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S: by Kaylyn Campbell 3rd Period

FC: My Geometry Scrapbook | by Kaylyn Campbell 3rd Period

1: Topics From Sections 1.1-1.5 2-3 Angles and Their Measures 4-5 Angle/Segment Bisectors 6-7 Comp., Supp., and Vertical Angles 8-9 Parallel Lines and Transversals 10-11 Perpendicular Lines 12-13 Triangles and Angle Measures 14-15 Pythagorean Theorem/Dist. Formula 16-17 Congruent Triangles 18-19 Ex: The Medians of a Triangle 4.6 20-21 Ex: Triangle Inequalities 7.2 22-23 | Table of Contents

2: Points, Lines, and Planes (1.3) | A point has no dimension. A line has one dimension and goes on forever in two directions. A plane has two dimensions. | Line | Plane

3: Real World Examples | A Pattern is a repeated sequence or design Example: Describe a pattern in the numbers. 5, 7, 10, 14, 19 Each number after the first is two more than the previous one. | Patterns | Real World Examples: | Example of Lines: The white lines on roads.

4: Angles and Their Measures | Right angles equal 90 degrees. | Acute angles measure between 0 and 90 degrees. | Obtuse angles measure between 90 and 180 degrees. | Straight angles measure 180. | Angles are made by two rays that have the same endpoint. Congruent Anlgles are angles that have the same measure. Angles are classified according to their measures, as shown in the picture on the left.

5: Real World Example: Architects frequently use right angles when building. The corner of a ceiling or window is a common example of a right angle. The edge of a window is a straight angle.

6: Segment and Angle Bisectors | To bisect is to divide into two equal parts. A Segment Bisector is a segment, ray, plane, or line that intersects a segment at its midpoint. An Angle Bisector is a ray that divides an angle into two congruent angles. | Angle Bisector | Segment Bisector

7: Example: | Find the length of -------- AB. Since segment AE= segment EB, the equation is: 6y=72 72 divided by 6=12 y=12 | Real World Example: | An example of a segment bisector are the bars of a fence. The bisect each other, as shown in the picture to the right.

8: Vertical | Supplementary | Complementary | Complementary, Supplementary, and Vertical Angles | Complementary Angles add up to equal 90 degrees. Supplementary Angles add up to equal 180 degrees. Vertical Angles are angles that are NOT adjacent, and their sides are formed by two intersecting lines (Vertical angles are Congruent).

9: Real World Example: | Example: Find the supplement of the angle. | Since supplementary angles add up to equal 180 degrees, and this angle is 45 degrees, its supplement would be 135 degrees. | This picture shows vertical angles in these tiles.

10: Parallel lines are lines that lie in the same plane and do not intersect. Transversals are lines that intersect two or more coplanar lines at different points. | Real World Example: The yellow lines on roads are parallel. | Corresponding Angles | Parallel Lines and Angles Formed by Transversals

11: Angles Formed by Transversals: Corresponding Angles are angles that occupy corresponding positions. Alternate Interior Angles are angles that lie between two lines on the opposite side of the transversal. Alternate Exterior Angles are angles that lie between two lines on the opposite side of the transversal. Same-Side Interior Angles are angles that lie in between the two lines on the same side of the transversal.

12: Lines are perpendicular if they intersect to form a right angle. A line perpendicular to a plane intersects a plane in a point and is perpendicular to every line in the plane that intersects it. | Perpendicular

13: Real World Example: When streets intersect they form right angles. | Lines

14: Triangles and Angle Measures | A triangle is a figure made by three segments joining three noncollinear points. | Classification of Triangles: By Sides: An Equilateral has 3 equal sides. An Isosceles has 2 equal sides. A Scalene has NO equal sides. By Angles: An Equiangular has 3 equal angles. | An Acute has 3 acute angles. A Right has 1 right angle. An Obtuse has 1 obtuse angle. | Isosceles Triangle | Right Triangle

15: Interior Angles are the angles between the adjacent sides of a triangle: Exterior Angles are angles that are adjacent to the interior angles: Theorem: m<1=m

16: The Pythagorean Theorem | The Pythagorean Theorem states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Formula: (hypotenuse)2 = (leg)2 + (leg)2 | Real World Example: You can use the Pythagorean Theorem to find measurements in a baseball field. The baseball diamond is a square, so when cut into, it becomes two triangles. If the length of two legs of a triangle are 90 ft, what is the measure of the hypotenuse? a2 + b2 = c2 902 + 902 = c2 8100 + 8100 = c2 16,200 = c2 ____________ _________ 16,200 = c2 = 127.3

17: The Distance Formula | The distance formula is used to show the distance between two points on a coordinate plane. Formula: AB= _________________ (x2-x1)2 + (y2-y1)2 | Real World Example: Sara wants to find the distance she will have to drive from the school to her friend Betty's house. She decides to use the distance formula to figure it out. ______________________ (5-1)2 + (5-2)2 ______________________ (4)2 + (3)2 _____________________ 16 + 9 ______________________ 25 = 5 | Points: (5,5) (1,2)

18: Congruent Triangles | Triangles are congruent if the pairs of corresponding angles and sides of one triangle are congruent with those of another triangle. | SSS | SAS | REAL WORLD EXAMPLE: | The triangles in these buildings are congruent.

19: 5 Ways to Prove Congruency: Side- Side-Side states that if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. Side-Angle-Side states that if there are two sides of one triangle with an angle in between them that are congruent to two sides of another triangle with an angle in between them, the triangles are congruent Angle-Side-Angle states that in a triangle, if two angles with a side in between them are congruent to another triangle with two angles and a side in between them, the triangles are congruent Angle-Angle-Side States that If two angles and a non-included side of one triangle is congruent to the same in another triangle, then the triangles are congruent Hypotenuse Leg States thatIf the hypotenuse and leg of one triangle is congruent to a hypotenuse and leg of another triangle, the triangles are congruent

20: Polygons | A Polygon is a plane figure that is made by 3 or more sides. A Parallelogram- is a quadrilateral whose paris of opposite sides are parallel. A Rhombus is a parallelogram with 4 equal sides. A Rectangle is a parallelogram with 4 right angles. A Square is a parallelogram with four equal sides and 4 right angles.

21: Example: Find _____ and ____ AB AC. Since the opposite sides of parallelograms are congruent, AB= 15 and AC= 9 | Real World Example: A door is an example of a rectangle.

22: Medians of a Triangle | The median of a triangle is a segment from a vertex to the midpoint of the opposite side. The centroid is the point where the three medians of a triangle intersect. The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side. | three cities

23: Example: Find the lengths of AD and AG, given that GD=48. Since AG is two thirds of the distance of AD, AG=32. 32 + 16= 48, so AD=48. | Real World Example: In the picture to the right, architects used the centroid of a triangle formed by three cities to find where to build a building in order for it to be in the center of the | three cities.

24: If one side of a triangle is longer than the other side, the angle opposite the longer side is larger than the angle opposite the shorter side. Example: Using the bottom picture, list the angles from largest to smallest. Since side AB is the longest,

25: Real World Example: | Inequalities | Architects use the Triangle Inequality Theorem when planning the Kitchen Triangle. Example: If the sum of the lengths of any two sides of a triangle is greater than the length of a third side, can the lengths in the picture below of the kitchen triangle form a triangle? No. 9 + 3 < 13 | __________________ | 9 | 13 | 3 | C

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