# Digital SAT Math Practice Questions -Medium : Lines, angles, and triangles

## SAT MAth Practice questions – all topics

• Geometry and Trigonometry Weightage: 15% Questions: 5-7
• Area and volume
• Lines, angles, and triangles
• Right triangles and trigonometry
• Circles

## SAT MAth and English  – full syllabus practice tests

[Calc]  Question   Medium

In the figure shown, points $$A$$ and $$B$$ lie on the circle with radius 1 centered at the origin, $$O$$. If the cosine of $$\angle A O B$$ is $$-\frac{1}{2}$$, what is the measure, in radians, of $$\angle A O B$$ ?
A) $$\frac{5 \pi}{12}$$
B) $$\frac{7 \pi}{12}$$
C) $$\frac{2 \pi}{3}$$
D) $$\frac{5 \pi}{6}$$

Ans:C

$$cos~ \angle \rm{AOB}=-\frac{1}{2}$$

$\rm cos~ \angle\rm{AOB}=\frac{\rm{OA}^2+\rm{OB}^2 – \rm{AB}^2}{2\times\rm{OA}\times \rm{OB}}$

$\frac{-1}{2}=\frac{\rm{1}^2+\rm{1}^2 – \rm{AB}^2}{2\times\rm{1}\times \rm{1}}$

$-1=2-\rm{AB}^2$

$3=\rm{AB}^2$

$\rm{AB}=\sqrt 3$

We are given $$\cos (\angle A O B)=-\frac{1}{2}$$.

Recall that $$\cos (\theta)=-\frac{1}{2}$$ for angles $$\theta$$ in the range from 0 to $$2 \pi$$ (or 0 to 360 degrees).
The standard angles where $$\cos (\theta)=-\frac{1}{2}$$ are:
$$120^{\circ}$$ (which is $$\frac{2 \pi}{3}$$ radians)
$$240^{\circ}$$ (which is $$\frac{4 \pi}{3}$$ radians)

[Calc]  Question Medium

The function $$f$$ is defined by $$f(x)=2 x+6$$. What is the graph of $$y=f(x)$$ ?

A

Y-intercept
The $$y$$-intercept occurs where the graph of the function crosses the $$y$$-axis. This happens when $$x=0$$.
$f(0)=2(0)+6=6$

Thus, the $$y$$-intercept is at $$(0,6)$$.
X-intercept
The $$\mathrm{x}$$-intercept occurs where the graph of the function crosses the $$\mathrm{x}$$-axis. This happens when $$f(x)=0$$.
$2 x+6=0$

Solve for $$x$$ :
\begin{aligned} & 2 x=-6 \\ & x=-3 \end{aligned}

Thus, the $$x$$-intercept is at $$(-3,0)$$.

[No calc]  Question  medium

Which of the following pieces of information is sufficient to prove that triangle ABC is an isosceles triangle?

1. $$\overline{AB}$$  is congruent to $$\overline{BC}$$
2. ∠A is congruent to ∠C

A) I is sufficient, but II is not

B) II is sufficient, but I is not

C) Either I or II is sufficient

D) Neither I nor II is sufficient

C)  Either I or II is sufficient

To determine which piece of information is sufficient to prove that triangle $$ABC$$ is an isosceles triangle, let’s analyze each statement:

Statement I: $$\overline{AB}$$ is congruent to $$\overline{BC}$$

If $$\overline{AB}$$ is congruent to $$\overline{BC}$$, then by definition, triangle $$ABC$$ has two sides that are equal in length. This is sufficient to prove that triangle $$ABC$$ is an isosceles triangle.

Statement II: $$\angle A$$ is congruent to $$\angle C$$

If $$\angle A$$ is congruent to $$\angle C$$, this implies that the angles opposite those angles ($$\overline{BC}$$ and $$\overline{AB}$$ respectively) are equal. According to the Isosceles Triangle Theorem, if two angles in a triangle are equal, then the sides opposite those angles are also equal. Therefore, this is sufficient to prove that triangle $$ABC$$ is an isosceles triangle.

Both statements independently provide sufficient information to prove that triangle $$ABC$$ is an isosceles triangle.

[Calc]  Question  Medium

Triangles $$\mathrm{ABC}$$ and $$\mathrm{DEF}$$ each have a corresponding angle measuring $$40^{\circ}$$. Which additional piece of information is sufficient to determine whether these two triangles are similar?

A) The length of line segment $$A C$$
B) The length of line segment $$\mathrm{DE}$$
C) The measure of another pair of corresponding angles in the two triangles.
D) The lengths of one pair of corresponding sides in the two triangles.

Ans:C

To determine if two triangles are similar, we need to check if their corresponding angles are equal or if their corresponding sides are proportional.

Given that triangles ABC and DEF each have a corresponding angle measuring $$40^\circ$$, the additional piece of information needed to determine similarity is:

C) The measure of another pair of corresponding angles in the two triangles.

By knowing two corresponding angles, we can use the Angle-Angle (AA) similarity criterion. If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

$\boxed{\text{C}}$

[Calc]  Question  Medium

In the figure shown, $$\bar{AE}$$ and $$\bar{BD}$$ intersect at point C. Which of the following additional pieces of information is NOT sufficient to prove that $$\bigtriangleup$$ABC is similar to $$\bigtriangleup$$EDC ?

A. $$\bar{AB}$$ is parallel to $$\bar{DE}$$ .

B. The measure of $$\angle D$$ is equal to the measure of  $$\angle B$$.

C. The length of $$\bar{AB}$$is equal to the length of $$\bar{DE}$$.

D. The measure of $$\angle A$$ is equal to the measure of $$\angle B$$, and the measure of $$\angle D$$is equal to the measure of $$\angle E$$.

Ans: C

The length of $$\bar{AB}$$ is equal to the length of $$\bar{DE}$$.”

If the lengths of two corresponding sides in two triangles are equal, it implies that all the corresponding sides are proportional with a ratio of 1:1. However, this alone does not guarantee that the corresponding angles are equal, which is another necessary condition for proving similarity between triangles.

[Calc]  Question   Medium

$3 x+4 y=16$

The given equation models the number of 3-credit-hour courses, $$x$$, and the number of 4-credit0hour courses, $$y$$, that Camila can take for a total of 16 credit hours next semester. Which graph models this relationship?

Ans:A

X-intercept
The $$x$$-intercept occurs when $$y=0$$.
Substitute $$y=0$$ into the equation:
\begin{aligned} & 3 x+4(0)=16 \\ & 3 x=16 \\ & x=\frac{16}{3} \end{aligned}

So, the $$x$$-intercept is $$\left(\frac{16}{3}, 0\right)$$.
Y-intercept
The $$y$$-intercept occurs when $$x=0$$.
Substitute $$x=0$$ into the equation:
\begin{aligned} & 3(0)+4 y=16 \\ & 4 y=16 \\ & y=4 \end{aligned}

[Calc]  Question Medium

In the figure shown, triangle $$A B C$$ is similar to triangle $$A D E$$ such that $$B$$ corresponds to $$D$$ and $$C$$ corresponds to $$E$$. The measure of angle $$A B C$$ is $$60^{\circ}$$. What is the measure of angle $$A D E$$ ?
A) $$15^{\circ}$$
B) $$20^{\circ}$$
C) $$45^{\circ}$$
D) $$60^{\circ}$$

D

• Triangle ABC is similar to triangle ADE.
• Angle ABC measures 60°.
• B corresponds to D, and C corresponds to E.

Since corresponding angles in similar triangles are equal, angle ADC must also be 60°.

Therefore, the measure of angle ADE is also 60°.

The correct answer is D) 60°.

[Calc]  Question  medium

Points A, B, and C lie on the circle as shown. What is the measure, in degrees, of arc $$\widehat{AC}$$ ?

A) 55

B) 110

C) 220

D) 305

B) 110

Therefore, the angle at the centre is twice the angle at the circumference.

So arc $$\widehat{AC}$$ = 110

Hint: Ac will subtend less the $180^{\circ}$ which is only in option B

[Calc]  Question  medium

In rectangle ABCD shown, $$\overline{AB}$$ = 6, $$\overline{CD}$$  = 6, and $$\overline{AC}$$ =10. What is the length of $$\overline{CE}$$ ?

A) 3.6

B) 4.2

C) 5

D) 8

A) 3.6

Using pythagoras theorem in $\triangle \rm{ADE}$ AND $\triangle \rm{DEC}$

In, $\triangle \rm{ADE}$

$d^2+(10-x)^2=8^2~~~-(1)$

In $$\triangle \rm{DEC}$$,
$d^2+x^2=6^2~~~~~-(2)$
Subtracting Eqation $$(1) – (2)$$
\begin{aligned} & (10-x)^2-(x)^2=8^2-6^2 \\ & 100-20 x=28 \\ & 20 x=72 \\ & x=3.6 \end{aligned}

[Calc]  Question   medium

In the figure shown, $$\overline{GE}$$ and $$\overline{DH}$$ intersect at point F. Which of the following additional statements is (are) sufficient to prove that triangle DEF is similar to triangle HGF?

1. The length of $$\overline{DE}$$ is $$\frac{1}{3}$$ the length of $$\overline{HG}$$
2. DE is parallel to HG

A)I is sufficient, but II is not.

B)II is sufficient, but I is not.

C)I is sufficient, and II is sufficient.

D)Neither I nor II is sufficient.

B)II is sufficient, but I is not.

Statement 1: The length of DE is 1/3 the length of HG. This statement alone does not guarantee similarity between the triangles DEF and HGF. It only establishes a proportional relationship between two sides, but does not provide any information about the angles.

Statement 2 (DE is parallel to HG) alone is sufficient to prove that the triangles DEF and HGF are similar. This is because when two lines (DE and HG) are parallel and intersected by a pair of transversals (GE and DH), the alternate interior angles formed (DEF and HGF) are congruent. Additionally, all triangles with one angle congruent have the other two angles congruent as well due to the angle sum theorem. Therefore, statement 2 establishes two pairs of congruent angles, which is sufficient to prove similarity.

[Calc]  Questions   medium

Triangle ABC and Triangle DEF each have an angle measuring $$29^{\circ }$$ and an angle measuring $$54^{\circ }$$, as shown above. Which of the following statements is sufficient to prove triangle ABC is congruent to triangle DEF ?
A) The length of $$\bar{EF}$$ is 10.
B) The measure of angle EDF is $$97^{\circ }$$.
C) The length of $$\bar{BC}$$ is equal to the length of $$\bar{EF}$$.
D) The measure of angle BAC is equal to the measure of angle EDF.

Ans: C

To determine which statement is sufficient to prove that triangle ABC is congruent to triangle DEF, we need to apply the principles of triangle congruence. Triangles can be proven congruent by several criteria, such as:

1. Side-Angle-Side (SAS)
2. Angle-Side-Angle (ASA)
3. Angle-Angle-Side (AAS)
4. Side-Side-Side (SSS)

Given triangles ABC and DEF both have angles measuring $$29^\circ$$ and $$54^\circ$$, the third angle in each triangle must be $$180^\circ – 29^\circ – 54^\circ = 97^\circ$$.

Now, let’s evaluate each statement:

A) The length of $$\overline{EF}$$ is 10.
This gives us one side length of triangle DEF but no corresponding information about triangle ABC. This is insufficient to prove congruence.

B) The measure of angle EDF is $$97^\circ$$.
This confirms that triangle DEF has angles of $$29^\circ$$, $$54^\circ$$, and $$97^\circ$$, which is already given. This does not provide any additional information about side lengths or other correspondences between the triangles. This is insufficient to prove congruence.

C) The length of $$\overline{BC}$$ is equal to the length of $$\overline{EF}$$.
Given that triangles ABC and DEF both have two angles of the same measures ($$29^\circ$$ and $$54^\circ$$), and knowing that a corresponding side ($$\overline{BC}$$) in one triangle is equal to a corresponding side ($$\overline{EF}$$) in the other triangle provides enough information. By the Angle-Side-Angle (ASA) postulate, knowing two angles and the included side, the triangles are congruent. This is sufficient to prove congruence.

D) The measure of angle BAC is equal to the measure of angle EDF.
This confirms that both triangles have angles of $$97^\circ$$, $$29^\circ$$, and $$54^\circ$$. This is already known from the given information and does not add any side length information. This is insufficient to prove congruence.

Therefore, the correct statement that is sufficient to prove that triangle ABC is congruent to triangle DEF is:

C) The length of $$\overline{BC}$$ is equal to the length of $$\overline{EF}$$.

[No- Calc]  Question   Medium

In the triangle RST, angle $$\mathrm{T}$$ measures 40 degrees and angle $$\mathrm{R}$$ measures 20 degrees. What is the measure, in degrees, of angle $$S$$ ?Ans:

Ans:120

In a triangle, the sum of the angles is always $$180^\circ$$. Given that $$\angle T = 40^\circ$$ and $$\angle R = 20^\circ$$, we can find $$\angle S$$ by using the following equation:
$\angle T + \angle R + \angle S = 180^\circ$

Substitute the given angles:
$40^\circ + 20^\circ + \angle S = 180^\circ$

Simplify to solve for $$\angle S$$:
$60^\circ + \angle S = 180^\circ$
$\angle S = 180^\circ – 60^\circ$
$\angle S = 120^\circ$

The measure of angle $$S$$ is:

120 degrees

[Calc]  Question   Medium

In triangle $\mathrm{ABC}$, the measure of angle $\mathrm{A}$ is $23^{\circ}$ and the measure of angle $B$ is $97^{\circ}$. In triangle DEF, the measure of angle $\mathrm{D}$ is $23^{\circ}$ and the measure of angle $\mathrm{E}$ is $97^{\circ}$. Which of the following additional pieces of information is needed to determine whether triangle $\mathrm{ABC}$ is similar to triangle DEF?
A) The measure of angle C
B) The measure of angle $\mathrm{F}$
C) The measure of angle $\mathrm{C}$ and the measure of angle $\mathrm{F}$
D) No additional information is needed.

D

[Calc]  Question Medium

Which of the following is an equation of the line in the $x y$-plane that contains the points $(1,3)$ and $(5,15)$ ?
A) $y=3 x$
B) $y=2 x+5$
C) $y=x+2$
D) $y=\frac{1}{3} x$

A

[Calc]  Question  Medium

An equation of the graph shown is $a x+b y=6$, where $a$ and $b$ are constants. What is the value of $b$ ?
A) -3
B) -1
C) 1
D) 3

A

[Calc]  Question   Medium

Line $p$ is defined by $2 y+4 x=9$. Line $r$ is perpendicular to line $p$ in the $x y$-plane. What is the slope of line $r$ ?

$.5,1 / 2$

Question

The function $$f$$ models Jack’s pulse, in beats per minute (bpm), as a function of his speed, in kilometers per hour (km/h), on a stationary bicycle. Based on the model, what was Jack’s pulse, in bpm, when his speed was 0 km/h?

1. 4
2. 15
3. 63
4. 123

C

Question

. The graph of the line $$y=-\frac{1}{2}x+3$$ in the $$xy$$-plane is translated 2 units to the right. What is the $$y$$-intercept of the translated line?

1. (0, 1)
2. (0, 2)
3. (0, 3)
4. (0, 4)

D

Question

The function A(t) = 10(1/2)t/30 represents the mass A(t) , in grams, of a certain radioactive isotope remaining in a substance after t seconds. Which of the following is the best interpretation of the value 10 in this context?

1. The initial mass, in grams, of the radioactive isotope in the substance when t = 0
2. The mass, in grams, of the radioactive isotope in the substance after 30 seconds
3. The number of seconds it takes for the radioactive isotope in the substance to completely disappear
4. The number of seconds it takes for half of the initial mass of radioactive isotope in the substance to disappear

A

Question

In the figure, lines $$l$$ and $$m$$ each intersect line $$k$$. Which of the following is sufficient to prove that lines $$l$$ and $$m$$ are parallel?

1. $$w$$=$$y$$
2. $$w$$=$$z$$
3. $$x$$=$$y$$
4. $$x$$=$$z$$

B

Question

A patio is to be made using square paving stones that are all the same size. There will be no gaps between the paving stones, and they will not overlap. The line in the $$xy$$-plane above represents the relationship between the area , in square feet, of the patio and the number of paving stones, , used to make the patio. The top surface of each paving stone is a square with side length feet. What is the value of

1. 1
2. 2
3. 3
4. 4

B

Question

Line is shown in the $$xy$$ plane above. Which of the following is an equation of line

D

Question

The graph of $y=\frac{3}{5}x-8$ in the $$xy$$- plane is a line. What is the slope of this line?

.6, 3/5

Question

In the figure, lines $$l$$ and $$m$$ each intersect line $$k$$. Which of the following is sufficient to prove that lines $$l$$ and $$m$$ are parallel?

1. $$w$$=$$y$$
2. $$w$$=$$z$$
3. $$x$$=$$y$$
4. $$x$$=$$z$$

B

Question

The graph of the equation 4$$x$$+3$$y$$=$$q$$, where $$q$$ is a constant, is a line in the         $$xy$$- plane. What are the coordinates of the point at which the line crosses the $$x$$-axis?

1. $(\frac{q}{3},0)$
2. $(\frac{q}{4},0)$
3. $(\frac{3}{q},0)$
4. $(\frac{4}{q},0)$

B

Question

Which of the following is the graph of $$y-5x=-6$$ in the $$xy$$-plane?

Ans: B

Questions

In the figure above, segments $A E$ and $B D$ are parallel. If angle $B D C$ measures $58^{\circ}$ and angle $A C E$ measures $62^{\circ}$, what is the measure of angle CAE?
A. $58^{\circ}$
B. $60^{\circ}$
C. $62^{\circ}$
D. $120^{\circ}$

Ans: B

Questions

Which of the following could be an equation for the graph shown in the $x y$-plane above?
A. $y=-\frac{2}{3} x+8$
B. $y=-\frac{3}{2} x+4$
C. $y=-\frac{1}{3} x+4$
D. $y=-\frac{4}{3} x+8$

Ans: C

Questions

Triangle $A B C$ and triangle $D E F$ are similar triangles, where $\overline{A B}$ and $\overline{D E}$ are corresponding sides. If $D E=2 A B$ and the perimeter of triangle $A B C$ is 20 , what is the perimeter of triangle $D E F$ ?
A. 10
B. 40
C. 80
D. 120

Ans: B

Questions

In the $x y$-plane, line $l$ has a slope of 2. If line $k$ is perpendicular to line $l$, which of the following could be an equation of line $k$ ? 1.4
A. $-10 x-5 y=20$
B. $3 x-6 y=14$
C. $4 x-2 y=17$
D. $6 x+12 y=36$

Ans: D

Questions

The graph of the exponential function $g$ in the $x y$-plane passes through the points $(0,1),(1,4)$, and $(2,16)$. Which of the following is NOT true?
A. A line can be drawn that does not intersect the graph of $g$.
B. A line can be drawn that intersects the graph of $g$ at exactly one point.
C. A line can be drawn that intersects the graph of $g$ at exactly two points.
D. A line can be drawn that intersects the graph of $g$ at exactly three points.

Ans: D

Questions

Which of the following is an equation of the line in the $x y$-plane that has slope 2 and passes through the point $(0,3)$ ?
A. $y=2 x+3$
B. $y=2 x-3$
C. $y=2(x+3)$
D. $y=2(x-3)$

Ans: A

Questions

The graph of the function $$f$$ is shown in the $$xy$$-plane above. The function $$f$$ is defined by the equation , $$f(x)=\frac{a}{b}x+ c$$ for positive constants $$a, b$$, and $$c$$, where $$\frac{a}{b}$$ is a fraction in lowest terms. Which of the following orders $$a, b$$, and $$c$$ from least to greatest?

1. $$a < b < c$$
2. $$a < c < b$$
3. $$b < c < a$$
4. $$c < a < b$$

Ans: B

Questions

$$x+5y=5$$
$$2x-y=-4$$

Which of the following graphs in the $$xy$$-plane could be used to solve the system of equations above?

Ans: C

Questions

In the figure above, $$\overline{AF}$$, $$\overline{BE}$$, and $$\overline{CD}$$ are parallel. Points $$B$$ and $$E$$ lie on $$\overline{AC}$$ and $$\overline{FD}$$, respectively. If AB=9, BC=18.5, and FE=8.5, what is the length of $$\overline{ED}$$, to the nearest tenth? 4.5

1. 16.8
2. 17.5
3. 18.4
4. 19.6

Ans: B

Questions

For the linear function $$f$$ ,the table above gives some values of $$x$$ and their corresponding values $$f (x)$$, where $$c$$ is a constant. Which of the following equations defines $$f$$?

1. $$f(x )=x+c$$
2. $$f(x )=x+3c$$
3. $$f(x)=cx+c$$
4. $$f(x) = 3cx+3c$$

Ans: C

Questions

The graph of a line in the $x y$-plane has a positive slope and intersects the $y$-axis at a point that has a negative $y$-coordinate. Which of the following could be an equation of the line?
A. $-3 x+2 y=-5$
B. $-3 x+2 y=5$
C. $3 x+2 y=-5$
D. $3 x+2 y=5$

Ans: A

Questions

The graph of the equation $a x+k y=6$ is a line in the $x y$-plane, where $a$ and $k$ are constants. If the line contains the points $(-2,-6)$ and $(0,-$ 3 ), what is the value of $k$ ?
A. -2
B. -1
C. 2
D. 3

Ans: A

Questions

In the $x y$-plane, the graph of line $l$ has slope 3 . Line $k$ is parallel to line $l$ and contains the point $(3,10)$. Which of the following is an equation of line $k$
A. $y=-\frac{1}{3} x+11$
B. $y=\frac{1}{3} x+9$
C. $y=3 x+7$
D. $y=3 x+1$

Ans: D

Questions

In triangle $$ABC$$ above, side $$\bar{AC}$$ is extended to point $$D$$. What is the value of $$y-x$$?

1. 40
2. 75
3. 100
4. 140

Ans: C

Questions

In the figure above, line $$t$$ intersects lines $$l$$ and $$k$$. Which of the following statements, if true, would imply that lines $$l$$ and $$k$$ are parallel?

1. $$w=y$$
2. $$w=z$$
3. $$x=z$$
4. $$x+y=l80$$

Ans: B

Questions

Line $$l$$ is shown in the $$xy$$-plane above. Line $$m$$ (not shown) is parallel to line $$l$$ and passes through the point (0,3). Which of the following is an equation of line $$m$$?

1. $$y=-\frac{2}{3}x+3$$
2. $$y=-\frac{3}{2}x+3$$
3. $$y=\frac{2}{3}x+3$$
4. $$y=\frac{3}{2}x+3$$

Ans: A

Questions

What are the slope and the $y$-intercept of the graph in the $x y$-plane of the equation $5 x+4 y+3=0$ ?
A. The slope is $-\frac{5}{4}$, and the $y$-intercept is $\left(0,-\frac{3}{4}\right)$.
B. The slope is $-\frac{5}{4}$, and the $y$-intercept is $\left(0, \frac{3}{4}\right)$.
C. The slope is $\frac{5}{4}$, and the $y$-intercept is $\left(0,-\frac{3}{4}\right)$.
D. The slope is $\frac{5}{4}$, and the $y$-intercept is $\left(0, \frac{3}{4}\right)$.