The measures of the acute angles in the isosceles trapezoid are one-half times the measures of the obtuse angles. Write and solve a system of equations to find the measures of all the angles.

Let x be the measure of one of the acute angles, and let y be the measure of one of the obtuse angles. Since the measures of the acute angles are one-half times the measures of the obtuse angles, we have the following system of equations:

x = 1/2 * y

x + y = 180

We can solve this system of equations using substitution or elimination. Using substitution, we can solve for x in the first equation and substitute that into the second equation to get a single equation in y:

x = 1/2 * y

1/2 * y = 1/2 * y

y = 180

Substituting this back into the first equation, we get:

x = 1/2 * 180

x = 90

Thus, the measure of each acute angle is 90 degrees, and the measure of each obtuse angle is 180 degrees.

semsee45 semsee45 · Answer 2 · 2022-12-09T22:11:33+01:00

Answer:

[tex]\textsf{System of equations: \quad }\begin{cases}x=\dfrac{1}{2}y\\\\2x+2y=360\end{cases}[/tex]

[tex]\begin{aligned} \textsf{Solutions}: \quad x &= 120^{\circ}\\y &= 60^{\circ}\end{aligned}[/tex]

Step-by-step explanation:

An obtuse angle is greater then 90° and less than 180°.

An acute angle is less than 90°.

Therefore, from inspection of the given diagram:

x = obtuse angle
y = acute angle

If the measures of the acute angles are one-half times the measures of the obtuse angles:

[tex]\implies x=\dfrac{1}{2}y[/tex]

Angles in a quadrilateral sum to 360°.

[tex]\implies 2x+2y=360[/tex]

Therefore, the system of equations is:

[tex]\begin{cases}x=\dfrac{1}{2}y\\\\2x+2y=360\end{cases}[/tex]

Substitute the first equation into the second equation and solve for y:

[tex]\implies 2\left(\dfrac{1}{2}y\right)+2y=360[/tex]

[tex]\implies y+2y=360[/tex]

[tex]\implies 3y=360[/tex]

[tex]\implies \dfrac{3y}{3}=\dfrac{360}{3}[/tex]

[tex]\implies y=120[/tex]

Substitute the found value of y into the first equation and solve for x:

[tex]\implies x=\dfrac{1}{2} \cdot 120[/tex]

[tex]\implies x=60[/tex]