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

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.

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 $m \angle BOD.$

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

• Types of Angles
• Finding Angle Measures
• Complementary and Supplementary Angles
• Vertical Angles
• Perpendicular Lines
• Simplifying Ratios
• Find Unknown Ratios
• SAT Ratios, Proportions, and Percents

• 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