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Geometry Notes Q – 1: Quadrilaterals Definition: A quadrilateral is a polygon with Theorem: The sum of the interior angles in a quadrilateral is A D Given: Quadrilateral ABCD Prove: mA + mB + mC + mD = 360 Statement Reason B C Ex: In quadrilateral CRAB, CR BC , mR = x 2 , mA = 7x and mB = (2x + 50). Find the measures of all four angles of CRAB. Review of Coordinate Geometry Formulas Slope formula: Distance Formula: Midpoint Formula: Four basic proofs To prove two segments congruent: To prove two segments parallel: To prove two segments perpendicular: To prove two segments bisect each other: Parts of a coordinate geometry proof Graph or diagram (optional but usually very helpful) Labeled calculations. For example: Appropriate conclusions and justifications. For example: Geometry Notes Q – 2: Parallelograms I Definition: A parallelogram is a quadrilateral with Properties of Parallelograms 1. 2. 3. 4. A B Given: Parallelogram ABCD. Prove: 2, 3 and 4 above. D Note: We have also proved the following theorem: A diagonal divides a parallelogram into C Ex: In parallelogram TOAD, TO = x2 – 1, OA = x + y, AD = 9x – 3y and DT = 4x – 8. a. Find the perimeter of TOAD. b. Find all the possible values of the height of TOAD measured from AD . c. If the height of TOAD measured from AD is 9, what is the area of TOAD. Ex: In parallelogram PIKA, mP = (3x + 2y), mI = (2x + y) and K = (3y + 24). Find the numerical measure of A. Ex: In quadrilateral HARD, mH = x 2 , mA= (x2 + x + 10), mR = (160 – 9x) and D = (190 – 10x). Is HARD a parallelogram? Geometry Notes Q – 3: Parallelograms II Theorem: The diagonals of a parallelogram A B Given: Parallelogram ABCD, diagonals AC and BD intersect at E. Prove: AC and BD E D C E Ex: Parallelogram EASY has diagonals intersecting at R. Find the lengths of the diagonals. A x 2y – 8 Y x+4 R y–2 S Geometry Notes Q – 5: Proving quadrilaterals are parallelograms Properties of Parallelograms: If a quadrilateral is a parallelogram, then 1. 2. 3. 4. 5. 6. Proving a quadrilateral is a parallelogram: A quadrilateral is a parallelogram if 1. or 2. or 3. or 4. or 5. Given: Quadrilateral ABCD with A C and B D. Prove: Quadrilateral ABCD is a parallelogram. A D B C Ex: Determine if each of the following is a parallelogram and give a reason. a. Given: a > b a2 b2 A B 25 25 D (a + b)(a b) b. x > 3 C 1 x 3 2 A B 40 40 D c. x3 2 C A x+1 20 x 10 3 x2 13 D B 3x 3 1 (4x + 2) 2x 4 C Geometry Notes Q – 6: Rectangles Theorem: If two supplementary angles are congruent, then Given: A supplementary to B; A B Prove: A and B are right angles Theorem: If two angles are supplementary and one is a right angle, then Given: A supplementary to B; A is a right angle Prove: B is a right angle Definition: A rectangle is a quadrilateral with Note: A rectangle is a Properties of rectangles: 1. 2. 3. Ex: In rectangle RECT, the diagonals intersect at M, RT = 6 and RM = 5. Find the perimeter of TMC. Theorem: A parallelogram with one right angle is a rectangle. A B D C A B D C Given: Parallelogram ABCD, A is a right angle Prove: ABCD is a rectangle Theorem: A parallelogram with congruent diagonals is a rectangle. Given: Parallelogram ABCD, AC BD Prove: ABCD is a rectangle Proving a quadrilateral is a rectangle: A quadrilateral is a rectangle if 1. or 2. or 3. Ex: Determine if each of the following is a rectangle and give a reason. a. b. A B D C A 25 B 65 D c. C A D B AC BD C Geometry Notes Q – 7: Rhombi & Squares Definition: A rhombus is a quadrilateral with Note: A rhombus is a Properties of rhombi: 1. 2. A 3. B 4. Given: Rhombus ABCD with diagonal AC Prove: AC bisects BAD and BCD D C A Given: Rhombus ABCD; diagonals AC and BD intersect at E. Prove: AC BD B E D C Ex: The diagram at right shows a rhombus with both diagonals drawn. Find the values of x, y and z. 25 y x z Proving a quadrilateral is a rhombus: A quadrilateral is a rhombus if 1. or 2. or 3. Ex: Quadrilateral RHOM has vertices R(2, 4), H(–5, 5), O(0, 0) and M(7, –1). Prove using coordinate geometry that RHOM is a rhombus. y H R O x M Square Definition: A square is a quadrilateral Note: A square is a Properties of squares: All of the properties of Geometry Notes Q – 10: Proofs practice Ex: Given: Quadrilateral WXYZ, diagonals WY and XZ intersect at V, WX || YZ , V is the midpoint of XZ , WZ YZ . X W Prove: WXYZ is a rectangle V Z Y Ex: Given: Rhombus RSTV, VTX , STW , RS SX , RV VW Prove: TX TW R S V T W X Geometry Notes Q - R: Review and Practice Quadrilaterals 1. C General: B A 2. D B Trapezoid: C A D 2.1. Isosceles trapezoid: B C A 3. Parallelogram: B A 3a. Proving a quadrilateral is a parallelogram D C D 3.1 Rectangle: B C A D 3.1a Proving a quadrilateral is a rectangle 3.2 Rhombus: C B A D B C A D 3.2a Proving a quadrilateral is a rhombus 3.3 Square: 3.3a Proving a quadrilateral is a square