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solid geometry - Angle

For COMPETITION
Number of Total Problems: 14.
FOR PRINT ::: (Book)

Problem Num : 1

From : NCTM

Type: None

Section:solid geometry

Theme:None
Adjustment# :

Difficulty: 1

Category Angle

Analysis

Solution/Answer

Problem Num : 2

From : NCTM

Type: None

Section:solid geometry

Theme:None
Adjustment# :

Difficulty: 1

Category Angle

Analysis

Solution/Answer

Problem Num : 3

From : NCTM

Type: None

Section:solid geometry

Theme:None
Adjustment# :

Difficulty: 1

Category Angle

Analysis

Solution/Answer

Problem Num : 4

From : NCTM

Type: None

Section:solid geometry

Theme:None
Adjustment# :

Difficulty: 1

Category Angle

Analysis

Solution/Answer

Problem Num : 5

From : NCTM

Type: None

Section:solid geometry

Theme:None
Adjustment# :

Difficulty: 2

Category Angle

Analysis

Solution/Answer

Problem Num : 6

From : AMC8

Type:

Section:solid geometry

Theme:
Adjustment# : 0

Difficulty: 1

'
The degree measure of angle $A$ is

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

Category Angle

Analysis

Solution/Answer
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 $180 - 70 - 80 = oxed{30^circ, B}$

Answer:

Problem Num : 7

From : AMC12

Type:

Section:solid geometry

Theme:
Adjustment# : 0

Difficulty: 1

'
Find the degree measure of an angle whose complement is 25% of its supplement.
$mathrm{(A) 48 } qquad mathrm{(B) 60 } qquad mathrm{(C) 75 } qquad mathrm{(D) 120 } qquad mathrm{(E) 150 }$

'

Category Angle

Analysis

Solution/Answer
$4(90-x)=(180-x)$
$360-4x=180-x$
$3x=180$
$x=60 Rightarrow mathrm {(B)}$

Answer:

Problem Num : 8

From : AMC10B

Type:

Section:solid geometry

Theme:
Adjustment# : 0

Difficulty: 1

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

Category Angle

Analysis

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

Answer:

Problem Num : 9

From : AMC10

Type:

Section:solid geometry

Theme:
Adjustment# : 0

Difficulty: 1

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

Category Angle

Analysis

Solution/Answer

We angle chase, and find out that:

$DAC=frac{180-140}{2} = 20$
$BAC=frac{180-40}{2} = 70$
$BAD=BAC-DAC=50 mathrm{(D)}$

Answer:

Problem Num : 10

From : AMC10

Type:

Section:solid geometry

Theme:
Adjustment# : 0

Difficulty: 1

'
Suppose that $frac{2x}{3}-frac{x}{6}$ is an integer. Which of the following statements must be true about $x$ ?
$mathrm{(A)} ext{It is negative.}\qquadmathrm{(B)} ext{It is even, but not necessarily a multiple of 3.}\qquadmat...$
'

Category Angle

Analysis

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

Answer:

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