Question

Algebra If \(m \angle R=48\) and \(m \angle S=2 x-10\), what values of \(x\) make \(m \angle R \geq m \angle S\) ? (Lesson 7-1) Complete each congruence statement. (Lesson 5-4)

Short answer

Any value of \(x\) less than or equal to 29 satisfies \(m \angle R \geq m \angle S\).

Step-by-step solution

Understanding the Problem

We are given that the measure of angle R is 48 degrees, and the measure of angle S is represented as \(2x - 10\). We need to find the values of \(x\) such that \(m \angle R \geq m \angle S\). This translates to the inequality \(48 \geq 2x - 10\).

Setting Up the Inequality

Start by setting up the inequality that represents the problem: \(48 \geq 2x - 10\).

Solving the Inequality

Add 10 to both sides of the inequality to isolate terms involving \(x\) on one side:\[48 + 10 \geq 2x\]This simplifies to \(58 \geq 2x\).

Isolating Variable x

Divide both sides by 2 to solve for \(x\):\[\frac{58}{2} \geq x\]Simplifying gives \(29 \geq x\).

Conclusion

Therefore, the solution is \(x \leq 29\), which means any value of \(x\) less than or equal to 29 will satisfy the inequality \(m \angle R \geq m \angle S\).

Key concepts

Angle Measure

Understanding how to measure angles is crucial since angles are fundamental elements in geometry. An angle is formed by two rays diverging from a common endpoint, called the vertex. Angles are usually measured in degrees, where a full circle is 360 degrees. In the given exercise, the measure of angle R is

• Fixed at 48 degrees, which sets a benchmark for comparison.
• Compared to angle S, represented as an expression involving a variable: \(2x - 10\).

The relationship or measure of angles helps us compare and solve problems involving angle sizes in figures.
Once you understand angle measurement in degrees, you can use algebra to relate different angles to each other through expressions or inequalities.

Inequality Solving

Solving inequalities, like equations, involves finding the value or range of values for the variable that makes the inequality true. In this exercise:

• We start with the inequality \(48 \geq 2x - 10\), meaning angle R is at least as large as angle S.
• Our aim is to find the values of \(x\) that satisfy this condition.

We begin by performing operations to isolate the variable term \(2x\) on one side:
Adding 10 to both sides simplifies the inequality to \(58 \geq 2x\).
Next, dividing both sides by 2 isolates \(x\), providing the final inequality \(29 \geq x\).
This solution tells us that any \(x\) less than or equal to 29 will keep angle S less than or equal to angle R, satisfying the original problem statement.

Congruence Statements

Congruence in geometry means that figures such as angles or segments are identical in shape and size. A congruence statement expresses that two geometrical figures are congruent, denoted with the symbol \( \cong \).
In the context of the exercise, it’s implied by comparing the measure of angle R with angle S to determine if they could be congruent at any point.
However, since the exercise is about finding when angle R is greater than or equal to angle S, congruence is achieved when the two angles are equal.
This can be expressed in a congruence statement as:

• If \(x = 29\), then \(m \angle R = m \angle S = 48\)

Understanding congruence is useful for recognizing when two elements in geometry are completely equivalent, enabling seamless transitions between different geometrical problems.

Variable Isolation

Variable isolation is a key technique in algebra, crucial for solving equations and inequalities. The goal is to manipulate the given problem to have the variable alone on one side of the equation or inequality.
In our exercise, we isolated \(x\) within the inequality \(48 \geq 2x - 10\) to understand when angle R would be greater than or equal to angle S.

• First, we added 10 to both sides: it was essential to balance the inequality.
• Then, dividing everything by 2 isolated the variable \(x\).

These operations help simplify and focus on the variable, making it easier to conclude that \(x \leq 29\).
Grasping how to isolate variables not only aids in solving inequalities but is also widely applicable in different branches of mathematics and problem-solving scenarios.