Formal Geometry
Chapter 6
Section 6.1 Angles of Polygons
Definitions Diagonal of a polygon-
Polygon # of Sides # of Triangles Sum of the Interior Angle Measures Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Dodecagon n-gon Regular Polygon-
Theorem List
Theorem 6.1-
Theorem 6.2-
Examples #1-2: Find the measure of each interior angle.
1. 2.
3. The Wonder Wheel at Coney Island in Brooklyn, New York, is a regular polygon with 16 sides. What is the measure of each interior angle of the polygon?
Examples #4-5: The measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon.
4. 150° 5. 170°
6. Find the value of 𝑥.
7. Find the measure of each interior angle in polygon ABCDE, in which the measures of the interior angles are 6𝑥, 4𝑥 + 13, 𝑥 + 9, 2𝑥 − 8 and 4𝑥 − 1.
Section 6.2 Parallelograms
Definitions Parallelogram- Theorem list Theorem 6.3- Theorem 6.4- Theorem 6.5- Theorem 6.6- Theorem 6.7- Theorem 6.8-Examples #1-4: Find the value of each variable, in the following parallelograms.
1. 2.
3. 4.
Example #5: Write a 2 column proof. Given: Prove: ABDH AC GC HDB G
Section 6.3 Day 1 Test for Parallelograms
Theorem list Theorem 6.9- Theorem 6.10- Theorem 6.11- Theorem 6.12-5 Ways to prove that a quadrilateral is a parallelogram
Examples #1-3: Find 𝒙 and 𝒚 so that the quadrilateral is a parallelogram.
1.
2.
3.
Example #4-5: Graph the quadrilateral with the given vertices. Determine whether the figure is a parallelogram. Justify your answer with the method indicated.
Section 6.4 Rectangles
Definitions Rectangle- Theorem List Theorem 6.13- Theorem 6.14-~List the 8 properties of a rectangle on your flashcard.
Examples #1-2: Use the figure of the figure at the right. Quadrilateral 𝑫𝑬𝑭𝑮 is a
rectangle.
1. If 𝐹𝐷 = 3𝑥 − 7 and 𝐸𝐺 = 𝑥 + 5, find 𝐸𝐺.
2. If 𝑚∠𝐸𝐹𝐷 = 2𝑥 − 3 and 𝑚∠𝐷𝐹𝐺 = 𝑥 + 12 find 𝑚∠𝐸𝐹𝐷
Example #3: Graph the quadrilateral with the given vertices. Determine whether the quadrilateral is a rectangle. Justify your answer with the method indicated.
3. 𝑊(−4,3), 𝑋(1, 5) , 𝑌(3, 1) , 𝑍(−2, −2); Slope Formula.
Example #4: Write a 2 column proof. 4. Given: ABDE is a rectangle
Prove: BC DC AC EC
Section 6.5 Rhombi and Squares
Definitions Rhombus- Theorem List Theorem 6.15- Theorem 6.16- Theorem 6.17- Theorem 6.18- Theorem 6.19- Definitions Square- Theorem List Theorem 6.20-Examples #1-2: Quadrilateral 𝑨𝑩𝑪𝑫 is a rhombus. Find each measure.
1. If 𝑚∠𝐵𝐶𝐷=64, find 𝑚∠𝐵𝐴𝐶
2. If 𝐴𝐵 = 2𝑥 + 3 and 𝐵𝐶 = 𝑥 + 7, find 𝐶𝐷
Example #3: Graph each set of vertices, determine whether JKLMis a rhombus, a rectangle, or a square. List all that apply.
3. 𝐽(−4, −1), 𝐾(1, −1) , 𝐿(4, 3) , 𝑀(−1, 3);
Example #4: Write a 2 column proof. 4.
~List the 9 properties of a rhombus on your flashcard. ~List the 11 properties of a square on your flashcard.
Given: ABCD is a rhombus Prove: APCP
Section 6.6 Trapezoids and Kites
Definitions Trapezoid- o Bases- o Legs- o Base angles- Isosceles Trapezoid- Theorem List Theorem 6.21- Theorem 6.22- Theorem 6.23-Definitions Midsegment of a trapezoid- Theorem List Theorem 6.24- Definitions Kite- Theorem List Theorem 6.25- Theorem 6.26-
Examples #1-2: Find each measure.
Example #3: In the figure at the right 𝒀𝒁 ̅̅̅̅̅is the midsegment of trapezoid 𝑻𝑾𝑹𝑽.
Determine the value of 𝒙.
3.
Examples #4-5: If 𝑨𝑩𝑪𝑫 is a kite, find each measure.
4. 𝐴𝐵 5. 𝑚∠𝐶
Example #6: Graph each set of vertices, verify that the quadrilateral is a trapezoid. Determine whether it is an isosceles trapezoid or not.
6. 𝐽(−4, −6), 𝐾(6, 2) , 𝐿(1, 3) , 𝑀(−4, −1);
~List the 6 properties of an isosceles trapezoid on your flashcard. ~List the 5 properties of a kite on your flashcard.
Quadrilateral Family Tree
Objectives: SWBAT identify special quadrilaterals.
Shape Description of Sides Description of Angles Information Interesting Quad Trapezoid Isosceles Trapezoid Kite Parallelogram Rectangle Rhombus Square
Fill in the table. Put an X in the box if the shape always has the property.
Property Parallelogram Rectangle Rhombus Square Trapezoid
Both pairs of opp. sides Exactly 1 pair of opp. sides Diagonals are Diagonals are Diagonals bisect each other
6.7 – COORDINATE QUADS
Objectives: SWBAT identify types of quads using a coordinate plane.
Characteristic
Definition
Formula
Congruent Perpendicular
Parallel
Midsegment / Midpoint
Coordinate Proof
1. Use the coordinate plane, and the Distance Formula to show that KLMN is a Rhombus. K(2, 5), L(-2, 3), M(2, 1), N(6, 3)
2. Use slope or the distance formula to determine the most precise name for the figure 𝐴(−1, −4), 𝐵(1, −1), 𝐶(4, 1), 𝐷(2, −2). A. Kite B. Rhombus C. Trapezoid D. Square
3. Given points 𝐵 (−3,3), 𝐶(3, 4), and 𝐷(4, −2). Which of the following points must be
point A in order for the quadrilateral 𝐴𝐵𝐶𝐷 to be a parallelogram?
A. 𝐴(−2, −1)
B. 𝐴(−1, −2)
C. 𝐴(−2, −3)
D. 𝐴(−3, −2)
4. Given a Trapezoid (−3,4), 𝐵(−5, −2), 𝐶(5, −2), and 𝐷(3,4). Find the following
a) Is the trapezoid Isosceles?
b) What are coordinates of the midsegment for the trapezoid?
6.8 – Proofs for QUADS
Objectives: SWBAT do Proofs involving Quadrilaterals
1. Given: Diagram at the right
Prove: ABCD is a parallelogram
Statements Reasons
1) 1) Given
2) 2) Given
3) ABCD is a parallelogram 3)
2. Given: 𝑃𝐺̅̅̅̅ ≅ 𝐺𝑈̅̅̅̅; 𝑃𝑈̅̅̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐺𝑃𝐿
Prove: Quad GULP is a trapezoid
Statements Reasons
1) 𝑃𝐺̅̅̅̅ ≅ 𝐺𝑈̅̅̅̅ 1)
2) 𝑃𝑈̅̅̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝐺𝑃𝐿 2)
3) 3) Definition of Isosceles Triangle
4) ∠𝑮𝑷𝑼 ≅ ∠𝑼𝑷𝑳 4)
5) 5) Substitution
3. Given: 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚; 𝐷𝐸̅̅̅̅ ≅ 𝐹𝐵̅̅̅̅
Prove: ∠1 ≅ ∠2
Statements Reasons
1) 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚 1)
2) 2)
3) 3) Opposite Angle Parallelogram Theorem
4) 𝐷𝐸̅̅̅̅ ≅ 𝐹𝐵̅̅̅̅ 4)
5) ∆𝑫𝑬𝑨 ≅ ∆𝑩𝑭𝑪 5)
6) ∠1 ≅ ∠2 6)
4. Given: 𝐷𝐸̅̅̅̅ ⊥ 𝐴𝐶̅̅̅̅; 𝐵𝐹̅̅̅̅ ⊥ 𝐴𝐶̅̅̅̅; 𝐴𝐸̅̅̅̅ ≅ 𝐹𝐶̅̅̅̅; 𝐷𝐸̅̅̅̅ ≅ 𝐹𝐵̅̅̅̅
Prove: ABCD is a parallelogram
Statements Reasons 1) 𝐷𝐸̅̅̅̅ ⊥ 𝐴𝐶̅̅̅̅ 1) 2) 𝐵𝐹̅̅̅̅ ⊥ 𝐴𝐶̅̅̅̅ 2) 3) 3) Definition of Perpendicular 4) ∠𝑫𝑬𝑨 ≅ ∠𝑩𝑭𝑪 4) 5) 𝐴𝐸̅̅̅̅ ≅ 𝐹𝐶̅̅̅̅ 5) 6) 𝐷𝐸̅̅̅̅ ≅ 𝐹𝐵̅̅̅̅ 6) 7) 7) SAS 8) 𝐴𝐷̅̅̅̅ ≅ 𝐵𝐶̅̅̅̅ 8) 9) 9) CPCTC 10) 𝐴𝐷̅̅̅̅ ∥ 𝐵𝐶̅̅̅̅ 10) 11) ABCD is a parallelogram 11)
5. Given: ABCD is a rectangle M is the midpoint of 𝐴𝐵̅̅̅̅ Prove: 𝐷𝑀̅̅̅̅̅ ≅ 𝐶𝑀̅̅̅̅̅ Statements Reasons 1) 1) 2) 𝐴𝐷̅̅̅̅ ≅ 𝐵𝐶̅̅̅̅ 2) 3) M is the midpoint of 𝐴𝐵̅̅̅̅ 3) 4) 𝑀𝐴̅̅̅̅̅ ≅ 𝑀𝐵̅̅̅̅̅ 4) 5) ∠𝑫𝑨𝑴 ≅ ∠𝑪𝑩𝑴 5) 6) ∆𝑫𝑨𝑴 ≅ ∆𝑪𝑩𝑴 6) 7) 𝐷𝑀̅̅̅̅̅ ≅ 𝐶𝑀̅̅̅̅̅ 7) 6. Given: 𝐻𝑆̅̅̅̅ ≅ 𝑆𝐵̅̅̅̅; 𝑅𝑆̅̅̅̅ ≅ 𝑆𝑂̅̅̅̅; 𝐻𝑅̅̅̅̅ ≅ 𝐻𝑂̅̅̅̅
Prove: RHOB is a rhombus
Statements Reasons 1) 𝐻𝑆̅̅̅̅ ≅ 𝑆𝐵̅̅̅̅ 1) 2) 𝑅𝑆̅̅̅̅ ≅ 𝑆𝑂̅̅̅̅ 2) 3) 𝐻𝑅̅̅̅̅ ≅ 𝐻𝑂̅̅̅̅ 3) 4) RHOB is a parallelogram 4) 5) 𝐻𝑅̅̅̅̅ ≅ 𝑂𝐵̅̅̅̅ 𝑅𝐵̅̅̅̅ ≅ 𝐻𝑂̅̅̅̅ 5) 6) RHOB is a rhombus 6)
7. Given: ABCD is a kite; 𝐴𝐵̅̅̅̅ ≅ 𝐴𝐷̅̅̅̅; 𝐵𝐶̅̅̅̅ ≅ 𝐷𝐶̅̅̅̅ Prove: ∠𝐵 ≅ ∠𝐷 Statements Reasons 1) 𝐴𝐵̅̅̅̅ ≅ 𝐴𝐷̅̅̅̅ 1) 2) 𝐵𝐶̅̅̅̅ ≅ 𝐷𝐶̅̅̅̅ 2) 3) 𝐴𝐶̅̅̅̅ ≅ 𝐴𝐶̅̅̅̅ 3) 4) ∆𝑨𝑩𝑪 ≅ ∆𝑨𝑫𝑪 4) 5) ∠𝑩 ≅ ∠𝑫 5)