FC: The Year in Review: Geometric Design Project | Ellie von Palko 2nd Period Geometry December 12th, 2012
1: Table of Contents pg. 2: Table of Contents, pt. 2 pg. 3: Table of Contents, pt. 3 pg. 4: Learning the Basics pg. 5: Learning the Basics—Real-World Examples pg. 6: At this Angle pg. 7: At this Angle—Real-World Examples pg. 8: The Two Bisectors pg. 9: The Two Bisectors—Real-World Examples pg. 10: Complements and Supplements pg. 11: Complements and Supplements—Real-World Examples pg. 12: Find the Parallels pg. 13: Find the Parallels—Real-World Examples
2: Table of Contents, pt. 2 pg. 14: Peculiar and Perpendicular pg. 15: Peculiar and Perpendicular—Real-World Examples pg. 16: Triangular Theorems pg. 17: Triangular Theorems—Real-World Examples pg. 18: Pythagorean Formula and Distance Theorem...wait... pg. 19: Pythagorean Formula and Distance Theorem...wait...—Real-World Examples pg: 20: Five Congruence Patterns pg: 21: Five Congruence Patterns—Real-World Examples pg. 22: Poly(Pocket)gons pg. 23: Poly(Pocket)gons—Real-World Examples
3: Table of Contents, pt. 3 pg. 24: *BONUS* The Midpoint Formula pg. 25: *BONUS* The Midpoint Formula—Real-World Examples pg. 26: *BONUS* New Patterns pg. 27: *BONUS* New Patterns—Real-World Examples
4: Learning the Basics | When observing terms such as point (no dimension), line (one dimension), and plane (two dimensions), they cannot be mathematically defined using other known words. They are called undefined terms. Collinear points are points that lie on the same line; coplanar points are points that lie on the same plane, and coplanar lines are lines that lie on the same plane. Figures often intersect if they have any points in common. The point(s) that the figures have in common are called the intersection.
5: Real-World Examples | EXAMPLE: Name the intersection of GH and FD. Answer: F | Real-World Relation: A real-world example of an intersection is a four-way intersection often found in Town Square. The center is the intersection and the roads are the lines.
6: At this Angle | An angle consists of two rays that have the same endpoint. The rays are the sides of the angle; the endpoint is the vertex of the angle. Acute angles have a measure between 0* and 90*. Obtuse angles have a measure between 90* and 180*. Right angles has an exact measure of 90*. Straight angles have an exact measure of 180*.
7: Real-World Examples | EXAMPLE: Name the angles in the figures on the left. Answers: Acute; obtuse; right | Real-World Relation: Many real-world objects have angles. A dresser has 90* angles and the top is a straight angle (180*).
8: The Two Bisectors | To bisect a segment means to divide the segment into two congruent segments. A segment bisector is a segment, ray, line, or plane that intersects a segment at its midpoint. An angle bisector is a ray that divides an angle into two angles that are congruent. The midpoint of a segment is the point on the segment that divides it into two congruent segments.
9: Real-World Examples | EXAMPLE: C (8) is the midpoint of AB. Find AC and CB. Answer: 4 | Real-World Relation: While it's hard to believe, a knife is both a segment and angle bisector. It divides food by the midpoint and produces even halves.
10: Complements and Supplements | Complementary angles are two angles with a sum that measures up to 90*. Supplementary angles are two angles with a sum that measures up to 180*. Vertical angles are two angles that are not adjacent and their sides are formed by two intersecting lines.
11: Real-World Examples | EXAMPLE: Is the angle complementary or supplementary? Answer: Supplementary | Real-World Relation: Though vague, a cardboard box closely resembles complementary angles.
12: Find the Parallels | Parallel lines are two lines that lie in the same plane and do not intersect. A transversal is a line that intersects two or more coplanar lines at different points. Corresponding Angles are two angles that occupy corresponding position. Alternate Interior Angles are two angles that lie between two lines on the opposite sides of the transversal. Alternate Exterior Angles are two angles that lie outside the two lines on the opposite sides of the transversal. Same-Side Interior Angles are two angles that lie between the two lines on the same side of the transversal.
13: Real-World Examples | EXAMPLE: Are AC and EG parallel to each other? Answer: Yes, the lines are parallel. Real-World Relations: The Twin Towers were parallel to each other.
14: Peculiar and Perpendicular | Perpendicular lines are two lines that intersect to form a right angle. Two planes are parallel planes if they do not intersect. A line perpendicular to a plane is a line that intersects a plane in a point and that is perpendicular to every line in the plane that intersects it.
15: Real-World Examples | EXAMPLE: Are the lines perpendicular? Answer: No, the lines are skew instead of perpendicular. | Real-World Relation: A compass has perpendicular lines because the direction patterns intersect and form right angles.
16: Triangular Theorems | A triangle is a figure formed by three segments joining three noncollinear points. Equilateral triangles have 3 congruent sides; isosceles triangles have at least 2 congruent sides, and scalene triangles have no congruent sides. Equiangular triangles have 3 congruent angles; acute triangles have 3 acute angles; right triangles have 1 right angle, and obtuse triangles have 1 obtuse angle. Interior angles are angles within a triangle; exterior angles are angles that are adjacent to the interior angles.
17: Real-World Examples | EXAMPLE: Classify the triangle based on sides and angles. Answer: Isosceles obsute Real-World Relation: This triangular prism is a good example of a triangle. It has three sides, three angles, and is equilateral because all sides are congruent.
18: Pythagorean Formula and Distance Theorem...wait... | The Pythagorean Theorem: 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. (hypotenuse)^2 = (leg)^2 + (leg)^2 The Distance Formula: If A(x1, y1) and B(x2, y2) are points in a coordinate plane, then the distance between A and B is. AB = *square rooted* (x2-x1)^2+(y2-y1)^2
19: Real-World Examples | EXAMPLE: If the triangle TYE has TY with 13 and YE with 9, what is ET? c^2 = 13^2 + 9^2 c^2 = 169 + 81 c^2 = 250 Answer: c = 15.9 Real-World Relation: The Pythagorean Theorem is often used in architectural context. Carpenters will want to level the floor, so they have to determine the measurements.
20: Five Congruence Patterns | SSS Congruence Postulate: If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. SAS Congruence Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. ASA Congruence Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent. AAS Congruence Postulate: If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. HL Congruence Postulate: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.
21: Real-World Examples | EXAMPLE: What postulate proves this triangle is congruent? Answer: ASA Congruence Postulate | Real-World Relation: Congruence Postulates also relate to architecture. Architects arrange certain triangular patterns in building houses, so any of these postulates help determine if it is congruent.
22: Poly(Pocket)gons | A polygon is a plane figure that is formed by three or more segments called sides. Each endpoint is a vertex of the polygon. A quadrilateral as a polygon with 4 sides and angles. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A rhombus is a parallelogram with four congruent sides. A rectangle is a parallelogram with four right angles. A square is a parallelogram with four congruent sides and four right angles.
23: Real-World Examples | EXAMPLE: Is this geometric figure below a polygon? If so, what kind? Answer: Yes, it is a pentagon. | Real-World Relation: This Polly Pocket house has many rectangles, but the house as a whole is a pentagon, a five-sided polygon.
24: The Midpoint Formula | The Midpoint Formula: the coordinates of the midpoint of a segment are the averages of the x-coordinates and the y-coordinates of the endpoints. M(x1+x2/2 , y1+y2/2)
25: Real-World Examples | EXAMPLE: A(6, 3), B(8, 4) (6 + 8)/2 (3 + 4)/2 Answer: (7, 3.5) Real-World Relation: Travelers often use the Midpoint Formula to track their distances.
26: New Patterns | In mathematics, a pattern is a sequence with a recurring motif. The point of mathematical patterns is to make the sequences longer. When you make a prediction, that means you presume what the next motif in the sequence will be. | 13, 16, 19, 22, 25, ...
27: Real-World Examples | EXAMPLE: What is the motif in the pattern on the left? Answer: The motif in the pattern is the additive number 3. Real-World Relation: Whenever you see the "Repeat" option on a DVD remote, you could say that the movie is a pattern when constantly replaying.