FC: Geometry Scrapbook 2nd Period Geometry By: Anthony Rivera
1: Table of Contents | 2-3 Geometric Basics 4-5 Angles and their Measures 6-7 Angle Bisectors, Segment Bisectors 8-9 Types of Angles 10-11 Parallel Lines 12-13 Perpendicular Lines 14-15 Triangles 16-17 Pythagorean Theorem, Distance Formula 18-19 Congruent Triangles 20-21 Polygons | Page
2: A line has one dimension and is represented by two arrowheads. They can have any number of points on them. Example of a line shown below. | Points in geometry are represented by a dot and have no dimension. A picture example is here shown by points A, P, and M
3: Figures intersect if they have any points in common. The intersection is the point that two figures have in common. In the example above, the intersection is marked by a brown circle. | In the real world, you could connect this to a map. Points being cities and the latitude lines are lines.
4: Angles and Measures | An angle consists of two rays that have the same point or vertex. | The measure of an angle is written in degrees. | Angles can either be acute, right, obtuse, or straight. To the right is an example of an obtuse angle, greater than 90 degrees and smaller than 180 degrees.
5: Angles and Measures | Angles are always greater than 0 degrees and smaller than or equal to 180 degrees. | To the right is a right angle, which are always 90 degrees | Below is an acute angle. Acute angles are angles that measure less than 90 degrees. | An example of angles in real life is the hands of a clock. At 3:00, the hour and minute hand form a right angle.
6: Angle and Segment Bisectors | An angle bisector is a ray that divides an angle into to equally measured angles. In this picture, the entire angle is 60 degrees, but ray A is placed in the middle | splitting the angle into two congruent angles. | An example of an angle bisector in real life is in architecture. The angle created by wood in this window is being bisected.
7: Angle and Segment Bisectors | A segment bisector is a ray, line, segment, or plane that intersects a segment at its midpoint. | In this picture, line l splits AB into two equal distances at point M. Because of this, AM and MB are congruent. | A segment bisector in real life can be found in some airplanes. In the picture to the right, the wingspan of 20 ft. is divided in half by the body of the plane.
8: Complementary, Supplementary, and Vertical Angles | Two angles are complementary when the sum of their angles is equal to 90 degrees, a picture of this is to the right. | Two angles are supplementary when the sum of their angles is equal to 180 degrees. | In this picture, the sum of the two angles is 180 degrees.
9: Complementary, Supplementary, and Vertical Angles | Vertical angles are not adjacent and their sides are formed by two intersecting lines. In this picture, the right angle's vertical is to the bottom left. Vertical angles are always congruent. | Complementary Angles can be found in many fences and decorations. A picture of this is to the right.
10: Parallel Lines and Transversal Angles | Parallel lines are lines that lie in the same plane and never intersect. | The transversal is a line that intersects two other lines at different points. | Corresponding angles are: formed by the transversal and congruent. In this example, 1 and 6 would be congruent.
11: Parallel Lines and Transversal Angles | Alternate Interior Angles lie between the two lines on opposite sides of the transversal. In this picture, 3 and 6 would be alternate interior angles. | Alternate Exterior Angles lie outside the two lines on opposite sides of the transversal. In this picture, 2 and 7 are alternate exterior angles. | An example of parallel lines in real life is train tracks. Train tracks are always parallel.
12: Perpendicular Lines | Perpendicular Lines are lines that intersect to form right angles. | If two lines are perpendicular to the same line, then they are parallel to each other. In this picture, lines R and S are perpendicular to line T. This makes lines R&S parallel.
13: Perpendicular Lines | Perpendicular Lines can be found in sidewalks where the concrete is separated to prevent cracking. | Also for perpendicular lines, if there is a line and a point not on a line in the same plane,the there is one line through that point perpendicular to the other line.
14: Triangles | A triangle is a figure formed by three non collinear points whose angles add up to 180 degrees. This triangle is a scalene triangle and doesn't have any congruent sides. | To the left is an obtuse triangle. Obtuse triangles have one obtuse angle. | The triangle to the right is a right triangle. It has only one right angle.
15: Triangles | Isosceles triangles have two congruent sides. | Equilateral triangles have three congruent sides. The triangle to the left is also an acute triangle, having three acute angles. | Triangles can be found in art in the real world. The picture to the right is a fan's replication of a symbol in a game.
16: Pythagorean Theorem & Distance Formula | The Pythagorean Theorem is: in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. | Pythagorean Theorem can be used to find the distance from the top of a tree to the end of its shadow. The tree's height forms one leg and the shadow length the other leg. Square the two legs, find the sum, then find the square root and you have the distance. | With the Pythagorean theorem, the longest side of a triangle can be found if you know the length of the two legs. The first leg (A) squared plus the second leg (B) squared equals the hypotenuse (C) squared. Then find the square root of C to get it's length.
17: Pythagorean Theorem & Distance Formula | The Distance formula is: if A(x1, y1) and B (x2, y2) are points in a coordinate plane, then the distance between A and B is AB= Square root of (x2-x1)2 + (y2-y1)2. | The distance formula is the square root of (x2-x1)2 + (y2-y1)2. In this picture x1,y1 is -3,6 and x2,y2 is 3,-3. Subtract (x2-x1): -3 -3 to get -6 and square it to get 36. Next subtract (y2-y1): -3 - 6 to get -9 and squared to end with 81. Add 36+81 for 117. Now you have to find the square root. The square root and distance of 117 equals 10.82.
18: Triangle Congruence | Triangles are congruent if all corresponding parts are congruent. The ways to find this out are Angle-Side-Angle, Hypotenuse-Leg for right triangles, Side-Angle-Side, Side-Side-Side, and Angle-Angle-Side. | Side-Side-Side congruence is possible when all sides are congruent, similar to the picture to the left. | With Hypotenuse-Leg, in a right triangle, if you know the length of the hypotenuse and the length of one leg, you can determine that two triangles are congruent. | Side-Angle-Side works when you know the congruent length of two sides of the triangle, and the measure of the angle between those two sides.
19: Triangle Congruence | Angle-Side-Angle is usable when the measure of two angles is known and the length of the side in between is also known. | Angle-Angle-Side can only be used when two angles are known and any side other than the one between the two angles is known. | Congruent triangles are used in real life for bridges. Triangles distribute weight better than quadrilaterals making it useful for construction.
20: Polygons | A polygon is a plane figure that is formed by three or more segments called sides. Each point in a polygon is a vertex of the polygon. | Quadrilaterals are 4 sided polygons. The angles within a quadrilateral always add up to 360 degrees. | Parallelograms are quadrilaterals with at least one pair of parallel lines. Both pairs of opposite angles are also congruent in most parallelograms, but not all. | Pentagons are polygons that have 5 sides.
21: Polygons | Polygons in real life can be found in virtually anything. Solar panels are a good example. They are rectangular quadrilaterals. | To call a figure a polygon, it cannot have any curved sides. The side must also be connected to each other. | The picture above is not a polygon because it has a curved side. The picture to the left is not a polygon because the sides do not connect.