Problem Solving in Geometry

Geometry Geometry is about the space you live in and the shapes that surround you. For thousands of years, people have studied geometry in some form to obtain a better understanding of the world in which they live. In this section, we will look at some basic geometric formulas for perimeter, area, and volume. Remember that perimeter is a linear measure, that is, a measure of length. It will be measured in units such as feet, or meters. Area is a measure of square units and it will be measured in units such as square feet, square miles, or square meters. How many floor tiles are in your kitchen? That’s an area measure. Volume is a measure of cubic units. It measures the number of cubic units in a three dimensional object or rather the amount of space occupied by the figure. A liter is a measure of volume. Cups and gallons are also measures of volume. The volume of a solid is the number of cubic units that can be contained in the solid.

Angles of a Triangle The sum of the measures of the three angles of a triangle is 180º. Important Fact about Triangles

Example : One angle of a triangle is three times another. The measure of the third angle is 10º more than the smallest angle. Find the angles. 1.Let x represent one of the quantities. x = measure of the first angle 2.Represent other quantities in terms of x. 3x = measure of the second angle x + 10 = measure of the third angle 3.Write an equation in x that describes the conditions. x + 3x + x + 10 = 180 (Sum of the angles of a triangle is 180.) Angles of a Triangle

Example continued : One angle of a triangle is three times another. The measure of the third angle is 10º more than the smallest angle. Find the angles. 4.Solve the equation and answer the question. x + 3x + x + 10 = 180 5x + 10 = 180 Combine like terms. 5x =170 Subtract 10 from both sides. x = 34 Divide both sides by 5. 3x = 102 measure of the second angle x + 10 = 44 measure of the third angle The angles are 34º, 102º and 44º. 5.Check the solution. 102º is three times 34º, thus one angle is three times another. The measure of the third, 44º is 10º more than the smallest angle. The angles sum to 180º Angles of a Triangle

Complementary and Supplementary Complementary Angles: Two angles whose sum is 90º. Supplementary Angles: Two angles whose sum is 180º. Special Angles

Algebraic Expressions for Angle Measures Measure of an angle: x Measure of the angle’s complement: 90 – x Measure of the angle’s supplement: 180 – x Complements and Supplements

Example : Find the measure of an angle that is 12 º less than its complement. 1.Let x represent one of the quantities. x = measure of the first angle 2.Represent other quantities in terms of x x = measure of the complement angle 3.Write an equation in x that describes the conditions. x = (90 – x) – 12 The angle is 12 less than the complement. Complement of an Angle

Example continued: Find the measure of an angle that is 12 º less than its complement. 4.Solve the equation and answer the question. x = (90 – x) – 12 x = 90 – x – 12 Remove parentheses. x = 78 – x Combine like terms. 2x = 78 Add x to both sides. x = 39 Divide both sides by 2. x = 39º measure of the first angle 90 – x = 51 measure of the complement angle The angles are 39º and 51º. 5.Check the solution. 39º is 12º less than 51º and the angles sum to 90º Complement of an Angle