In the figure, lines and are parallel. Points and lie on line , and points and lie on line . Segments , , , and form a trapezoid, and diagonals and intersect at point . Given that and , diagonal bisects , and diagonal bisects . Which choice is the measure of ? Note: Figure not drawn to scale.

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Super-Hard SAT Math Question 152 | Free SAT Prep | Aniko

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Question 152·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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In the figure, lines $a$ and $b$ are parallel. Points $A$ and $D$ lie on line $a$ , and points $B$ and $C$ lie on line $b$ . Segments $\overline{AB}$ , $\overline{BC}$ , $\overline{CD}$ , and $\overline{DA}$ form a trapezoid, and diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$ .

Given that $\angle ABC = 70°$ and $\angle BCD = 50°$ , diagonal $\overline{AC}$ bisects $\angle DAB$ , and diagonal $\overline{BD}$ bisects $\angle CDA$ .

Which choice is the measure of $\angle AEB$ ?

Note: Figure not drawn to scale.

When you see parallel lines in a quadrilateral, first use supplementary (same-side interior) angle relationships to convert given bottom angles into top angles. Then apply any angle-bisector information to split those vertex angles cleanly. Finally, for an angle formed by two diagonals at their intersection, decide whether the question is asking for the acute or obtuse angle and combine the diagonal inclinations accordingly.

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Use supplementary angles with parallel lines

Because $a \parallel b$ , look at $\angle ABC$ with $\angle DAB$ , and $\angle BCD$ with $\angle CDA$ . What relationship do those pairs have?

Apply the bisector information

Once you have $\angle DAB$ and $\angle CDA$ , cut each of them in half to get the angles that the diagonals make with $\overline{AD}$ .

Decide whether the asked angle is acute or obtuse

The diagonals form both an acute and an obtuse angle at $E$ . Make sure $\angle AEB$ matches the rays that go toward $A$ and toward $B$ .

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Compute the top angles

Enter 180-70 to get $\angle DAB$ , and enter 180-50 to get $\angle CDA`.

Halve each angle for the bisectors

Enter (180-70)/2 and (180-50)/2 to represent the two angles made with $\overline{AD}$ by the diagonals.

Add the two halves

Enter (180-70)/2 + (180-50)/2. The value shown is the measure of $\angle AEB$ .

Step-by-step Explanation

Find the top angles using parallel lines

Since $a \parallel b$ , each leg of the trapezoid acts as a transversal.

Along transversal $\overline{AB}$ , $\angle DAB$ and $\angle ABC$ are supplementary:

\angle DAB = 180° - 70° = 110°

Along transversal $\overline{CD}$ , $\angle CDA$ and $\angle BCD$ are supplementary:

\angle CDA = 180° - 50° = 130°

Use the angle bisectors

Because $\overline{AC}$ bisects $\angle DAB$ ,

\angle CAD = \frac{110°}{2} = 55°

Because $\overline{BD}$ bisects $\angle CDA$ ,

\angle ADB = \frac{130°}{2} = 65°

Determine $\angle AEB$ from the diagonals’ directions

Ray $\overrightarrow{EA}$ lies on diagonal $\overline{AC}$ , and ray $\overrightarrow{EB}$ lies on diagonal $\overline{BD}$ .

At the top base $\overline{AD}$ , diagonal $\overline{AC}$ makes a $55°$ angle with $\overline{AD}$ , and diagonal $\overline{BD}$ makes a $65°$ angle with $\overline{AD}$ on the other side.

Therefore, the obtuse angle between rays $\overrightarrow{EA}$ and $\overrightarrow{EB}$ is the sum:

\angle AEB = 55° + 65° = 120°

So, the measure of $\angle AEB$ is 120°.

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