pGeometry: Measurement of Angles Level 8 of 9 Examples Vp p Measurement of Angles Level 8 In this article we will go over a problem that involves inequalities and a problem that has an algebraic expression as a solution.
Lets take a look at the first problem. In the diagram below angle B is acute. What are the restrictions on the measurement of angle B? What are the restrictions on x? In this problem we are given a diagram of an angle that has an algebraic expression for its angle measurement. We need to answer two distinct questions. The first question is asking us to determine the restrictions on the measurement of angle B. Whenever we are dealing with restrictions in geometry, it will most likely require the use of inequalities since we are now interested in a range or interval of numbers that will make an algebraic expression true as oppose to a particular value that will make an algebraic expression true.
So make sure you brush up on your inequalities if you are a little rusty. Alright to determine the restrictions on the measurement of angle B we will start with the definition of acute angles. Recall that acute angles are greater than 0 degrees but less than 90 degrees. This means that angle B has to be greater than 0 degrees and less than 90 degrees. In other words the measurement of angle B is between 0 and 90 degrees exclusive.
Now, the next question is asking us to determine the restrictions on the variable x. In other words what are the values of x that will make the angle acute? We can determine this by using the inequalities from the previous question these inequalities are the geometric relations of the problem. All that is left to do is to replace angle B with the algebraic expression that is given and solve both inequalities. So our first inequality would be 2x plus 14 is greater than 0 and the second inequality would be 2x plus 14 is less than 90, these inequalities were obtained by using the definition of acute angles.
Now it is just a matter of solving for x. The method for solving inequalities is essentially the same as the method for solving equations, the only difference is that you have to make sure you flip the inequality symbol when you divide or multiply by a negative number. Solving for x we obtain x is greater than 7 and x is less than 38, so the value of x has to be a number between negative 7 and 38 in order for this angle to be acute. Alright lets try the next example. Given that angle 1 is congruent to angle 2, the measure of angle 1 is equal to x plus 14, and the measure of angle 2 is equal to y minus 3, solve for y in terms of x. In this problem we are provided with a diagram of two angles and we are provided with geometric and algebraic relations for both angles.
We are asked to solve for the variable y in terms of the variable x. Now in this problem we are not going to be solving for a particular number or a range of numbers, we are simply going to find an expression for y in terms of x, so our final answer will have numerical and algebraic expressions. We can start by setting up the geometric relations. We know that angle 1 is congruent to angle 2 so we can set the measurement of the angles equal to one another. Next we go ahead and substitute the algebraic relations for each angle.
Now it is just a matter of solving for the variable y, so we add 3 to both sides and simplify the expression in the end y is equal to x plus 17 and this is our final answer, notice that this problem is asking us to solve for y in terms of x. We do no need to find a numerical answer. Alright in our final article on measurement of angles we will go over 2 final examples that require the use of algebra to solve.p