from: category_eng

1.Calculation

Ans:

The degree measure of angle $A$ is

$ext{(A)} 20 qquad ext{(B)} 30 qquad ext{(C)} 35 qquad ext{(D)} 40 qquad ext{(E)} 45$

Find the degree measure of an angle whose complement is 25% of its supplement.

$mathrm{(A) 48 } qquad mathrm{(B) 60 } qquad mathrm{(C) 75 } qquad mathrm{(D) 120 } qquad mathrm{(E) 150 }$

The angles of quadrilateral $ABCD$ satisfy $angle A=2 angle B=3 angle C=4 angle D.$ What is the degree measure of $angle A,$ rounded to the nearest whole number?

$extbf{(A) } 125 qquad extbf{(B) } 144 qquad extbf{(C) } 153 qquad extbf{(D) } 173 qquad extbf{(E) } 180$

Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$ . Point $D$ is inside triangle $ABC$ , angle $ABC$ measures 40 degrees, and angle $ADC$ measures 140 degrees. What is the degree measure of angle $BAD$ ?

$mathrm{(A)} 20qquad mathrm{(B)} 30qquad mathrm{(C)} 40qquad mathrm{(D)} 50qquad mathrm{(E)} 60$

10.

Suppose that $frac{2x}{3}-frac{x}{6}$ is an integer. Which of the following statements must be true about $x$ ?

$mathrm{(A)} ext{It is negative.}\qquadmathrm{(B)} ext{It is even, but not necessarily a multiple of 3.}\qquadmat...$

11.

In the given circle, the diameter $overline{EB}$ is parallel to $overline{DC}$ , and $overline{AB}$ is parallel to $overline{ED}$ . The angles $AEB$ and $ABE$ are in the ratio $4 : 5$ . What is the degree measure of angle $BCD$ ?

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$extbf{(A)} 120 qquad extbf{(B)} 125 qquad extbf{(C)} 130 qquad extbf{(D)} 135 qquad extbf{(E)} 140$

12.

Triangle $ABC$ has $AB=2 cdot AC$ . Let $D$ and $E$ be on $overline{AB}$ and $overline{BC}$ , respectively, such that $angle BAE = angle ACD$ . Let $F$ be the intersection of segments $AE$ and $CD$ , and suppose that $riangle CFE$ is equilateral. What is $angle ACB$ ?

$extbf{(A)} 60^circ qquad extbf{(B)} 75^circ qquad extbf{(C)} 90^circ qquad extbf{(D)} 105^circ qquad ex...$

13.

Point B is due east of point A. Point C is due north of point B. The distance between points A and C is $10sqrt 2$ , and $angle BAC= 45^circ$ . Point D is 20 meters due north of point C. The distance AD is between which two integers?

$extbf{(A)} 30 ext{and} 31 qquad extbf{(B)} 31 ext{and} 32 qquad extbf{(C)} 32 ext{and} 33 qquad extbf{...$

1. 60

2. 18

3. 32

4. 15

5. 15

Angle-chasing using the small triangles:

Use the line below and to the left of the $110^circ$ angle to find that the rightmost angle in the small lower-left triangle is $180 - 110 = 70^circ$ .

Then use the small lower-left triangle to find that the remaining angle in that triangle is $180 - 70 - 40 = 70^circ$ .

Use congruent vertical angles to find that the lower angle in the smallest triangle containing $A$ is also $70^circ$ .

Next, use line segment $AB$ to find that the other angle in the smallest triangle contianing $A$ is $180 - 100 = 80^circ$ .

The small triangle containing $A$ has a $70^circ$ angle and an $80^circ$ angle. The remaining angle must be $180 - 70 - 80 = oxed{30^circ, B}$

$4(90-x)=(180-x)$

$360-4x=180-x$

$3x=180$

$x=60 Rightarrow mathrm {(B)}$

8.

The sum of the interior angles of any quadrilateral is $360^circ.$ $egin{align*}360 &= angle A + angle B + angle C + angle D&= angle A + frac{1}{2}A + frac{1}{3}A + frac{1}{...$ $angle A = 360 cdot frac{12}{25} = 172.8 approx oxed{mathrm{(D) } 173}$

We angle chase, and find out that:

10.

$frac{2x}{3}-frac{x}{6}quadLongrightarrowquadfrac{4x}{6}-frac{x}{6}quadLongrightarrowquadfrac{3x}{6}quadLongright...$ For $frac{x}{2}$ to be an integer, $x$ must be even, but not necessarily divisible by $3$ . Thus, the answer is $mathrm{(B)}$ .

11.

12.

Let $angle BAE = angle ACD = x$ .

$egin{align*}angle BCD &= angle AEC = 60^circ angle EAC + angle FCA + angle ECF + angle AEC &= angle EAC +...$

Since $frac{AC}{AB} = frac{1}{2}$ , $angle BCA = oxed{90^circ extbf{(C)}}$

13.