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CHAPTER 6 0 Honors Geometry Section 6.1 Notes: Angles of Polygons Learning Targets Question, Topics and Vocabulary 1. I can find and use the sum of the measures of the interior and exterior angles of a polygon. Problems, Definitions and Work Diagonal of a Polygon The vertices of polygon PQRST that are not consecutive with vertex P are vertices R and S. Therefore, polygon PQRST has two diagonals from vertex P, PR and PS . Notice that the diagonals from vertex P separate the polygon into three triangles. The _______ of the angle measures of a polygon is the __________ of the angle measures of the triangles formed by drawing all the possible diagonals from one vertex. Sum of the Angles of a Polygon Polygon Number of Sides Number of Triangles Sum of Interior Angle Measures Triangle Quadrilateral Pentagon Hexagon n-gon Polygon Interior Angle Sum Example 1: Find the sum of the measures of the interior angles of a convex nonagon. You can use the Polygon Interior Angles Sum Theorem to find the sum of the interior angles of a polygon and to find missing measures in polygons. 1 Example 2: Find the measures of each interior angle of parallelogram RSTU. Example 3: A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the measure of one of the interior angles of the pentagon. Given the interior angle measure of a regular polygon, you can also use the Polygon Interior Angles Sum Theorem to find a polygon’s number of sides. Example 4: a) The measure of an interior angle of a regular polygon is 150. Find the number of sides in the polygon. b) The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. Using the polygons below, does a relationship exist between the number of sides and sum of its exterior angles? 2 Polygon Exterior Angles Sum Example 5: Find the value of x in the diagram. Example 6: Find the measure of each exterior angle of a regular decagon. 3 Honors Geometry Section 6.2 Notes: Parallelograms Learning Targets Question, Topics and Vocabulary 1. I can recognize and apply properties of the sides, angles, and diagonals of parallelograms. Problems, Definitions and Work Parallelogram *To name a parallelogram, use the symbol definition. . In ABCD, BC || AD and AB || DC by Other Properties of Parallelograms Property: Theorem 6.3 Ex: Property: Theorem 6.4 Ex: Property: Theorem 6.5 Ex: Property: Theorem 6.6 Ex: 4 Example 1: In parallelogram ABCD, suppose mB = 32, CD = 80 inches, and BC = 15 inches. a) Find AD. b) Find mC. c) Find mD. Example 2: ABCD is a parallelogram. a) Find AB. b) Find mC. c) Find mD. Example 3: Write a two-column proof. Given: Parallelogram ABCD AC and BD are diagonals P is the intersection of AC and BD Prove: AC and BD bisect each other Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 9. 9. 10. 10. 11. 11. 5 Example 4: Given: WXYZ is a parallelogram Prove: WXZ ≅ YZX Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. Diagonals of a Parallelogram Theorem 6.7 Ex: Theorem 6.8 Ex: Example 5: If WXYZ is a parallelogram… a) Find the value of r. b) Find the value of s. c) Find the value of t. 6 Example 6: a) What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with vertices M(–3, 0), N(–1, 3), P(5, 4) and R(3, 1)? b) What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with vertices L(0, –3), M(–2, 1), N(1, 5) and O(3, 1)? 7 Honors Geometry Section 6.3 Notes: Tests for Parallelograms Learning Targets Question, Topics and Vocabulary 1. I can recognize the conditions that ensure a quadrilateral is a parallelogram. Problems, Definitions and Work We already learned that if a quadrilateral has opposite sides parallel, it is a parallelogram by definition. However there are more tests to determine if a quadrilateral is a parallelogram. Conditions for Parallelograms Theorem 6.9 Ex: Theorem 6.10 Ex: Theorem 6.11 Ex: Theorem 6.12 Ex: 8 Example 1: a) Determine whether the quadrilateral is a parallelogram. Justify your answer. b) Which theorem would prove the quadrilateral is a parallelogram? Example 2: Scissor lifts, like the platform lift shown, are commonly applied to tools intended to lift heavy items. In the diagram, A C and B D. Explain why the consecutive angles will always be supplementary, regardless of the height of the platform. Example 3: a) Find x and y so that the quadrilateral is a parallelogram. b) Find m so that the quadrilateral is a parallelogram. 9 How to Prove that a Quadrilateral is a Parallelogram: Definition: Theorem 6.9: Theorem 6.10: Concept Summary Theorem 6.11: Theorem 6.12: Example 4: Quadrilateral QRST has vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula. Coordinate Geometry We can use the Distance, Slope, and Midpoint Formulas to determine whether a quadrilateral in the coordinate plane is a parallelogram. 10 Example 5: Graph quadrilateral EFGH with vertices E(–2, 2), F(2, 0), G(1, 5), and H(–3, –2). Determine whether the quadrilateral is a parallelogram. 11 Honors Geometry Section 6.4 Notes: Rectangles Learning Targets Question, Topics and Vocabulary 1. I can recognize and apply properties of rectangles. Problems, Definitions and Work By definition, a rectangle has the following properties: Rectangle All four angles are right angles. Opposite sides are parallel and congruent. Opposite angles are congruent Consecutive angles are supplementary. Diagonals bisect each other. Diagonals of a Rectangle (Theorem 6.13) Example 1: A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet, and LN = 6.5 feet, find KM. Example 2: Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ. Example 3: Quadrilateral RSTU is a rectangle. If mRTU = (8x + 4) and mSUR = (3x – 2), solve for x. 12 Example 4: Quadrilateral EFGH is a rectangle. If mFGE = (6x – 5) and mHFE = (4x – 5), solve for x. Diagonals of a Rectangle (Theorem 6.14) *This is the converse of theorem 6.13 Example 5: Some artists stretch their own canvas over wooden frames. This allows them to customize the size of canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides and the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, DA = 35 inches, BD = 37 inches, and AC = 37 inches, explain how an artist can be sure that the frame is rectangular. 13 Example 6: Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3). Determine whether JKLM is a rectangle using the Distance Formula. Coordinate Geometry You can also use the properties of rectangles to prove that a quadrilateral positioned on a coordinate plane is a rectangle given the coordinates of the vertices. Example 7: Graph the quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. A(–3, 1), B(–3, 3), C(3, 3), D(3, 1); Distance Formula 14 15 Honors Geometry Section 6.5 Notes: Rhombi and Squares Learning Targets Question, Topics and Vocabulary 1. I can recognize and apply the properties of rhombi and squares. Problems, Definitions and Work Rhombus Diagonals of a Rhombus Theorem 6.15 Theorem 6.16 Example 1: The diagonals of rhombus WXYZ intersect at V. If mWZX = 39.5, find mZYX. Example 2: The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, solve for x. Example 3: ABCD is a rhombus. Find mCBD if mABC = 126°. 16 Square Parallelograms (Opp. Sides are Parallel) Relationship between Parallelograms, Rhombi, Rectangles, and Squares Properties to Recognize Recall that a parallelogram with four right angles is a rectangle, and a parallelogram with four congruent sides is a rhombus. Therefore, a parallelogram that is both a rectangle and a rhombus is also a square. All of the properties of parallelograms, rectangles, and rhombi apply to squares. For example, the diagonals of a square bisect each other (parallelogram), are congruent (rectangle), and are perpendicular (rhombus). Conditions for Rhombi and Squares Theorem 6.17 17 Theorem 6.18 Theorem 6.19 Theorem 6.20 Example 4: Write a two-column proof Given: LMNP is a parallelogram 1 2 2 6 Prove: LMNP is a rhombus. Statements Reason 1. 1.Given 2. LM || PN 2. 3. 3. Alternate Interior Angles 4. 1 2 4. 5. 5. Given 18 6. 5 6 6. 7. 7. Example 5: Hector is measuring the boundary of a new garden. He wants the garden to be square. He has set each of the corner stakes 6 feet apart. What does Hector need to know to make sure that the garden is square? Example 6: Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for the given vertices: A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain. Coordinate Geometry 19 Honors Geometry Section 6.6 Trapezoids Today’s Objectives: Question, Topics and Vocabulary 1. I can apply properties of trapezoids and kites. 2. I can use the trapezoid midsegment theorem. Problems, Definitions and Work Trapezoid Bases Legs Base Angles Isosceles Trapezoid Theorem 6.21 Theorem 6.22 Theorem 6.23 20 Example 1: Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and LN = 3.6 feet. a) Find mMJK. b) Find MN Example 2: Quadrilateral ABCD has vertices A(5, 1), B(–3, –1), C(–2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid. Midsegment Trapezoid Midsegment Theorem 21 (6.24) Example 3: In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x? Example 4: WXYZ is an isosceles trapezoid with median. Find XY if JK = 18 and WZ = 25. Kite Theorem 6.25 Theorem 6.26 Example 5: If WXYZ is a kite, find mXYZ. 22 Example 6: If MNPQ is a kite, find NP. Example 7: If BCDE is a kite, find mCDE. 23 Honors Geometry Systems of Equations Worksheet Name: _____________________________________ 1. QUAD is a parallelogram. Determine the values of x and y. U A 18 Q D 2. ABCD is a parallelogram. Determine the values of x and y B C y A D 3. MNOP is a parallelogram. Determine the values of x and y. N M O P 4. Draw the parallelogram ABCD with sides AB = y, DC = 3x, BC = x, and AD = 4 – y. Then, find the values of x and y. 24 5. Given: parallelogram PSTM mP = (2x + y)° mM = (3x + 5y)° mT = (4x – 3y + 8)° Find the values of x, y, mP, and mS. S T P M M 6. Given: Parallelogram KMOP, mM = (x + 3y)° mO = (x – 4)° mP = (4y – 8)° Find: mK K O P 25 26