Modeling with Geometry
|
|
- Jayson Turner
- 6 years ago
- Views:
Transcription
1 Modeling with Geometry
2
3 6.3 Parallelograms Properties of Parallelograms Sides A parallelogram is a quadrilateral with both pairs of opposite sides parallel. If a quadrilateral is a parallelogram, the 2 pairs of opposite sides are congruent. If a quadrilateral is a parallelogram, the 2 pairs of opposite angles are congruent. Angles If a quadrilateral is a parallelogram, the consecutive angles are supplementary. If a quadrilateral is a parallelogram and one angle is a right angle, then all angles are right angles. Diagonals If a quadrilateral is a parallelogram, the diagonals bisect each other. If a quadrilateral is a parallelogram, the diagonals form two congruent triangles. Example 1: Given: ABCD is a parallelogram. Prove: AB = CD and BC = DA. Statement Reason 1. ABCD is a parallelogram Definition of a parallelogram 3. <1 = <4, <3 = < AC = AC ABC = CDA CPCTC 1
4 Example 2: Given: ABCD is a parallelogram. Prove: AC and BD bisect each other at E. Statement Reason 1. ABCD is a parallelogram 1. Given 2. AB DC <1 = <4, <2 = < AB = DC ASA 6. AE = CE, BE = DE Definition of bisector Example 3: For what values of x and y must each figure be a parallelogram? a) b) c) d) e) f) 2
5 Homework 6.3: Parallelograms Math 3 Name: 1. Use the diagram below to solve for x and y if the figure is a parallelogram. a) PT = 2x, QT = y + 12, TR = x + 2, TS = 7y b) PT = y, TR = 4y -15, QT = x + 6, TS = 4x Find the measure of each angle if the figure is a rhombus. a) Find the m 1. b) Find the m 2. c) Find the m 3. d) Find the m Solve for x if the figure is a rhombus. 4. Solve for x if the figure is a rectangle. 5. What is the length of LN if the figure is a rectangle? 6. Solve for the missing angle measures if the figure is a rhombus. 7. What is the length of SW? 8. Solve for x if the figure is a rhombus. 3
6 6.4 Quadrilaterals Rectangle Rhombus Square A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four congruent sides. A square is a parallelogram with four congruent sides and four right angles. A rectangle has all the properties of a parallelogram PLUS: 4 right angles Diagonals are congruent A rhombus has all the properties of a parallelogram PLUS: 4 congruent sides Diagonals bisect angles Diagonals are perpendicular A square has all the properties of a parallelogram PLUS: All the properties of a rectangle All the properties of a rhombus Example 1: Solve for x and the measure of each angle if DGFE is a rectangle. Example 2: ABCD is a rectangle whose diagonals intersect at point E. a) If AE = 36 and CE = 2x 4, find x. b) If BE = 6y + 2 and CE = 4y + 6, find y. Example 3: Using the diagram to the right to answer the following if ABCD is a rhombus. a) Find the m 1. b) Find the m 2. c) Find the m 3. d) Find the m 4. Example 4: Solve for each variable if the following are rhombi. a) b) 4
7 Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides, called bases, and two nonparallel sides, called legs. Isosceles Trapezoids An isosceles trapezoid is a trapezoid with congruent legs. A trapezoid is isosceles if there is only: One set of parallel sides Base angles are congruent Legs are congruent Diagonals are congruent Opposite angles are supplementary Trapezoid Midsegment The median (also called the midsegment) of a trapezoid is a segment that connects the midpoint of one leg to the midpoint of the other leg. Theorem: If a quadrilateral is a trapezoid, then a) the midsegment is parallel to the bases and b) the length of the midsegment is half the sum of the lengths of the bases Example 5: CDEP is an isosceles trapezoid and m<c = 65. What are m<d, m<e, and m<f? Example 6: What are the values of x and y in the isosceles triangle below if DE DC? Example 7: QR is the midsegment of trapezoid LMNP. What is x and the length of LM? You Try! TU is the midsegment of trapezoid WXYZ. What is x and the length of TU? Kite A kite is a quadrilateral with two pairs of adjacent, congruent sides. Its diagonals are perpendicular. If a quadrilateral is a kite, then: Its diagonals bisect the opposite angles. One pair of opposite angles are congruent. One diagonal bisects the other. Example 4: Quadrilateral DEFG is a kite. What are m<1, m<2, and m<3? You Try! Quadrilateral KLMN is a kite. What are m<1, m<2, and m<3? 5
8 Homework 6.4: Quadrilaterals Directions: For questions #1-2, find the measure of each missing angle Directions: For questions #3-4, find x and the length of EF Directions: For questions #5-6, find the measures of the numbered angles in each kite Challenge Question: Solve for the unknown angle measures in the kite shown below. 6
9 Math Tangent Lines of Circles Unit 6 SWBAT solve for unknown variables using theorems about tangent lines of circles. Tangent to a Circle Ex: (AB) A line in the plane of the circle that intersects the circle in exactly one point. Ex: Segment AB is a tangent to Circle O. Point of Tangency The point where a circle and a tangent intersect. Ex: Point P is a point of tangency on Circle O. Tangent Theorem 1: Converse Theorem 1: If a line is tangent to a circle, then it is perpendicular to the radius draw to the point of tangency. If a line is perpendicular to the radius of a circle at its endpoint on a circle, then the line is tangent to the circle. Example: If RS is tangent, then PR RS. Example 1: Find the measure of x. a) b) Example 2: Find x. All segments that appear tangent are tangent to Circle O. a) b) Example 3: Is segment MN tangent to Circle O at P? Explain. 7
10 Tangent Theorem 2: If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent. Example 4: Solve for x. To circumscribe is when you draw a figure around another, touching it at points as possible. Ex: The circle is circumscribed about the triangle. Circumscribed vs. Inscribed To inscribe is to draw a figure within another so that the inner figure lies entirely within the boundary of the outer. Ex: The triangle is inscribed in the circle. Tangent Theorem 3: (Circumscribed Polygons) When a polygon is circumscribed about a circle, all of the sides of the polygon are tangent to the circle. Example 5: Triangle ABC is circumscribed about O. Find the perimeter of triangle ABC. You Try! Find x. Assume that segments that appear to be tangent are tangent. a) b b) c) 8
11 Practice 6.7: Tangents of Circles Directions: Assume that lines that appear to be tangent are tangent. O is the center of each circle. What is the value of x? Directions: In each circle, what is the value of x to the nearest tenth? TY and ZW are diameters of S. TU and UX are tangents of S. What is msyz? Directions: Each polygon circumscribes a circle. What is the perimeter of each polygon?
12 6.8 Chords & Arcs of Circles Any segment with that are the center and a point on the circle is a. A that passes through the center is a of a circle. Example 1: Name the circle, a radius, a chord, and a diameter of the circle. Circle: Radius: Chord: Diameter: The given point is called the. This point names the circle. Any segment with that are on a circle is called a. Circle: Radius: Chord: Diameter: Since a is composed of two radii, then d = 2r and r = d/2 Theorem 1: Converse Theorem 1: Within a circle or in congruent circles, chords equidistant from the center or centers are congruent. Within a circle or in congruent circles, congruent chords are equidistant from the center (or centers). Theorem 2: Converse Theorem 2: Within a circle or in congruent circles, congruent central angles have congruent arcs. Within a circle or in congruent circles, congruent arcs have congruent central angles. Theorem 3: Converse Theorem 3: Within a circle or in congruent circles, congruent central angles have congruent chords. Within a circle or in congruent circles, congruent chords have congruent central angles. Theorem 4: Converse Theorem 4: Within a circle or in congruent circles, congruent chords have congruent arcs. Within a circle or in congruent circles, congruent arcs have congruent chords. Example 2: The following chords are equidistant from the center of the circle. a) What is the length of RS? b) Solve for x. 10
13 Theorem 5: In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. Theorem 6: In a circle, if a diameter bisects a chord that is not a diameter, then it is perpendicular to the chord. Theorem 7: In a circle, the perpendicular bisector of a chord contains the center of the circle. Example 3: In O, CD OE, OD = 15, and CD = 24. Find x. Example 4: Find the value of x to the nearest tenth. You Try! Find the value of x to the nearest tenth. a) b) 11
14 Practice 6.8: Chords & Arcs of Circles 6. A student draws X with a diameter of 12 cm. Inside the circle she inscribes equilateral ABC so that AB, BC, and CA are all chords of the circle. The diameter of X bisects AB. The section of the diameter from the center of the circle to where it bisects AB is 3 cm. To the nearest whole number, what is the perimeter of the equilateral triangle inscribed in X? 12
15 6.9 Inscribed Angles Major Arc: Minor Arc: Semicircle: An arc of a circle measuring more than or equal to 180 An arc of a circle measuring less than 180 An arc of a circle measuring 180 Central Angle: Central Angle Theorem: A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle. In a circle, or congruent circles, congruent central angles have congruent arcs. Example 1: Identify the following in P at the right. For parts d-f, find the measure of each arc in P. a) A semicircle b) A minor arc c) A major arc d) ST e) STQ f) RT Inscribed Angle: Inscribed Angle Theorem: An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords. The measure of an inscribed angle is half the measure of its intercepted arc. Example 2: What are the values of a and b? You Try! What are the m A, m B, m C, and m D? 13
16 Corollary 1: Corollary 2: Corollary 3: Two inscribed angles that intercept the same arc are congruent. An angle inscribed in a semicircle is a right angle. The opposite angles of a quadrilateral inscribed in a circle are supplementary. Example 3: What is the measure of each numbered angle? a) b) You Try! Find the measure of each numbered angle in the diagram to the right. a) m 1 = b) m 2 = c) m 3 = d) m 4 = Tangent Chord Angle: An angle formed by an intersecting tangent and chord has its vertex "on" the circle. Tangent Chord Angle Theorem: The tangent chord angle is half the measure of the intercepted arc. Tangent Chord Angle = ½ (Intercepted Arc) Example 4: In the diagram, SR is tangent to the circle at Q. If mpmq = 212, what is the m PQR? You Try! In the diagram, KJ is tangent to O. What are the values of x and y? Practice: Find the value of each variable. For each circle, the dot represents the center
17 Homework 6.9: Inscribed Angles Math 3 Name: Directions: Find the value of each variable. For each circle, the dot represents the center Directions: Find the value of each variable. Lines that appear to be tangent are tangent Directions: Find each indicated measure for M. 10. mb 11. mc Directions: Find the value of each variable. For each circle, the dot represents the center
18 6.9 Angle Measures and Segment Lengths Theorem 1: Theorem 2: The measure of an angle formed by two lines that intersect inside a circle is half the sum of the measures of the intercepted arcs. The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs. Example 1: Find each measure. a) b) c) Example 2: Find the following angles. a) m MPN b) c) You Try! Find the following angles. a) b) x c) Arc AB 16
19 Theorem 3: For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and the circle. Example 4: Find the value of the variable in O. a) b) c) You Try! What is the value of the variable to the nearest tenth? 17
20 Homework 6.9: Angles and Segments Directions: Solve for x Directions: Solve for each variable listed There is a circular cabinet in the dining room. Looking in from another room at point A, you estimate that you can see an arc of the cabinet of about 100. What is the measure of A formed by the tangents to the cabinet? Directions: Find the diameter of O. A line that appears to be tangent is tangent. If your answer is not a whole number, round to the nearest tenth Directions: CA and CB are tangents to variable. O. Write an expression for each arc or angle in terms of the given 11. using x 12. using y 13. mc using x 18
21 6.10 Equations of Circles Standard Form of Circles Center: Radius: Point on the circle: Example 1: Write the equation of a circle with the given information. a) Center (0,0), Radius=10 b) Center (2, 3), Diameter=12 h = k = r = h = k = r = Example 2: Determine the center and radius of a circle the given equation b) ( x 3) ( y 5) 81 a) x y c) ( x 4) ( y 6) 1 Example 3: Use the center and the radius to graph each circle a) ( x 2) y 64 b) x ( y 4) 36 Center: Radius: Center: Radius: Writing an Equation with a Pass-Thru Point Example 4: Write the equation of a circle with a given center (2, 5) that passes through the point (5,-1). Step 1: Substitute the center (h, k) into the equation Step 2: Substitute the pass through point (x, y) into the equation for x and y. Step 3: Simplify and solve for r 2. Step 4: Substitute r 2 back into the equation from Step 1. 19
22 Writing an Equation with Two Points on the Circle Midpoint Formula Find the midpoint (radius) between the two endpoints, and then follow steps 1-4. Example 5: Write the equation of a circle with endpoints of diameter at (-6, 5) and (4, -3). Writing the Equation of a Circle in Standard Form Step 1: Step 2: Step 3: Step 4: Group x s and group y s together. Move any constants to the right side of the equation. Use complete the square to make a perfect square trinomial for the x s and then again for the y s. *Remember, whatever you do to one side of the equation, you must do to the other! Simplify factors into standard form of a circle! Example 5: Write the equation of a circle in standard form. Then, state the center and the radius. a) x 2 + y 2 + 4x - 8y + 16 = 0 b) x 2 + y 2 + 6x - 4y = 0 c) x 2 + y 2-6x - 2y + 4 = 0 d) x 2 + y 2 + 8x - 10y - 4 = 0 20
23 Homework 6.10: Equations of Circles Note: If r 2 is not a perfect square then leave r in simplified radical form but use the decimal equivalent for graphing. Example: ) Graph the following circle: a. (x - 3) 2 + (y + 1) 2 = 4 b. (x 2) 2 + (y 5) 2 = 9 c. (y + 4) 2 + (x + 2) 2 = 16 2) For each circle, identify its center and radius. a. (x + 3) 2 + (y 1) 2 = 4 b. b. x 2 + (y 3) 2 = 18 c. (y + 8) 2 + (x + 2) 2 = 72 Center: Radius: Center: Radius: Center: Radius: 3) Write the equation of the following circles: 4) Give the equation of the circle that is tangent to the y-axis and center is (-3, 2). 5) Compare and contrast the following pairs of circles a. Circle #1: (x - 3) 2 + (y +1) 2 = 25 Circle #2: (x + 1) 2 + (y - 2) 2 = 25 b. Circle #1: (y + 4) 2 + (x + 7) 2 = 6 Circle #2: (x + 7) 2 + (y + 4) 2 = 36 21
24 6) Find the standard form, center, and radius of the following circles: a. x 2 + y 2 4x + 8y 5 = 0 b. 4x 2 + 4y y + 5 = 0 Center: Radius: Center: Radius: 7) Graph the following circles. a. x 2 2x + y 2 + 8y 8 = 0 b. x 2 + y 2 6x + 4y 3 = 0 8) Give the equation of the circle whose center is (5,-3) and goes through (2,5) 9) Give the equation whose endpoints of a diameter at (-4,1) and (4, -5) 10) Give the equation of the circle whose center is (4,-3) and goes through (1,5) 11) Give the equation whose endpoints of a diameter at (-3,2) and (1, -5) 22
25 Length of a Circular Arc Arcs have two properties. They have a measurable curvature based upon the corresponding central angle (measure of arc = measure of central angle). Arcs also have a length as a portion of the circumference. portion of circle whole circle central angle in degrees 360 central angle in radians 2 arc length circumference x length CB 360 2r -or - x (radians) 2 length CB 2r Remember: circumference of a circle = 2πr For a central angle θ in radians, and arc length s - the proportion can be simplified to a formula: 2 s 2r s2 2r s r Length of an Arc: for θ in radians s = rθ Examples: 1) For a central angle of π/6 in a circle of radius 10 cm, find the length of the intercepted arc. 2) For a central angle of 4π/7 in a circle of radius 8 in, find the length of the intercepted arc. 3) For a central angle of 40 in a circle of radius 6 cm, find the length of the intercepted arc. 4.) Find the degree measure to the nearest tenth of the central angle in a circle that has an arc length of 87 and a radius of 16 cm. 23
26 Area of a Sector Sector of a circle: a region bounded by a central angle and the intercepted arc Sectors have an area as a portion of the total area of the circle. portion of circle whole circle central angle in deg rees 360 central angle in radians 2 area of sector area of circle x 360 area of sector r 2 Remember: area of a circle = πr 2 -or - x (radians) 2 area of sec tor r 2 For a central angle θ in radians, and area of sector A, the proportion can be simplified to a formula: 2 A r 2 Area of a Circular Sector: A=½r 2 θ Examples: A2 r 2 A 1 2 r2 for θ in radians 5) Find the area of the sector of the circle that has a central angle measure of π/6 and a radius of 14 cm. 6) Find the area of the sector of the circle that has a central angle measure of 60 and a radius of 9 in. HONORS 7) A sector has arc length 12 cm and a central angle measuring 1.25 radians Find the radius of the circle and the area of the sector. 24
27 Practice: Arc Length & Area of Sectors 25
28 26
29 Review: 1. Find x What is x? Find x. 9. Find x 10. Find HG 11. Find measure of arc x 12. Find CE 27
30 13. Which is the equation of a circle with r=11 and center (0,6)? 14. Find the arc length 15. Find x 16. Find the area of the sector 17. Find the center and radius 18. Find x 19. Find x and y 20. What is the length of RS? 21. Find WS 22. What is the measure of angle 1? 23. Find CA 24. Find x 28
31 29
32 30
33 31
Videos, Constructions, Definitions, Postulates, Theorems, and Properties
Videos, Constructions, Definitions, Postulates, Theorems, and Properties Videos Proof Overview: http://tinyurl.com/riehlproof Modules 9 and 10: http://tinyurl.com/riehlproof2 Module 9 Review: http://tinyurl.com/module9livelesson-recording
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY. 3 rd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for
More informationPeriod: Date Lesson 13: Analytic Proofs of Theorems Previously Proved by Synthetic Means
: Analytic Proofs of Theorems Previously Proved by Synthetic Means Learning Targets Using coordinates, I can find the intersection of the medians of a triangle that meet at a point that is two-thirds of
More informationLesson 9: Coordinate Proof - Quadrilaterals Learning Targets
Lesson 9: Coordinate Proof - Quadrilaterals Learning Targets Using coordinates, I can find the intersection of the medians of a triangle that meet at a point that is two-thirds of the way along each median
More informationPerimeter. Area. Surface Area. Volume. Circle (circumference) C = 2πr. Square. Rectangle. Triangle. Rectangle/Parallelogram A = bh
Perimeter Circle (circumference) C = 2πr Square P = 4s Rectangle P = 2b + 2h Area Circle A = πr Triangle A = bh Rectangle/Parallelogram A = bh Rhombus/Kite A = d d Trapezoid A = b + b h A area a apothem
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY. 3 rd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES HONORS GEOMETRY 3 rd Nine Weeks, 2016-2017 1 OVERVIEW Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationPostulates, Theorems, and Corollaries. Chapter 1
Chapter 1 Post. 1-1-1 Through any two points there is exactly one line. Post. 1-1-2 Through any three noncollinear points there is exactly one plane containing them. Post. 1-1-3 If two points lie in a
More informationGeometry Final Exam - Study Guide
Geometry Final Exam - Study Guide 1. Solve for x. True or False? (questions 2-5) 2. All rectangles are rhombuses. 3. If a quadrilateral is a kite, then it is a parallelogram. 4. If two parallel lines are
More informationAngles. An angle is: the union of two rays having a common vertex.
Angles An angle is: the union of two rays having a common vertex. Angles can be measured in both degrees and radians. A circle of 360 in radian measure is equal to 2π radians. If you draw a circle with
More informationUnit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with
Unit 10 Circles 10-1 Properties of Circles Circle - the set of all points equidistant from the center of a circle. Chord - A line segment with endpoints on the circle. Diameter - A chord which passes through
More informationChapter 8. Quadrilaterals
Chapter 8 Quadrilaterals 8.1 Find Angle Measures in Polygons Objective: Find angle measures in polygons. Essential Question: How do you find a missing angle measure in a convex polygon? 1) Any convex polygon.
More informationGeometry Definitions and Theorems. Chapter 9. Definitions and Important Terms & Facts
Geometry Definitions and Theorems Chapter 9 Definitions and Important Terms & Facts A circle is the set of points in a plane at a given distance from a given point in that plane. The given point is the
More informationGeometry/Trigonometry Unit 5: Polygon Notes Period:
Geometry/Trigonometry Unit 5: Polygon Notes Name: Date: Period: # (1) Page 270 271 #8 14 Even, #15 20, #27-32 (2) Page 276 1 10, #11 25 Odd (3) Page 276 277 #12 30 Even (4) Page 283 #1-14 All (5) Page
More informationGeometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review
Geometry Unit 6 Properties of Quadrilaterals Classifying Polygons Review Polygon a closed plane figure with at least 3 sides that are segments -the sides do not intersect except at the vertices N-gon -
More informationMANHATTAN HUNTER SCIENCE HIGH SCHOOL GEOMETRY CURRICULUM
COORDINATE Geometry Plotting points on the coordinate plane. Using the Distance Formula: Investigate, and apply the Pythagorean Theorem as it relates to the distance formula. (G.GPE.7, 8.G.B.7, 8.G.B.8)
More informationGeometry. Geometry is one of the most important topics of Quantitative Aptitude section.
Geometry Geometry is one of the most important topics of Quantitative Aptitude section. Lines and Angles Sum of the angles in a straight line is 180 Vertically opposite angles are always equal. If any
More informationChapter 11 Areas of Polygons and Circles
Section 11-1: Areas of Parallelograms and Triangles SOL: G.14 The student will use similar geometric objects in two- or three-dimensions to a) compare ratios between side lengths, perimeters, areas, and
More informationName Class Date. Lines that appear to be tangent are tangent. O is the center of each circle. What is the value of x?
12-1 Practice Tangent Lines Lines that appear to be tangent are tangent. O is the center of each circle. What is the value of x? 1. To start, identify the type of geometric figure formed by the tangent
More information6.1 Circles and Related Segments and Angles
Chapter 6 Circles 6.1 Circles and Related Segments and Angles Definitions 32. A circle is the set of all points in a plane that are a fixed distance from a given point known as the center of the circle.
More informationU4 Polygon Notes January 11, 2017 Unit 4: Polygons
Unit 4: Polygons 180 Complimentary Opposite exterior Practice Makes Perfect! Example: Example: Practice Makes Perfect! Def: Midsegment of a triangle - a segment that connects the midpoints of two sides
More information22. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
Chapter 4 Quadrilaterals 4.1 Properties of a Parallelogram Definitions 22. A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. 23. An altitude of a parallelogram is the
More informationSecondary Math II Honors. Unit 4 Notes. Polygons. Name: Per:
Secondary Math II Honors Unit 4 Notes Polygons Name: Per: Day 1: Interior and Exterior Angles of a Polygon Unit 4 Notes / Secondary 2 Honors Vocabulary: Polygon: Regular Polygon: Example(s): Discover the
More informationMath 3315: Geometry Vocabulary Review Human Dictionary: WORD BANK
Math 3315: Geometry Vocabulary Review Human Dictionary: WORD BANK [acute angle] [acute triangle] [adjacent interior angle] [alternate exterior angles] [alternate interior angles] [altitude] [angle] [angle_addition_postulate]
More informationEQUATIONS OF ALTITUDES, MEDIANS, and PERPENDICULAR BISECTORS
EQUATIONS OF ALTITUDES, MEDIANS, and PERPENDICULAR BISECTORS Steps to Find the Median of a Triangle: -Find the midpoint of a segment using the midpoint formula. -Use the vertex and midpoint to find the
More informationGEOMETRY COORDINATE GEOMETRY Proofs
GEOMETRY COORDINATE GEOMETRY Proofs Name Period 1 Coordinate Proof Help Page Formulas Slope: Distance: To show segments are congruent: Use the distance formula to find the length of the sides and show
More informationTopic 7: Properties of Circles
This Packet Belongs to (Student Name) Topic 7: Properties of Circles Unit 6 Properties of Circles Module 15: Angles and Segments in Circles 15.1 Central Angles and Inscribed Angles 15.2 Angles in Inscribed
More information6-1 Study Guide and Intervention Angles of Polygons
6-1 Study Guide and Intervention Angles of Polygons Polygon Interior Angles Sum The segments that connect the nonconsecutive vertices of a polygon are called diagonals. Drawing all of the diagonals from
More informationTENTH YEAR MATHEMATICS
10 The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION TENTH YEAR MATHEMATICS Wednesday, August 16, 1967-8 :30 to 11 :30 a.m., only The last page of the booklet is the answer sheet,
More informationChapter 2 Diagnostic Test
Chapter Diagnostic Test STUDENT BOOK PAGES 68 7. Calculate each unknown side length. Round to two decimal places, if necessary. a) b). Solve each equation. Round to one decimal place, if necessary. a)
More informationGeometry Third Quarter Study Guide
Geometry Third Quarter Study Guide 1. Write the if-then form, the converse, the inverse and the contrapositive for the given statement: All right angles are congruent. 2. Find the measures of angles A,
More informationUnit 2 Study Guide Topics: Transformations (Activity 9) o Translations o Rotations o Reflections. o Combinations of Transformations
Geometry Name Unit 2 Study Guide Topics: Transformations (Activity 9) o Translations o Rotations o Reflections You are allowed a 3 o Combinations of Transformations inch by 5 inch Congruent Polygons (Activities
More informationChapter 10 Similarity
Chapter 10 Similarity Def: The ratio of the number a to the number b is the number. A proportion is an equality between ratios. a, b, c, and d are called the first, second, third, and fourth terms. The
More informationGet Ready. Solving Equations 1. Solve each equation. a) 4x + 3 = 11 b) 8y 5 = 6y + 7
Get Ready BLM... Solving Equations. Solve each equation. a) 4x + = 8y 5 = 6y + 7 c) z+ = z+ 5 d) d = 5 5 4. Write each equation in the form y = mx + b. a) x y + = 0 5x + y 7 = 0 c) x + 6y 8 = 0 d) 5 0
More informationGEOMETRY is the study of points in space
CHAPTER 5 Logic and Geometry SECTION 5-1 Elements of Geometry GEOMETRY is the study of points in space POINT indicates a specific location and is represented by a dot and a letter R S T LINE is a set of
More informationGeometry Rules. Triangles:
Triangles: Geometry Rules 1. Types of Triangles: By Sides: Scalene - no congruent sides Isosceles - 2 congruent sides Equilateral - 3 congruent sides By Angles: Acute - all acute angles Right - one right
More informationSection Congruence Through Constructions
Section 10.1 - Congruence Through Constructions Definitions: Similar ( ) objects have the same shape but not necessarily the same size. Congruent ( =) objects have the same size as well as the same shape.
More information10-1 Circles & Circumference
10-1 Circles & Circumference Radius- Circle- Formula- Chord- Diameter- Circumference- Formula- Formula- Two circles are congruent if and only if they have congruent radii All circles are similar Concentric
More information2.1 Length of a Line Segment
.1 Length of a Line Segment MATHPOWER TM 10 Ontario Edition pp. 66 7 To find the length of a line segment joining ( 1 y 1 ) and ( y ) use the formula l= ( ) + ( y y ). 1 1 Name An equation of the circle
More informationNAME DATE PERIOD. Areas of Parallelograms and Triangles. Review Vocabulary Define parallelogram in your own words. (Lesson 6-2)
11-1 Areas of Parallelograms and Triangles What You ll Learn Skim Lesson 11-1. Predict two things you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary
More informationMadison County Schools Suggested Geometry Pacing Guide,
Madison County Schools Suggested Geometry Pacing Guide, 2016 2017 Domain Abbreviation Congruence G-CO Similarity, Right Triangles, and Trigonometry G-SRT Modeling with Geometry *G-MG Geometric Measurement
More informationMADISON ACADEMY GEOMETRY PACING GUIDE
MADISON ACADEMY GEOMETRY PACING GUIDE 2018-2019 Standards (ACT included) ALCOS#1 Know the precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined
More information1. AREAS. Geometry 199. A. Rectangle = base altitude = bh. B. Parallelogram = base altitude = bh. C. Rhombus = 1 product of the diagonals = 1 dd
Geometry 199 1. AREAS A. Rectangle = base altitude = bh Area = 40 B. Parallelogram = base altitude = bh Area = 40 Notice that the altitude is different from the side. It is always shorter than the second
More informationCURRICULUM GUIDE. Honors Geometry
CURRICULUM GUIDE Honors Geometry This level of Geometry is approached at an accelerated pace. Topics of postulates, theorems and proofs are discussed both traditionally and with a discovery approach. The
More informationPRACTICE TEST ANSWER KEY & SCORING GUIDELINES GEOMETRY
Ohio s State Tests PRACTICE TEST ANSWER KEY & SCORING GUIDELINES GEOMETRY Table of Contents Questions 1 30: Content Summary and Answer Key... iii Question 1: Question and Scoring Guidelines... 1 Question
More informationadded to equal quantities, their sum is equal. Same holds for congruence.
Mr. Cheung s Geometry Cheat Sheet Theorem List Version 6.0 Updated 3/14/14 (The following is to be used as a guideline. The rest you need to look up on your own, but hopefully this will help. The original
More informationMoore Catholic High School Math Department
Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during
More informationLesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms
Lesson 4.3 Ways of Proving that Quadrilaterals are Parallelograms Getting Ready: How will you know whether or not a figure is a parallelogram? By definition, a quadrilateral is a parallelogram if it has
More informationA closed plane figure with at least 3 sides The sides intersect only at their endpoints. Polygon ABCDEF
A closed plane figure with at least 3 sides The sides intersect only at their endpoints B C A D F E Polygon ABCDEF The diagonals of a polygon are the segments that connects one vertex of a polygon to another
More informationCongruent triangles/polygons : All pairs of corresponding parts are congruent; if two figures have the same size and shape.
Jan Lui Adv Geometry Ch 3: Congruent Triangles 3.1 What Are Congruent Figures? Congruent triangles/polygons : All pairs of corresponding parts are congruent; if two figures have the same size and shape.
More informationfall08ge Geometry Regents Exam Test Sampler fall08 4 The diagram below shows the construction of the perpendicular bisector of AB.
fall08ge 1 Isosceles trapezoid ABCD has diagonals AC and BD. If AC = 5x + 13 and BD = 11x 5, what is the value of x? 1) 8 4 The diagram below shows the construction of the perpendicular bisector of AB.
More informationReview: What is the definition of a parallelogram? What are the properties of a parallelogram? o o o o o o
Geometry CP Lesson 11-1: Areas of Parallelograms Page 1 of 2 Objectives: Find perimeters and areas of parallelograms Determine whether points on a coordinate plane define a parallelogram CA Geometry Standard:
More informationFGCU Invitational Geometry Individual 2014
All numbers are assumed to be real. Diagrams are not drawn to scale. For all questions, NOTA represents none of the above answers is correct. For problems 1 and 2, refer to the figure in which AC BC and
More informationALLEGHANY COUNTY SCHOOLS CURRICULUM GUIDE
GRADE/COURSE: Geometry GRADING PERIOD: 1 Year Course Time SEMESTER 1: 1 ST SIX WEEKS Pre-Test, Class Meetings, Homeroom Chapter 1 12 days Lines and Angles Point Line AB Ray AB Segment AB Plane ABC Opposite
More informationCommon Core Specifications for Geometry
1 Common Core Specifications for Geometry Examples of how to read the red references: Congruence (G-Co) 2-03 indicates this spec is implemented in Unit 3, Lesson 2. IDT_C indicates that this spec is implemented
More informationNotes Circle Basics Standard:
Notes Circle Basics M RECALL EXAMPLES Give an example of each of the following: 1. Name the circle 2. Radius 3. Chord 4. Diameter 5. Secant 6. Tangent (line) 7. Point of tangency 8. Tangent (segment) DEFINTION
More informationRussell County Pacing Guide
August Experiment with transformations in the plane. 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance
More informationMoore Catholic High School Math Department
Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during
More informationMidpoint of a Line Segment Pg. 78 # 1, 3, 4-6, 8, 18. Classifying Figures on a Cartesian Plane Quiz ( )
UNIT 2 ANALYTIC GEOMETRY Date Lesson TOPIC Homework Feb. 22 Feb. 23 Feb. 24 Feb. 27 Feb. 28 2.1 2.1 2.2 2.2 2.3 2.3 2.4 2.5 2.1-2.3 2.1-2.3 Mar. 1 2.6 2.4 Mar. 2 2.7 2.5 Mar. 3 2.8 2.6 Mar. 6 2.9 2.7 Mar.
More informationGeometry Vocabulary Math Fundamentals Reference Sheet Page 1
Math Fundamentals Reference Sheet Page 1 Acute Angle An angle whose measure is between 0 and 90 Acute Triangle A that has all acute Adjacent Alternate Interior Angle Two coplanar with a common vertex and
More informationDEFINITIONS. Perpendicular Two lines are called perpendicular if they form a right angle.
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
More informationUnit Number of Days Dates. 1 Angles, Lines and Shapes 14 8/2 8/ Reasoning and Proof with Lines and Angles 14 8/22 9/9
8 th Grade Geometry Curriculum Map Overview 2016-2017 Unit Number of Days Dates 1 Angles, Lines and Shapes 14 8/2 8/19 2 - Reasoning and Proof with Lines and Angles 14 8/22 9/9 3 - Congruence Transformations
More information3.1 Investigate Properties of Triangles Principles of Mathematics 10, pages
3.1 Investigate Properties of Triangles Principles of Mathematics 10, pages 110 116 A 1. The area of PQR is 16 square units. Find the area of PQS. the bisector of F the right bisector of side EF the right
More informationTheorems & Postulates Math Fundamentals Reference Sheet Page 1
Math Fundamentals Reference Sheet Page 1 30-60 -90 Triangle In a 30-60 -90 triangle, the length of the hypotenuse is two times the length of the shorter leg, and the length of the longer leg is the length
More informationIndex COPYRIGHTED MATERIAL. Symbols & Numerics
Symbols & Numerics. (dot) character, point representation, 37 symbol, perpendicular lines, 54 // (double forward slash) symbol, parallel lines, 54, 60 : (colon) character, ratio of quantity representation
More informationGeometry. Geometry. Domain Cluster Standard. Congruence (G CO)
Domain Cluster Standard 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance
More informationTest #1: Chapters 1, 2, 3 Test #2: Chapters 4, 7, 9 Test #3: Chapters 5, 6, 8 Test #4: Chapters 10, 11, 12
Progress Assessments When the standards in each grouping are taught completely the students should take the assessment. Each assessment should be given within 3 days of completing the assigned chapters.
More informationA. 180 B. 108 C. 360 D. 540
Part I - Multiple Choice - Circle your answer: REVIEW FOR FINAL EXAM - GEOMETRY 2 1. Find the area of the shaded sector. Q O 8 P A. 2 π B. 4 π C. 8 π D. 16 π 2. An octagon has sides. A. five B. six C.
More informationStandards to Topics. Common Core State Standards 2010 Geometry
Standards to Topics G-CO.01 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance
More information2 Formula (given): Volume of a Pyramid V = 1/3 BH What does B represent? Formula: Area of a Trapezoid. 3 Centroid. 4 Midsegment of a triangle
1 Formula: Area of a Trapezoid 2 Formula (given): Volume of a Pyramid V = 1/3 BH What does B represent? 3 Centroid 4 Midsegment of a triangle 5 Slope formula 6 Point Slope Form of Linear Equation *can
More informationPearson Mathematics Geometry Common Core 2015
A Correlation of Pearson Mathematics Geometry Common Core 2015 to the Common Core State Standards for Bid Category 13-040-10 A Correlation of Pearson, Common Core Pearson Geometry Congruence G-CO Experiment
More informationGeometry SOL Review Packet QUARTER 3
Geometry SOL Review Packet QUARTER 3 Arc Length LT 10 Circle Properties Important Concepts to Know Sector Area It is a fraction of. It is a fraction of. Formula: Formula: Central Angle Inscribed Angle
More informationDISTANCE FORMULA: to find length or distance =( ) +( )
MATHEMATICS ANALYTICAL GEOMETRY DISTANCE FORMULA: to find length or distance =( ) +( ) A. TRIANGLES: Distance formula is used to show PERIMETER: sum of all the sides Scalene triangle: 3 unequal sides Isosceles
More informationInstructional Unit CPM Geometry Unit Content Objective Performance Indicator Performance Task State Standards Code:
306 Instructional Unit Area 1. Areas of Squares and The students will be -Find the amount of carpet 2.4.11 E Rectangles able to determine the needed to cover various plane 2. Areas of Parallelograms and
More informationParallelograms. MA 341 Topics in Geometry Lecture 05
Parallelograms MA 341 Topics in Geometry Lecture 05 Definitions A quadrilateral is a polygon with 4 distinct sides and four vertices. Is there a more precise definition? P 1 P 2 P 3 09-Sept-2011 MA 341
More informationGeometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never
1stSemesterReviewTrueFalse.nb 1 Geometry (H) Worksheet: 1st Semester Review:True/False, Always/Sometimes/Never Classify each statement as TRUE or FALSE. 1. Three given points are always coplanar. 2. A
More informationCircles - Probability
Section 10-1: Circles and Circumference SOL: G.10 The student will investigate and solve practical problems involving circles, using properties of angles, arcs, chords, tangents, and secants. Problems
More informationAppendix E. Plane Geometry
Appendix E Plane Geometry A. Circle A circle is defined as a closed plane curve every point of which is equidistant from a fixed point within the curve. Figure E-1. Circle components. 1. Pi In mathematics,
More informationFSA Geometry End-of-Course Review Packet. Circles Geometric Measurement and Geometric Properties
FSA Geometry End-of-Course Review Packet Circles Geometric Measurement and Geometric Properties Table of Contents MAFS.912.G-C.1.1 EOC Practice... 3 MAFS.912.G-C.1.2 EOC Practice... 5 MAFS.912.G-C.1.3
More informationGeometry/Pre AP Geometry Common Core Standards
1st Nine Weeks Transformations Transformations *Rotations *Dilation (of figures and lines) *Translation *Flip G.CO.1 Experiment with transformations in the plane. Know precise definitions of angle, circle,
More informationGeometry: Traditional Pathway
GEOMETRY: CONGRUENCE G.CO Prove geometric theorems. Focus on validity of underlying reasoning while using variety of ways of writing proofs. G.CO.11 Prove theorems about parallelograms. Theorems include:
More informationGeometry First Semester Practice Final (cont)
49. Determine the width of the river, AE, if A. 6.6 yards. 10 yards C. 12.8 yards D. 15 yards Geometry First Semester Practice Final (cont) 50. In the similar triangles shown below, what is the value of
More informationSelect the best answer. Bubble the corresponding choice on your scantron. Team 13. Geometry
Team Geometry . What is the sum of the interior angles of an equilateral triangle? a. 60 b. 90 c. 80 d. 60. The sine of angle A is. What is the cosine of angle A? 6 4 6 a. b. c.. A parallelogram has all
More informationName Date Class. 6. In JKLM, what is the value of m K? A 15 B 57 A RS QT C QR ST
Name Date Class CHAPTER 6 Chapter Review #1 Form B Circle the best answer. 1. Which best describes the figure? 6. In JKLM, what is the value of m K? A regular convex heptagon B irregular convex heptagon
More informationStudent Name: Teacher: Date: Miami-Dade County Public Schools. Test: 9_12 Mathematics Geometry Exam 2
Student Name: Teacher: Date: District: Miami-Dade County Public Schools Test: 9_12 Mathematics Geometry Exam 2 Description: GEO Topic 5: Quadrilaterals and Coordinate Geometry Form: 201 1. If the quadrilateral
More informationProving Theorems about Lines and Angles
Proving Theorems about Lines and Angles Angle Vocabulary Complementary- two angles whose sum is 90 degrees. Supplementary- two angles whose sum is 180 degrees. Congruent angles- two or more angles with
More informationGeometry 1 st Semester Exam REVIEW Chapters 1-4, 6. Your exam will cover the following information:
Geometry 1 st Semester Exam REVIEW Chapters 1-4, 6 Your exam will cover the following information: Chapter 1 Basics of Geometry Chapter 2 Logic and Reasoning Chapter 3 Parallel & Perpendicular Lines Chapter
More informationMATH II SPRING SEMESTER FINALS REVIEW PACKET
Name Date Class MATH II SPRING SEMESTER FINALS REVIEW PACKET For 1 2, use the graph. 6. State the converse of the statement. Then determine whether the converse is true. Explain. If two angles are vertical
More informationm 6 + m 3 = 180⁰ m 1 m 4 m 2 m 5 = 180⁰ m 6 m 2 1. In the figure below, p q. Which of the statements is NOT true?
1. In the figure below, p q. Which of the statements is NOT true? m 1 m 4 m 6 m 2 m 6 + m 3 = 180⁰ m 2 m 5 = 180⁰ 2. Look at parallelogram ABCD below. How could you prove that ABCD is a rhombus? Show that
More informationProblems #1. A convex pentagon has interior angles with measures (5x 12), (2x + 100), (4x + 16), (6x + 15), and (3x + 41). Find x.
1 Pre-AP Geometry Chapter 10 Test Review Standards/Goals: G.CO.11/ C.1.i.: I can use properties of special quadrilaterals in a proof. D.2.g.: I can identify and classify quadrilaterals, including parallelograms,
More informationMathematics Standards for High School Geometry
Mathematics Standards for High School Geometry Geometry is a course required for graduation and course is aligned with the College and Career Ready Standards for Mathematics in High School. Throughout
More informationGeometry Semester 1 Model Problems (California Essential Standards) Short Answer
Geometry Semester 1 Model Problems (California Essential Standards) Short Answer GE 1.0 1. List the undefined terms in Geometry. 2. Match each of the terms with the corresponding example a. A theorem.
More information2) Find the value of x. 8
In the figure at the right, ABC is similar to DEF. 1) Write three equal ratios to show corresponding sides are proportional. D 16 E x 9 B F 2) Find the value of x. 8 y A 16 C 3) Find the value of y. Determine
More informationChapter 6. Sir Migo Mendoza
Circles Chapter 6 Sir Migo Mendoza Central Angles Lesson 6.1 Sir Migo Mendoza Central Angles Definition 5.1 Arc An arc is a part of a circle. Types of Arc Minor Arc Major Arc Semicircle Definition 5.2
More informationGiven a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G- CO.1 Identify Definitions Standard 1 Experiment with transformations in the plane. Know precise definitions of angle, circle, perpendicular line, parallel line, or line segment, based on the undefined
More information6.1: Date: Geometry. Polygon Number of Triangles Sum of Interior Angles
6.1: Date: Geometry Polygon Number of Triangles Sum of Interior Angles Triangle: # of sides: # of triangles: Quadrilateral: # of sides: # of triangles: Pentagon: # of sides: # of triangles: Hexagon: #
More informationName: Date: Period: Chapter 11: Coordinate Geometry Proofs Review Sheet
Name: Date: Period: Chapter 11: Coordinate Geometry Proofs Review Sheet Complete the entire review sheet (on here, or separate paper as indicated) h in on test day for 5 bonus points! Part 1 The Quadrilateral
More informationGeometry Practice. 1. Angles located next to one another sharing a common side are called angles.
Geometry Practice Name 1. Angles located next to one another sharing a common side are called angles. 2. Planes that meet to form right angles are called planes. 3. Lines that cross are called lines. 4.
More informationtheorems & postulates & stuff (mr. ko)
theorems & postulates & stuff (mr. ko) postulates 1 ruler postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of
More informationProving Triangles and Quadrilaterals Satisfy Transformational Definitions
Proving Triangles and Quadrilaterals Satisfy Transformational Definitions 1. Definition of Isosceles Triangle: A triangle with one line of symmetry. a. If a triangle has two equal sides, it is isosceles.
More informationGeometry. Cluster: Experiment with transformations in the plane. G.CO.1 G.CO.2. Common Core Institute
Geometry Cluster: Experiment with transformations in the plane. G.CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of
More information