Angle Addition Postulate - TutorOne USA

Angle Addition Postulate

Chapter 43

Concept Explanation:

The Angle Addition Postulate states that if a point $B$ lies in the interior of $\angle AOC$ , then:

$m\angle AOB + m\angle BOC = m\angle AOC$

This means that the sum of the measures of two smaller angles formed by a point on the interior of an angle is equal to the measure of the entire angle.

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\begin{tikzpicture}
\draw<a href="0,0">thick</a> -- (4,0) node[anchor=north] {C};
\draw<a href="0,0">thick</a> -- (3,3) node[anchor=south] {A};
\draw<a href="0,0">thick</a> -- (4,2) node[anchor=south] {B};
\node at (-0.2,-0.2) {O};
\draw<a href="0,0">fill</a> circle [radius=0.05];
\end{tikzpicture}


*** Error message:
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In the diagram, the angles $\angle AOB$ and $\angle BOC$ together form $\angle AOC$ . If $m\angle AOB = 30^\circ$ and $m\angle BOC = 50^\circ$ , then:

$m\angle AOC = 30^\circ + 50^\circ = 80^\circ$

Common Mistakes:

Incorrectly Adding Angles: Sometimes students incorrectly add angles that are not adjacent or fail to check whether the angles share a common vertex.
Overlooking Angle Units: Be sure to check that all angles are in the same unit (degrees or radians) before adding.

Helpful Tips:

Label the Angles: Marking all angles clearly on diagrams can help ensure the correct angles are being added.
Use a Protractor: When working with diagrams, use a protractor to verify the angle measures.

Hard Questions:

Q1: Given $m\angle AOB = 40^\circ$ and $m\angle BOC = 60^\circ$ , find $m\angle AOC$ .

Step-by-Step Solution:

Apply the Angle Addition Postulate:

$m\angle AOC = m\angle AOB + m\angle BOC = 40^\circ + 60^\circ = 100^\circ$

Answer:

$m\angle AOC = 100^\circ$

Q2: If $m\angle AOC = 120^\circ$ and $m\angle AOB = 70^\circ$ , find $m\angle BOC$ .

Step-by-Step Solution:

Use the Angle Addition Postulate:

$m\angle AOC = m\angle AOB + m\angle BOC$

Rearrange to find $m\angle BOC$ :

$m\angle BOC = m\angle AOC - m\angle AOB = 120^\circ - 70^\circ = 50^\circ$

Answer:

$m\angle BOC = 50^\circ$

Q3: In a triangle, if the exterior angle at vertex $A$ is $150^\circ$ and $m\angle B = 60^\circ$ , find $m\angle C$ .

Step-by-Step Solution:

Use the exterior angle theorem, which states:

$m\angle A = m\angle B + m\angle C$

Rearrange to find $m\angle C$ :

$m\angle C = m\angle A - m\angle B = 150^\circ - 60^\circ = 90^\circ$

Answer:

$m\angle C = 90^\circ$

Q4: Given that $m\angle AOB = 45^\circ$ and $m\angle AOC = 100^\circ$ , find $m\angle BOC$ .

Step-by-Step Solution:

Use the Angle Addition Postulate:

$m\angle AOC = m\angle AOB + m\angle BOC$

Rearrange to find $m\angle BOC$ :

$m\angle BOC = m\angle AOC - m\angle AOB = 100^\circ - 45^\circ = 55^\circ$

Answer:

$m\angle BOC = 55^\circ$

Q5: Find the measure of $m\angle XYZ$ if $m\angle XYW = 35^\circ$ and $m\angle WYZ = 60^\circ$ .

Step-by-Step Solution:

Apply the Angle Addition Postulate:

$m\angle XYZ = m\angle XYW + m\angle WYZ = 35^\circ + 60^\circ = 95^\circ$

Answer:

$m\angle XYZ = 95^\circ$

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