In , the measures of and are and , respectively. If the triangle is isosceles with , what is the value of ? (Disregard the degree symbol when entering your answer.)

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 44 | Free SAT Prep | Aniko

00:00

Question 44·Medium·Lines, Angles, and Triangles

0 / 50

In $\triangle RST$ , the measures of $\angle R$ and $\angle S$ are $(2x-10)^\circ$ and $(x+35)^\circ$ , respectively. If the triangle is isosceles with $\angle R = \angle T$ , what is the value of $x$ ? (Disregard the degree symbol when entering your answer.)

Your answer

(Express the answer as an integer)

For isosceles triangle angle problems, first translate the word description into equal-angle relationships (for example, if $\angle R = \angle T$ , then write both in terms of the same expression). Next, use the fact that the three interior angles of a triangle sum to 180 degrees to write one equation involving all three angle expressions. Simplify to a single linear equation in $x$ , then solve carefully and, if time allows, plug the value back into the angle expressions to ensure the two specified angles are indeed equal.

Hints

Click me!

Use the definition of an isosceles triangle

In an isosceles triangle, two angles are equal. Here, you are told $\angle R = \angle T$ . How can you write the measure of $\angle T$ in terms of $x$ using this information?

Apply the triangle angle sum

The three interior angles of any triangle always add up to $180^\circ$ . Try writing an equation that adds $\angle R$ , $\angle S$ , and $\angle T$ and sets their sum equal to 180.

Form a single equation in x

After substituting the expressions for $\angle R$ , $\angle S$ , and $\angle T$ into your sum-to-180 equation, combine like terms so that you get an equation of the form $ax + b = 180$ .

Solve the linear equation carefully

Once you have an equation like $ax + b = 180$ , isolate $x$ by undoing the addition/subtraction first, then the multiplication/division. Check your arithmetic when subtracting and dividing.

Desmos Guide

Set up the angle-sum equation in Desmos

In Desmos, type the equation 2(2x - 10) + (x + 35) = 180. Desmos will interpret this as the relationship that comes from adding $\angle R$ , $\angle S$ , and $\angle T$ and setting their sum equal to 180.

Find the solution for x

After entering the equation, look at the left panel where Desmos lists solutions. It will show the value of $x$ that makes the equation true; that value is the solution to the problem.

Step-by-step Explanation

Translate the geometry into algebra

We are told:

$\angle R = (2x - 10)^\circ$
$\angle S = (x + 35)^\circ$
Triangle RST is isosceles with $\angle R = \angle T$ .

So $\angle T$ must also measure $(2x - 10)^\circ$ .

Use the triangle angle sum

The sum of the interior angles of any triangle is $180^\circ$ .

So for $\triangle RST$ :

\angle R + \angle S + \angle T = 180

Substitute the expressions for the three angles:

(2x - 10) + (x + 35) + (2x - 10) = 180

Now all three angles are written in terms of $x$ .

Simplify the equation

Combine like terms on the left side.

First, add the $x$ terms:

$2x$ (from $\angle R$ )
$x$ (from $\angle S$ )
$2x$ (from $\angle T$ )

So the total $x$ term is $2x + x + 2x = 5x$ .

Next, combine the constants:

$-10$ (from $\angle R$ )
$+35$ (from $\angle S$ )
$-10$ (from $\angle T$ )

So the total constant is $-10 + 35 - 10 = 15$ .

This gives the simpler equation:

5x + 15 = 180

Solve for x

Now solve the linear equation $5x + 15 = 180$ :

Subtract 15 from both sides:

5x = 180 - 15 = 165

Divide both sides by 5:

x = \frac{165}{5} = 33

So the value of $x$ is 33.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation