# SAT Lines and Angles

To successfully solve problems about lines and angles on the SAT, you need to know:

- angles on the SAT are always measured in degrees
- the types of angles
- angle arithmetic
- the definition of complementary and supplementary angles
- the definition of vertical angles
- the definition of an angle bisector
- the definition of a midpoint
- the segment addition postulate
- the angle addition postulate

## Examples for Lines

In the figure above, if $AB=x-1, BC=2x+1, CD=x,$ and $AD=12,$ what is the length of $\overline{BD}?$

(A) $\ \ 2$

(B) $\ \ 3$

(C) $\ \ 7$

(D) $\ \ 9$

(E) $\ \ 10$

Correct Answer: E

Solution:By the segment addition postulate, we have:

$\begin{array}{r c l} AB + BC + CD &=& 12\\ x-1 + 2x + 1 + x &=& 12\\ 4x &=& 12\\ x &=&3\\ \end{array}$

We are looking for $BD.$

$BD = BC + CD = 2x+1 + x = 3x + 1 = 3\cdot 3 + 1 = 9 + 1 =10.$

Incorrect Choices:

(A)

Tip: Read the entire question carefully.

Tip: If a diagram is drawn to scale, trust it.

If you solve for $AB$ instead of $BD,$ you will get this wrong answer. Note that we can eliminate this answer since the diagram is drawn to scale, and since $BD$ seems to be greater than half of $AD,$ or greater than 6.

(B)

Tip: Read the entire question carefully.

Tip: If a diagram is drawn to scale, trust it.

If you solve for $x$ instead of $BD,$ you will get this wrong answer. Note that we can eliminate this answer since the diagram is drawn to scale, and since $BD$ seems to be greater than half of $AD,$ or greater than 6.

(C)

Tip: Read the entire question carefully.

If you solve for $BC$ instead of $BD,$ you will get this wrong answer.

(D)

Tip: Read the entire question carefully.

If you solve for $AC,$ instead of $BD,$ you will get this wrong answer.

$A$ and $B$ have coordinates $a$ and $b,$ such that $b>a.$ If $M$ is the midpoint of segment $\overline{AB},$ and $M$ has the coordinate $x,$ all of the following are true EXCEPT:

In the diagram above, points(A) $\ \ AM = \frac{1}{2}AB$

(B) $\ \ MB = b-x$

(C) $\ \ \overline{AM} \cong \overline{MB}$

(D) $\ \ 2x = a+b$

(E) $\ \ x = 2b-a$

## Examples for Angles

The ratio of $m\angle AOB$ to $m\angle BOC$ to $m\angle COD$ is 1 : 3 : 2. What is the measure of $\angle BOD?$

(A) $\ \ 30$

(B) $\ \ 90$

(C) $\ \ 100$

(D) $\ \ 120$

(E) $\ \ 150$

Correct Answer: E

Solution 1:

Tip: Angles on a line sum to $180^\circ.$

If $\angle AOD$ were divided into 1 + 3 + 2 = 6 parts, then $\angle BOC$ would equal to 3 out of 6 parts, and $\angle COD$ would equal to 2 out of 6 parts.Since $m\angle AOD = 180^\circ,$ it follows that

$m\angle BOC = \frac{3}{6} \cdot 180^\circ = 90^\circ$ and $m\angle COD = \frac{2}{6} \cdot 180^\circ = 60^\circ.$

Therefore $m\angle BOD = m\angle BOC + m\angle COD = 90^\circ + 60^\circ = 150^\circ.$

Solution 2:

Tip: Angles on a line sum to $180^\circ.$

Let $m\angle AOB = x.$ Then, according to the given ratio, $m \angle BOC = 3x$ and $m\angle COD = 2x.$$\begin{array}{r c l} m \angle AOB + m \angle BOD + m \angle DOC &=& 180^\circ\\ x + 3x + 2x &=& 180^\circ\\ 6x &=& 180^\circ\\ x &=& 30^\circ\\ \end{array}$

Therefore $m\angle BOD = m\angle BOC + m\angle COD = 3x + 2x = 5x = 5\cdot 30^\circ = 150^\circ.$

Solution 3:

Tip: If a diagram is drawn to scale, trust it.

It is evident from the diagram that $\angle BOD$ is obtuse. Therefore, we can eliminate options (A) and (B).

We draw segment $\overline{OE},$ perpendicular to $\overline{AD}$ and we divide $\angle AOE$ in two $45^\circ$ angles. Because $m\angle BOD > 90^\circ+45^\circ=135^\circ,$ we eliminate choices (C) and (D), and select choice (E) as the correct answer.

Incorrect Choices:

(A)

Tip: Read the entire question carefully.

This is $m\angle AOB,$ not $m \angle BOD.$

(B)

Tip: Read the entire question carefully.

This is $m\angle BOC,$ not $m \angle BOD.$

(C)

This answer choice is just meant to confuse you.

(D)

Tip: Read the entire question carefully.

This is $m\angle AOC,$ not $m \angle BOD.$

## Review

If you thought these examples difficult and you need to review the material, these links will help:

## SAT Tips for Lines and Angles

- Angles at a point sum to $360^\circ.$
- Angles on a line sum to $180^\circ.$
- $\angle A$ and $\angle B$ are complementary if $m\angle A + m\angle B=90^\circ.$
- $\angle A$ and $\angle B$ are supplementary if $m\angle A + m\angle B=180^\circ.$
- Vertical angles are congruent.
- The angle bisector divides an angle in half.
- The midpoint of a segment divides it in half.
- If a diagram is drawn to scale, trust it.
- SAT General Tips

**Cite as:**SAT Lines and Angles.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/sat-lines-and-angles/