Tangent Ratio
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This document discusses the tangent ratio and how to use it to solve problems involving right triangles. It provides examples of finding the tangent of various angles, using the tangent ratio to find missing side lengths of triangles, and calculating the angle of a road given its grade. Key terms introduced are tangent ratio, sine, cosine, acute angle, opposite leg, and adjacent leg. Examples are provided to illustrate using the tangent ratio to solve problems.
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This document discusses three rules for determining if triangles are similar: the AA Similarity Postulate, SAS Similarity Theorem, and SSS Similarity Theorem. It provides examples of applying each rule to determine if triangles are similar, and if so, writing the similarity statement and ratio.
7-3 similar polygons
Two polygons are similar if their corresponding angles are congruent and the lengths of corresponding sides are proportional. This means the ratio of corresponding sides is the same. Similar polygons can be labeled by writing one polygon name ~ the other, listing corresponding vertices in the same order. The ratio of the lengths of two corresponding sides when two polygons are similar is called the scale factor. An example shows quadrangle WARD is similar to quadrangle W'A'R'D' with a scale factor of 2:3, and uses the scale factor to find missing side lengths, perimeters, and the ratio of perimeters of the two quadrilaterals.
7 1 7-2 ratios and proportions
This document discusses ratios, proportions, and their properties. It begins by defining a ratio as a comparison of two quantities using division, and provides examples of expressing ratios in simplest form using fractions. It then discusses proportions, which state that two ratios are equal. The key properties of proportions are: (1) the product of the means equals the product of the extremes, (2) if several ratios are equal the sums of the corresponding terms are in the same ratio, and (3) the inverse of a ratio is found by flipping its terms. Examples demonstrate using these properties to solve proportion problems.
6 2 Parallelograms
This document discusses properties of parallelograms. It defines a parallelogram as a quadrilateral with both pairs of opposite sides parallel. It then presents proofs of several theorems about parallelograms: that opposite sides are congruent, consecutive angles are supplementary, opposite angles are congruent, and that the diagonals of a parallelogram bisect each other. Students are prompted to complete proofs of these properties and theorems. The document concludes by listing the key properties of parallelograms.
6 3 Proving Parallelograms
This document discusses different ways to prove that a quadrilateral is a parallelogram, including: if both pairs of opposite sides are congruent; if an angle is supplementary to its consecutive angles; if both pairs of opposite angles are congruent; if the diagonals of a quadrilateral bisect each other; or if one pair of opposite sides is both congruent and parallel. It provides examples of each theorem and assigns practice problems for students to complete.
Midpoint and distance
The document discusses formulas for finding the midpoint and distance between two points in the coordinate plane. It provides examples of using the midpoint and distance formulas to find the midpoint and length of a segment. Students are assigned homework problems using these formulas and are notified that the next class will involve additional practice finding midpoints and distances in the coordinate plane using the given formulas.
1 6 construction
Construction in geometry: congruent segments and angles, perpendicular bisectors and angle bisectors.
Geometry Angle Pairs
This document discusses different types of angles and their relationships using a diagram. It defines adjacent angles, vertical angles, complementary angles, and supplementary angles. It then lists four statements about the angles in the diagram and asks the reader to determine if each one is true or false and email the answers.
1 4angles
This document defines geometric concepts like congruent objects, segments, midpoints, and bisectors. It also introduces angles, describing them as dividing the plane into interior, exterior, and points on the angle. It discusses naming angles using different notation and measuring angles. Finally, it mentions classifying angles by their measures and the important Angle Addition Postulate.
1 3
This document contains a true/false quiz about geometry concepts like lines, planes, and coplanar points. It provides the answers and encourages the reader to re-examine any questions they missed. It then introduces the concept of postulates, or accepted statements without proof, and emphasizes that every word of postulates and theorems is important. It notes that some postulates should be memorized and that the Segment Addition Postulate will be used over 1000 times and should be memorized.
Geo final exam review
This geometry exam review covers topics that will be on the final exam. It includes true/false questions, multiple choice, matching, and free response problems involving geometry concepts like triangles, circles, polygons, and three-dimensional shapes. Calculators may be used but the exam may have non-calculator sections, so students should prepare with and without calculators. The review is due before the scheduled final exam date.
Equations of circles
This document discusses graphing circles and converting between graphing form and standard form equations of circles. It provides examples of writing the equation of a circle given its center and radius in graphing form. It also shows how to find the center, radius, and equation of a circle given the standard form equation. The key points are:
- In graphing form, the center is denoted as (h, k) and the equation is (x - h)2 + (y - k)2 = r2
- To convert to standard form, complete the square and factor the left side, moving any constant to the right side.
6 5 Inequalities in Two Triangles
If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle. Similarly, if two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.
6 4 Inequalities in One Triangle
This document discusses properties of triangles related to side lengths and angles. It states that if one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. It also states that if one angle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle. The document then asks the reader to identify the longest and shortest sides of a triangle based on these properties.
Chapter 6 and 6-2
This document discusses inequalities and indirect proofs in chapter 6 of a mathematics textbook. It introduces properties of inequalities, such as if a > b and c > 0, then ac > bc. It presents the exterior angle inequality theorem, which states that the measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. It explores how to use properties of inequalities to prove this theorem indirectly. The document also discusses conditional statements and their converses, inverses, and contrapositives. It explains how to use Venn diagrams to represent conditionals and their logical equivalents. Finally, it introduces the law of syllogism for making indirect logical deductions.
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Tangent Ratio
- 1. 8-5 The Tangent Ratio
- 2. Greek for “Triangle Measurement” You will need to use a scientific calculator to solve some of the problems. (You can find one online if you don’t have one.) Your calculator needs to be in “degree” mode when we do trigonometry.
- 3. Right triangle trigonometry Three ratios: Tangent of an angle (tanA) Sine of an angle (sinA) Cosine of an angle (cosA) Tangent of A (one of the acute angles) A B C opposite leg adjacent leg
- 4. leg opposite Atangent of A = leg adjacent to A A B C adjacent leg opposite leg
- 5. 1. Find tan X 2. Find tan Y 1. Find tan 30° 2. Find tan 60° Y 30° 6 X Z 815 17
- 6. 1. Find tan X 2. Find tan Y 1. Find tan 30° 2. Find tan 60° Y 30° 6 X Z 815 17 tan 𝑌 = 15 8 tan 𝑋 = 8 15 tan 60 = 6 3 6 = 3 tan 30 = 6 6 3 = 3 3
- 7. Find the value of y to the nearest tenth. 56 32 y tan 56 = 𝑦 32 𝑦 = 32 ∗ tan 56 𝑦 ≈ 32 ∗ 1.4826 𝑦 ≈ 47.4
- 8. The grade of a road is the ratio of its rise to its run and is usually given as a decimal or percent. Find the angle that the road makes with the horizontal if its grade is 4%. rise run x grade = rise run tan x = 0.0400