the correct solution. A student is trying to find the sum of the angle measures of a regular 27-gon. They write (n 2)180 n (27 2)180 = 27 25 180 = 27 2 =166 3 Sum 3/9/2015 8:22 PM 10. 14 Geometry Unit 11 Quadrilaterals Properties of Parallelograms – Proof Emphasis CW 11.3 HW 11.3 on pgs: 17 & 18 Warm-up: __________1. Given parallelogram ABCD where diagonals AC and BD intersect at E, what congruence postulate shows that ABE CDE ? __________2. Given parallelogram EFGH, if mE 62 , what is mH ? Parallelogram Proofs: Tell why each quadrilateral ABCD is a parallelogram. _________________ _________________ _________________ _________________ _______________ To Prove a Parallelogram: - May need to prove two triangles congruent - May need to use facts about parallel lines - MUST SHOW EITHER: 1. ↔ Diagonals bisect each other 2. 3. ↔ 2 pairs opp sides 4. 5. ↔ 2 pairs opp angles 6.* → consec ' s supp (additional proof reason, not used to prove a Proof Example: ↔ 2 pairs opp sides || 1 pair opp sides and || → ) Given: Quadrilateral ABCD 1 2 BD bisects AC Prove: ABCD is a parallelogram D C 2 4 A 1 E B 3/9/2015 8:22 PM 3 15 Proof Practice: 1. Given: E is the midpoint of BD BD bisects AC Prove: ABCD is a parallelogram 2. Given: BC || AD , A C Prove: ABCD is a parallelogram From the August 2009 Regents… [6 pts] Given: Quadrilateral ABCD, diagonal AFEC , AE FC , BF AC , DE AC , 1 2 Prove: ABCD is a parallelogram. 3/9/2015 8:22 PM 3. 16 Geometry Unit 11 Quadrilaterals TWO PAGES!! Properties of Parallelograms – Proof Emphasis HW 11.3 __________1. In parallelogram EFGH, mG is 25 degrees less than mH . Find mH . __________2. Which statement is not always true about a parallelogram? (1) The diagonals are congruent. (2) The opposite sides are congruent. (3) The opposite angles are congruent. (4) The opposite sides are parallel. __________3. In parallelogram QRST, diagonals QS and RT intersect at point E. Which statement is always true? (1) (3) Prove: (2) (4) RQS SQT TQE RQE PQRS PE SQ , RF SQ SE QF OVER 3/9/2015 8:22 PM 4. Given: QS RT RES TEQ 17 5. Given: ABCD is a quadrilateral AB CD 1 2 Prove: ABCD is a parallelogram Statement 1. ABCD is a quadrilateral AB CD 1 2 2. DC AB 3. ABCD is a parallelogram Reason 1. 2. 3. 6. The vertices of quadrilateral ABCD are given. Draw ABCD in the coordinate plane and show that it is a parallelogram. A(-2, 3), B(-5, 7), C(3,6), and D(6,2) 7. Find the error: What is the mT in pentagon PQRST? The student writes: 350 mT 540 mT 190 3/9/2015 8:22 PM 130 60 160 mT (5 2)180 18 Geometry Unit 11 Quadrilaterals Properties of Parallelograms – Proof Emphasis CW 11.4 HW 11.4 on pgs: 21 & 22 Warm-up: __________a.) Which of the following does not prove a parallelogram? (1) Quadrilateral with two pairs of opposite sides congruent (2) Quadrilateral with diagonals congruent (3) Quadrilateral with one pair opposite sides congruent and parallel (4) Quadrilateral with diagonals that bisect each other __________b.) Which of the following is false about a parallelogram? (1) It has opposite angles congruent (2) It has consecutive angles that are complementary (3) It has diagonals that bisect each other (4) It has opposite sides parallel 1. Quadrilateral ABCD has vertices A 2, 2 , B(1, 4) , C 2,8 , and D 1, 6 . Use midpoints to prove that ABCD is a parallelogram. 3/9/2015 8:22 PM 2. Quadrilateral ABCD has vertices A 2, 1 , B(1,3) , C 6,5 , and D 7,1 . Prove that ABCD is a parallelogram. 19 Algebraic Proof: For what value of the variables must ABCD be a parallelogram? 3 4. Summary: 5. To show a parallelogram by distances, you must show the _____ distances of the _________. To show a parallelogram by midpoints, you must show the _____ midpoints of the __________. To show a parallelogram by slopes, you must show the _____ slopes of the __________. Challenging Proofs: 6. Given: Parallelogram ABCD FG bisects DB Prove: DB bisects FG Given: FDEC , ABCD BE FC , AF FC Prove: ABEF is a parallelogram 3/9/2015 8:22 PM 7. 20 Geometry Unit 11 Quadrilaterals TWO PAGES!! Properties of Parallelograms – Proof Emphasis HW 11.4 For what value of the variables must ABCD be a parallelogram? 1. 2. 3. Given: PQRS is a quadrilateral 1 2 3 4 4. Given: Prove: PQRS is a parallelogram LM is a median in GKL LM MJ GJKL is a parallelogram OVER 3/9/2015 8:22 PM Prove: 21 5. Three of the vertices of parallelogram ABCD are A(-4, 1), B(-1, 5), C(6, 5), and D(x, y). Find the coordinates of point D. Show your method. 6. The vertices of quadrilateral ABCD are given. Draw ABCD in the coordinate plane and show that it is a parallelogram. A(0,1), B(4,4), C(12,4), and D(8,1) 7. Find the error in the student’s conclusion: In the two pictures below, can you prove the quadrilateral is a parallelogram based on the given information? The student writes: Yes, One pair of opposite sides is congruent and one pair is parallel b. The student writes: No, One pair of opposite sides is congruent but that is not enough information. 3/9/2015 8:22 PM a. 22 Geometry Unit 11 Quadrilaterals Properties of Rhombi, Rectangles, Squares Warm-up: Properties: CW 11.5 HW 11.5 on pgs: 25 & 26 *Review the Quadrilateral Family Tree List the 5 properties of a parallelogram List the 2 extra properties for a Rectangle 1.________________________________ 1.__________________________ 2.________________________________ 2.__________________________ 3.________________________________ List the 3 extra properties for a Rhombus 4.________________________________ 1.__________________________ 5.________________________________ 2.__________________________ Square 3.__________________________ Practice: __________1. Given rhombus ABCD, if AD 3w 7 and AB 2(w 8) , find AD and DC. __________ Justify your answer. __________2. Given rhombus ABCD such that diagonals AC and DB intersect at E, what is mAEB ? __________3. Given rhombus ABCD with AB 3k 1 and the perimeter of rhombus ABCD is 13k 1 , find AB. 3/9/2015 8:22 PM __________4. In the diagram below of rhombus ABCD, diagonals AC and DB intersect at E. If mDAB 7 x 14 and m2 5x 5 , find m1 . 23 __________5. Given rectangle RSTW, what is mRWT ? __________6.Given rectangle PQRS such that diagonals PR and QS intersect at T, if PR 7a 2 and ST 4a 3 , find PT . __________7. Using the diagram to the right of rectangle RSTW, if m1 61 , find mRZW . Cool Square Shortcut: **remember the special right triangle from Trig Unit 10.** __________8.If the length of the side of a square is 6, find the length of a diagonal of the square in simplest radical form. Fill in the measurements on your diagram. __________9. If the length of the side of a square is 8, find the length of a diagonal of the square in simplest radical form. Fill in the measurements on your diagram. What is the shortcut for finding the length of a diagonal of a square? __________10.The diagonals of square DEFG intersect at H. Given that EH 5 , find the perimeter of the square in simplest radical form. If UV z 2 , what kind of parallelogram is TUVW and why? 3/9/2015 8:22 PM __________11.In parallelogram TUVW, TU 3z 14 and WV 2 z 6 . Find the value of z. 24 Geometry Unit 11 Quadrilaterals TWO PAGES!! Properties of Rhombi, Rectangles, Squares HW 11.5 1. True or False: __________a.) All parallelograms are quadrilaterals. __________b.) All quadrilaterals are parallelograms. __________c.) All rhombuses are parallelograms. (i.e., they come from parallelograms on the family tree) __________d.) All rectangles are squares. __________e.) All squares are rectangles. __________f.) All squares are parallelograms. __________2. In the diagram below of rhombus ABCD, diagonals AC and __________ DB intersect at E. If m1 50 , find m2 and m3 . __________3. Given rhombus ABCD such that diagonals AC and DB intersect at E, if AC 2 y 8 and EC 2 y 1 , find EC. 4. Using the diagram to the right of rectangle RSTW, __________a.) Find m1 m2 . __________b.) If m1 3x 12 and m2 2 x 7 , find m1 . 6. Fill in all of the angle measures in the diagram of the square: 3/9/2015 8:22 PM __________5. Given rectangle PQRS, if PR 5 y 2 , SQ 11y 10 , find SQ . 25 7. Based upon the markings on each figure below, determine the most precise name for each quadrilateral. a)_____________ b)________________ c)__________________ d) __________________ 8. In the table below, check the quadrilateral(s) that have the following properties Property Parallelogram Rhombus Rectangle a. All sides b. Opposite sides // c. All ’s are rt. ’s d. Diag.s bisect ea other e. Diags are f. Opposite sides g. Consec ’s are sup. h. Diags are Square Review: a)_________9. a) What is the sum of the interior angle measures of a regular octagon? b)_________ b) What is one exterior angle of a regular octagon? ___________10. What is the measure of one interior angle of a regular 12-gon? ___________11. What is the value of x in the regular polygon at the right? ___________12. If the measure of an exterior angle of a regular polygon is 24, how many sides does the polygon have? 13. Find the errors:. A student was asked to find the measures of the numbered angle in rhombus ABCD. They wrote: m1 = 90, Diags of rhom. BDC ACD, so m2 = 40 b) A student was asked to find the value of x in parallelogram PQRS that makes it a rectangle. They wrote: 5x – 2 = 4x + 1 9x = 3 x = 1/3 3/9/2015 8:22 PM a) 26 Geometry Unit 11 Quadrilaterals Properties of Rhombi, Rectangles, Squares – Proof Emphasis Warm-up: CW 11.6 HW 11.6 on pgs: 29 & 30 The diagonals of a rhombus measure 16 cm and 30 cm. Find the perimeter of the rhombus. 1. Quadrilateral ABCD has vertices A 4, 2 , B(7,3) , C 8, 6 , and D 5,5 . Prove that ABCD is a rhombus. 3/9/2015 8:22 PM 2. The vertices of quadrilateral PQRS are P 0, 2 , Q(4,8) , R 7, 6 , and S 3, 0 . Use slopes to prove that PQRS is a rectangle. 27 3. Quadrilateral MATH has vertices M 1, 4 , A(4, 7) , T 7, 2 , and H 2, 1 . Prove that MATH is a square. 4. Given: ABCD is a rhombus AE CE Prove: ADE CDE In a parallelogram ABCD, AB 2 x 3 , BC 4 x 5 , and CD 5x 9 . Show that ABCD is a rhombus. 3/9/2015 8:22 PM 5. 28 Geometry Unit 11 Quadrilaterals TWO PAGES!! Properties of Rhombi, Rectangles, Squares – Proof Emphasis 1. HW 11.6 Given: ABCD is a rectangle M is the midpoint of AB Prove: DM CM OVER 3/9/2015 8:22 PM 2. Using the appropriate formulas, prove quadrilateral PQRS is a rectangle. P( -1, -2 ), Q( 5 , 2 ), R( 7, -1 ), S( 1, -5 ) 29 __________3. The measures of five of the interior angles of a hexagon are 150°, 100°, 80°, 165°, and 150°. What is the measure of the sixth interior angle? (1) (2) (3) (4) 75° 105° 180° 80° __________4. The measures of two consecutive angles of a parallelogram are in the ratio of 5:4. What is the measure of an obtuse angle of the parallelogram? (1) (2) (3) (4) 20° 80° 100° 160° __________5. In parallelogram ABCD, diagonals AC and BD intersect at E. If BE ED x 10 , what is the value of x? (1) (2) (3) (4) 2 x and 3 30 -6 6 -30 __________6. In the diagram below of parallelogram ABCD with diagonals AC and BD , m1 45 and mDCB 120 . What is the measure of 2 ? (1) (2) (3) (4) 30° 15° 60° 45° (1) (2) (3) (4) 12 37 40 50 3/9/2015 8:22 PM __________7. In the accompanying diagram of rectangle ABCD, mBAC 3x 4 and mACD x 28 . What is mCAD ? 30 Geometry Unit 11 Quadrilaterals Properties of Rhombi, Rectangles, Squares – Proof Emphasis Warm-up: CW 11.7 HW 11.7 on pgs: 33 & 34 Given: ABCD is a rhombus Prove: BFA DFC 1. Quadrilateral ABCD has vertices A 3, 6 , B(6, 0) , C 9, 9 , and D 0, 3 . Prove that ABCD is a parallelogram but NOT a rhombus. 2. Quadrilateral ABCD has vertices A 0, 2b , B 0, 0 , C 4a, 0 , and D 4a, 2b . Prove that ABCD is a rectangle using slopes. 3/9/2015 8:22 PM Practice: 31 3. 4. Quadrilateral ABCD has vertices A 2, 1 , B(2,3) , C 4,1 , and D 0, 3 . Prove that ABCD is a rectangle. Given: DFEC , AGE , BGF Rectangle ABCD DF CE Prove: ADE BCF Given: Square ABCD Prove: Diagonals are congruent (By means of proving triangles congruent) 3/9/2015 8:22 PM 5. 32 Geometry Unit 11 Quadrilaterals Properties of Rhombi, Rectangles, Squares – Proof Emphasis 1. HW 11.7 Given: Rectangle ABCD Isosceles ALD with vertex L Prove: L is the midpoint of BC Given: Rhombus ABCD CB bisects DF Prove: BF AD Statement 1. Rhombus ABCD CB bisects DF 2. DC AB 3. DCE FBE 4. DEC FEB 5. DE FE 6. DCE FBE OVER Reason 1. 2. 3. 4. 5. 6. 7. BF CD 7. 8. CD AD 8. 9. BF AD 9. 3/9/2015 8:22 PM 2. 33 Given rectangle ABCD where diagonals AC and BD intersect at E, complete the following: __________a.) If AC 4 x 60 and BD 30 x , find BD . __________b.) If AC 4 x 60 and AE x 5 , find EC . __________c.) If mBAC 4 x 5 and mCAD 5x 14 , find mCAD . __________d.) If AE 2 x 3 and BE 12 x , find BD . 4. Given square ABCD where diagonals AC and BD intersect at E, complete the following: __________a.) If mBAC 9 x , find x. __________b.) If AB x2 15 and BC 2 x , find the perimeter of the square. 5. In a parallelogram TUVW with diagonals that intersect at X, TX 2 y 11 , VX y 9 , WU y 18 . __________a.) b.) Find TV . Can TUVQ be a rectangle? Why? 3/9/2015 8:22 PM 3. 34 Geometry Unit 11 Quadrilaterals Properties of Trapezoids and Kites CW 11.8 HW 11.8 on pgs:37 & 38 Warm-up: In the diagram to the right, XY || AB . Solve for angles x and y. If AX and BY intersect at P, what kind of triangle is XPY ? Properties: *Review Family Tree Midsegment (Median) of Trapezoid: ________________________________________________________ _______________________________________________________________________________________ __________1. The measures of the bases of a trapezoid measure 64 and 82. Find the length of the midsegment of the trapezoid. a._________2. Find EF in the given trapezoid below. b._________ a) b) HG________ EF________ CD________ CD________ EF________ HG________ __________4. In an isosceles trapezoid, the smaller base measures 2 x 4 , the larger base measures 4 x 8 , and __________ the midsegment measures 24. Find the measures of the bases. 3/9/2015 8:22 PM 3. Find the lengths of the segments with variable expressions.. 35 __________5. In trapezoid RSUT, RS TU , x is the midpoint of RT , and v is the midpoint of SU . If RS = 30 and XV = 44, find TU. __________6. In isosceles trapezoid FJHG with JH || FG and where JH is the smaller base, if mJ 110 , __________ find mF , mG , and mH . __________ Find the value(s) of the variable(s) in each kite. 7._________ 8. _________ _________ __________9.Given kite STUV with ST SV , VU TU , diagonals SU and VT intersect at R, VR RT , __________ VR 9 , UR 7 ,and SR 8 , find VU and SV in simplest radical form. b.) Determine if MATH is an isosceles trapezoid. 3/9/2015 8:22 PM 10. Quadrilateral MATH has vertices M 1,1 , A(2,5) , T 5, 7 , and H 7,5 . a.) Prove that MATH is a trapezoid. 36 Geometry Unit 11 Quadrilaterals Properties of Trapezoids and Kites TWO PAGES!!! HW 11.8 Find EF in each trapeoid __________1. _________2. __________3.Given kite JKLM with JM JK and ML KL , if mM 88 and mL 120 , find mK __________ and mJ . __________4.Given kite JKLM with JM JK and ML KL , if mJ 60 and mL 50 , find mK . __________5. Given kite STUV with ST SV , VU TU , diagonals SU and VT intersect at R, SR TR VR , __________ UR 12 ,and SR 5 , find VU and SV in simplest radical form. 6. Find the Errors: A) The student is given: segment MS is the midsegment of trapezoid WXYZ. What is the value of x? B) The student is given: Quad TUVW is a kite. What the mTUV and mTWY? They write: They write: mTUV + mUTW + mUVW = 180 x + 40 + 60 = 180 x = 80 mTUV = mTWV = 80 3/9/2015 8:22 PM MS = WX + ZY 4x – 4 = 18 + 3x + 4 x – 4 = 22 x = 26 37 7. Use the given vertices to graph parallelogram JKLM. Classify the figure and explain your reasoning. Then, find the perimeter of parallelogram JKLM. 3/9/2015 8:22 PM J( -4, 2 ), K( 0 , 3 ), L( 1 , -1 ), M( -3 , -2 ) 38 Geometry Unit 11 Quadrilaterals Properties of Trapezoids and Kites – Proof Emphasis Warm-up: CW 11.9 HW:11.9 on pgs: 41 & 42 Name each type of quadrilateral for which the statement is true (parallelogram, rhombus, rectangle, square). a.) Both pairs of opposite angles are congruent. _____________________________________________ b.) The quadrilateral is equilateral. ________________________________________________________ c.) Given rhombus MNOP, the diagonals intersect at Q and mQNO 48 , find mNPO . Proof Reasons & Theorems: Trap quad w/ only 1 pr // sides Isos Trap Trap w/ non // sides Isos Trap Trap w/ base ’s Isos Trap Trap w/ diags Median of trap = average of bases it’s || to Kite → diags Kite → only 1 pr opp ’s Kite 2 pr adjacent sides Median of trap meets midpts of non-|| legs Practice: Given: TRAP is a trapezoid TA RP Prove: RPA TAP 3/9/2015 8:22 PM 1. 39 2. Given: ABCD is a kite Prove: BE bisects CBA Quadrilateral ABCD has vertices A 0, 4 , B(0, 8) , C 3, 4 , and D 3,1 . Prove that ABCD is a trapezoid but NOT an isosceles trapezoid. 3/9/2015 8:22 PM 3. 40 10R Unit 11 Quadrilaterals TWO PAGES!!! Properties of Trapezoids and Kites – Proof Emphasis 1. Complete the following proof: DE || AV DAV EVA Prove: DAVE is an isosceles trapezoid 2. DAVE is a trapezoid 3. 4. DAVE is an isosceles trapezoid 2. Reason 1. 2. 3. CPCTC 4. Show that quadrilateral A 0, 2 , B 9,1 , C 4,6 , D 1,5 is an isosceles trapezoid. SHOW ALL WORK. OVER 3/9/2015 8:22 PM Given: Statement 1. DE || AV DAV EVA HW 11.9 41 3. Given: Isosceles trapezoid ABCD with AB || CD Prove: 1 2 Points P, Q, R, S are the vertices of a quadrilateral. Give the most specific name for PQRS. Justify your answer: P 1,0 , Q 1, 2 , R 6,5 , S 3,0 3/9/2015 8:22 PM 4. 42 Geometry Unit 11 Quadrilaterals Special Quadrilaterals Warm-up: # 2 3 4 5 6 7 8 9 10 Property Rectangle Rhombus Square Kite Trapezoid Isosceles Trapezoid Both pairs opposite sides congruent Both pairs opposite angles congruent Exactly one pair of opposite sides congruent Exactly one pair of opposite sides parallel Exactly one pair of opposite angles congruent Consecutive angles are supplementary Diagonals bisect each other Diagonals are congruent Diagonals are perpendicular Diagonals bisect the angles Practice: 1. Put an X in the box if the quadrilateral always has the given property. Give the most specific name for each quadrilateral. Explain your reasoning. 2. 3. 4. 3/9/2015 8:22 PM 1 CW 11.10 HW: 11.10 on pgs: 43 Tell whether enough information is given in the diagram to classify the quadrilateral by the indicated name. 5. Rectangle 6. Isosceles Trapezoid 7. Rhombus 8. Kite Which two segments or angles must be congruent so that you can prove that FGHJ is the indicated quadrilateral? There may be more than one correct answer. 9. Kite 10. Isosceles Trapezoid Directions: In #11-16, use the information in and below each diagram and the properties of various quadrilaterals to find x, y, and z, as required. Label answers with appropriate units. C 120° A 70° U 12. x 60° B R Quadrilateral ABCD D 13. A Rectangle ABCD y T S Parallelogram RSTU C x y z 70° O 14. z 50° x N 130° x B z y L M Isosceles Trapezoid LMNO 3/9/2015 8:22 PM D 11. 44 15. 16. T S H y 120° 30° x G x z Q E R Rhombus QRST y 20° F Trapezoid EFGH __________17. For what value of x is the quadrilateral a parallelogram? __________18. Find the length of the midsegment or find the value of x. __________ __________19. JKLM is a kite. Find mK . Which statement is true? (1) (2) (3) (4) __________21. Which quadrilateral does not necessarily have congruent diagonals? (1) (3) __________22. All parallelograms are quadrilaterals. All parallelograms are rectangles. All quadrilaterals are trapezoids. All trapezoids are parallelograms. isosceles trapezoid rhombus (2) (4) square rectangle True or False: All squares are similar to each other. (1) True (2) False 3/9/2015 8:22 PM __________20. 45 __________23. To prove that a parallelogram is a rectangle, it is sufficient to show that the (1) (3) __________24.* diagonals are congruent (2) opposite sides are congruent (4) A parallelogram must be a rectangle if the opposite angles are (1) congruent (3) supplementary __________25. congruent and bisect the angles to which they are drawn congruent and do not bisect the angles to which they are drawn not congruent and bisect the angles to which they are drawn not congruent and do not bisect the angles to which they are drawn A quadrilateral must be a square if (1) (2) (3) (4) 27. (2) equal in measure (4) complementary A parallelogram must be a square if the diagonals are (1) (2) (3) (4) __________26. diagonals are perpendicular adjacent sides are congruent diagonals are congruent sides and angles are congruent opposite sides and opposite angles are congruent diagonals bisect each other and are perpendicular to each other In quadrilateral ABCD, mA x 10 , mB 2x 10 , mC 2 x 70 , and mD 3x 50 . What kind of quadrilateral is ABCD and why? Type of Quadrilateral: ____________________ Why? ____________________________________________________________________________ In parallelogram ABCD, AB 3x 2 , DC 10 x 12 , and AD 5x 2 . What kind of parallelogram is ABCD? Justify your answer. Type of Parallelogram: ____________________ Justification: _____________________________________________________________________ 3/9/2015 8:22 PM 28. 46 29. __________30. In rectangle ABCD, diagonals AC and BD intersect at P. If CP 40 and BD 2 x 12 . What is the value of x? __________31. WXYZ is a parallelogram, YA is an altitude to WX , and YA AX . Find mZ . W X Y __________32. The lengths of the diagonals of a rhombus are 10 and 24. Find the perimeter of the rhombus. __________33. The lengths of the diagonals of a rhombus are 14 and 48. Find the perimeter of the rhombus. __________34. __________ In an isosceles trapezoid, the smaller base measures 2 x 4 , the larger base measures 4 x 8 , and the midsegment measures 24. Find the measures of the bases. 3/9/2015 8:22 PM Z A 47 __________35. __________ The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon. a.) 1080° b.) __________37. 2520° __________36. Solve for x. Solve for x. __________38. __________ In an isosceles trapezoid, the smaller base measures 3x 1 , the larger base measures 6 x 8 , and the midsegment measures 36. Find the measures of the bases. __________39. In isosceles trapezoid ABCD, diagonal AC measures 2 x 4 and diagonal BD measures 4 x 8 . Find the measure of AC . Study Polygons: Triangle Quadrilateral Pentagon Hexagon Heptagon Regular Polygon: 40. Pentagon 41. Heptagon 42. Decagon 43. Nonagon 44. Octagon 8 sides 9 sides 10 sides 11 sides 12 sides Sum of its interior angles: Octagon Nonagon Decagon 11-gon Dodecagon Measure of an interior angle: 3/9/2015 8:22 PM 3 sides 4 sides 5 sides 6 sides 7 sides 48 Geometry Unit 11 Quadrilaterals Quadrilaterals Review Packet – Proofs Practice 1. Quadrilateral QUAD has vertices Q 1,1 , U (3, 4) , A 1,5 , and D 3, 2 . Prove that QUAD is a parallelogram. 2. Given: PQRS is a parallelogram PS QT Prove: QRT is isosceles Quadrilateral ABCD has vertices A 5, 0 , B(2,9) , C 4, 7 , and D 1, 2 . Prove that ABCD is a rectangle. 3/9/2015 8:22 PM 3. 49 4. Given: Rectangle ABCD E is the midpoint of DC Prove: 1 2 6. Quadrilateral ABCD has vertices A 3, 2 , B(2, 6) , C 2, 7 , and D 1,3 . Prove that ABCD is a rhombus. Given: Rhombus ABCD E is the midpoint of DF Prove: AD BF 3/9/2015 8:22 PM 5. 50 7. Quadrilateral PQRS has vertices P 0, 0 , Q(4,3) , R 7, 1 , and S 3, 4 . Show that PQRS is a square. 8. Quadrilateral DEFG has vertices D 4, 0 , E (0,1) , F 4, 1 , and G 4, 3 . Prove that ABCD is a trapezoid but NOT an isosceles trapezoid 9. Given: Trapezoid ABCD, BC || AD , BE and CF are altitudes, AE DF 3/9/2015 8:22 PM Prove: Trapezoid ABCD is isosceles 51 Geometry Reasons for Proofs: Transitive Reflexive Symmetric Substitution Add Post of Seg Add Post of ’s Sub Post of Seg Sub Post of ’s //lines alt int ’s //lines alt ext ’s //lines corresp ’s //lines co-int ’s supp //lines to same line Seg bisector 2 seg Midpt 2 seg bisector 2 ’s bisector 2 seg & 2 rt ’s lines 2 rt ’s 10R: 2 steps: lines rt ’s AND All rt ’s Median 2 seg Altitude 2 rt ’s 10R: 2 steps Altitude 2 rt ’s AND All rt ’s Vertical ’s Halves of ’s are SSS SSS SAS SAS ASA ASA AAS AAS HL(R) HL(R) CPCTC Isosc w/ 2 sides & 2 ’s Sides opp ’s ’s opp sides 3/9/2015 8:22 PM Comp ’s + to 90 Comp of ’s Rt ’s ’s Rt has 1 rt All rt ’s are Supp ’s + to 180 Linear pairs supp ’s Supp of ’s 52