Perfect1600

Digital SAT · Solve it in Desmos

How to Solve Angle Relationships on Desmos (Digital SAT)

Desmos only comes in after the geometry is done; it solves the equation the angle rule produces. Reading the figure and choosing the rule stay entirely on your side of the screen.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for angle relationships

The Desmos steps

  1. 1

    Build the equation from the rule

    Apply the angle fact yourself, ending with something like 3x + 2x + 40 = 180.

  2. 2

    Solve it in Desmos

    Enter that equation with x and read the value, or type the arithmetic in one line.

  3. 3

    Recover the angle

    Put x back into the expression to get the measure that was asked.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

Medium example
In triangle NOPNOP, the measure of angle NN is (3x)°(3x)\degree, the measure of angle OO is (4x)°(4x)\degree, and the measure of angle PP is (5x)°(5x)\degree. What is the measure of angle PP?
A
15°15\degree
B
45°45\degree
C
60°60\degree
75°75\degree

A: Incorrect. This reports the value of xx instead of angle PP.

B: Incorrect. This reports angle NN's measure (3×15=453\times15=45) instead of angle PP.

C: Incorrect. This reports angle OO's measure (4×15=604\times15=60) instead of angle PP.

D: Correct. 3x+4x+5x=1803x+4x+5x=180, so 12x=18012x=180 and x=15x=15. Angle P=5×15=75°P=5\times15=75\degree.

Explanation

3x + 4x + 5x = 180, so 12x = 180 and x = 15. Angle P = 5 x 15 = 75 degrees.

Medium example
In triangle KLMKLM, the exterior angle at vertex MM measures (7x−5)°(7x-5)\degree. The measures of interior angles KK and LL are (2x+15)°(2x+15)\degree and (3x−10)°(3x-10)\degree, respectively. What is the value of xx?
A
0
5
C
203\frac{20}{3}
D
15

A: Incorrect. This makes a sign slip setting up the equation, 7x−5=5x−57x-5=5x-5, giving x=0x=0.

B: Correct. An exterior angle equals the sum of the two remote interior angles: 7x−5=(2x+15)+(3x−10)7x-5=(2x+15)+(3x-10), so 7x−5=5x+57x-5=5x+5, 2x=102x=10, and x=5x=5.

C: Incorrect. This subtracts the two sides in the wrong order, 5x−5=2x+155x-5=2x+15, giving x=203x=\frac{20}{3}.

D: Incorrect. This mistakenly sums all three given expressions as if they were a triangle's interior angles: (7x−5)+(2x+15)+(3x−10)=180(7x-5)+(2x+15)+(3x-10)=180 gives x=15x=15.

Explanation

The exterior angle at M equals the sum of the two remote interior angles: 7x - 5 = (2x + 15) + (3x - 10), so 7x - 5 = 5x + 5, 2x = 10, and x = 5.

Hard example
Triangle STUSTU is similar to triangle VWXVWX, where the ratio of the areas of the two triangles is 44 to 99. The measure of angle SS is (7x−11)°(7x-11)\degree and the measure of angle VV is (5x+9)°(5x+9)\degree, where SS corresponds to VV. What is the measure of angle VV?
A
10°10\degree
B
48°48\degree
59°59\degree
D
118°118\degree

A: Incorrect. This reports the value of xx instead of angle VV.

B: Incorrect. This subtracts 1111 a second time after already finding the angle: 59−11=4859-11=48.

C: Correct. Since SS corresponds to VV, 7x−11=5x+97x-11=5x+9, so x=10x=10 and angle V=5(10)+9=59°V=5(10)+9=59\degree. The area ratio does not affect corresponding angle measures.

D: Incorrect. This doubles the correct angle measure by mistake: 59×2=11859\times2=118.

Explanation

Since S corresponds to V, 7x-11 = 5x+9, so x = 10 and angle V = 5(10)+9 = 59 degrees. The area ratio between the two triangles does not affect corresponding angle measures.

Try it with the calculator

Question 1easy
Angle diagram
mn110°?

Two angles are supplementary. The measure of one angle is 110∘110^\circ. What is the measure of the other angle?

Question 2easy

In a triangle, two of the angles measure 45∘45^\circ and 55∘55^\circ. What is the measure of the third angle?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Desmos questions

Can Desmos solve angle relationships questions on the SAT?

Desmos only comes in after the geometry is done; it solves the equation the angle rule produces. Reading the figure and choosing the rule stay entirely on your side of the screen.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master angle relationships, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

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