Question

lines, angles, and triangles practice set
directions: keep a calculator handy as you work through this practice set, but reserve it for those questions that
have a calculator icon next to them—and even on those questions, use it only if you really need it. questions without
the icon are intended to be done without a calculator, to help you prepare for the no - calculator section of the sat.

1. in the figure above, ( d = 105 ). if ( a = 2c ), what is the

value of ( a + b )?
a) 105
b) 120
c) 130
d) 145

1. in the given figure, lines ( a ), ( b ), and ( c ) intersect at one

point. what is the value of ( x )?
a) 33
b) 43
c) 57
d) 90

1. in the given figure, ( ab ) and ( de ) are parallel. what is

the length of ( ac )?
a) ( 2sqrt{181} )
b) 36
c) 54
d) ( 4sqrt{181} )

1. what is the value of ( c ) if the number of radians in a

630 - degree angle can be written as ( cpi ), where ( c ) is a
constant?
a) ( \frac{1}{6} )
b) ( \frac{5}{6} )
c) ( \frac{3}{2} )
d) ( \frac{7}{2} )

Explanation:

Step1: Use the property of supplementary angles

Since \(d = 105\), then \(b=180 - d=180 - 105 = 75\) (supplementary angles).

Step2: Use the triangle - angle sum property

In the triangle, \(a + b + c=180\). Substitute \(b = 75\), so \(a + c=180 - 75=105\).

Step3: Substitute \(a = 2c\) into \(a + c=105\)

If \(a = 2c\), then \(2c + c=105\), \(3c=105\), \(c = 35\).

Step4: Calculate \(a\)

Since \(a = 2c\), \(a=2\times35 = 70\).

Step5: Calculate \(a + b\)

\(a + b=70+75 = 145\)

Step1: Use the property of vertical angles and right - angle sum

We know that the sum of angles around a point is \(360^{\circ}\). Also, we have a right - angle (\(90^{\circ}\)) and a \(57^{\circ}\) angle.
Since \(x\) and the angle adjacent to \(57^{\circ}\) (in the non - right - angle part) are related.
We know that \(x+90 + 57=180\) (linear pair with the other half of the full - circle angles around the intersection point).
\(x=180-(90 + 57)=33\)

Step1: Use the similarity of triangles

Since \(AB\parallel DE\), \(\triangle ABC\sim\triangle DEC\).
The ratio of corresponding sides is \(\frac{DE}{AB}=\frac{EC}{BC}\). Let \(BC=x\), then \(\frac{20}{40}=\frac{18}{x}\), \(20x=40\times18\), \(x = 36\). So \(BC=BE + EC=36\), \(BE=36 - 18 = 18\).

Step2: Use the Pythagorean theorem in \(\triangle ABC\)

\(AB = 40\), \(BC = 36\). By the Pythagorean theorem \(AC=\sqrt{AB^{2}+BC^{2}}=\sqrt{40^{2}+36^{2}}=\sqrt{1600 + 1296}=\sqrt{2896}=4\sqrt{181}\)

Step1: Use the degree - radian conversion formula

We know that \(180^{\circ}=\pi\) radians.
If \(\theta = 630^{\circ}\), then \(\theta\) (in radians) is \(\frac{630}{180}\pi\).
\(\frac{630}{180}=\frac{7}{2}\)

Answer:

D. 145