Angles in Pre-algebra: Understanding Key Concepts

Using the Vertex	Using the Vertex and One Point on Each Ray	Using a Number	Using Greek Letters
$∠ B$	$∠ A B C$ or $∠ C B A$	$∠ 1$	$∠ α$ or $∠ β$ or $∠ θ$

Using the Vertex

Using the Vertex and One Point on Each Ray

Using a Number

Using Greek Letters

∠ B

∠ A B C

∠ C B A

∠ 1

∠ α

∠ β

∠ θ

Congruent Angles

Adjacent Angles

Vertical Angles

∠ A X B

and

∠ B X C

∠ C X D

and

∠ E X A

∠ D X E

and

∠ A X C

∠ A X B

and

∠ B X C

∠ B X C

and

∠ C X D

∠ C X D

and

∠ D X E

∠ D X E

and

∠ E X A

∠ E X A

and

∠ A X B

∠ C X D

and

∠ E X A

∠ D X E

and

∠ A X C

a Start by identifying whether there are congruent angles, or angles with the same measure. In the diagram, only one angle measure is shown. However, congruent angles are also denoted with the same number of angle markers.

Since $∠ A X B$ and $∠ B X C$ have the same number of markers, the angles are congruent.

Congruent Angles
$∠ A X B$ and $∠ B X C$

Next, focus on identifying adjacent angles. Adjacent angles have the same vertex, share one side, and they do not overlap. These three conditions are met by $∠ D X E$ and $∠ E X A .$

In the diagram there are five pairs of adjacent angles.

Adjacent Angles
$∠ A X B$ and $∠ B X C$
$∠ B X C$ and $∠ C X D$
$∠ C X D$ and $∠ D X E$
$∠ D X E$ and $∠ E X A$
$∠ E X A$ and $∠ A X B$

Finally, look for vertical angles. Vertical angles are opposite angles formed when two lines or line segments intersect. Since lines $A D$ and $C E$ intersect at $X,$ they form two pairs of vertical angles. To make it easier to see, ignore the unnecessary parts of the diagram and focus on just these two lines.

From the diagram, $∠ C X D$ and $∠ E X A$ are vertical angles, as are $∠ D X E$ and $∠ A X C .$

Vertical Angles
$∠ C X D$ and $∠ E X A$
$∠ D X E$ and $∠ A X C$

Because vertical angles are always congruent, the last pairs of angles are also congruent angles. All the information obtained from the diagram is summarized in the following table.

Congruent Angles	Adjacent Angles	Vertical Angles
$∠ A X B$ and $∠ B X C$ $∠ C X D$ and $∠ E X A$ $∠ D X E$ and $∠ A X C$	$∠ A X B$ and $∠ B X C$ $∠ B X C$ and $∠ C X D$ $∠ C X D$ and $∠ D X E$ $∠ D X E$ and $∠ E X A$ $∠ E X A$ and $∠ A X B$	$∠ C X D$ and $∠ E X A$ $∠ D X E$ and $∠ A X C$

b To check whether

∠ B X C

and

∠ D X E

are related, ignore the irrelevant parts for a moment.

The angles have the same vertex but they do not have a common side. Therefore, they are not adjacent angles. Notice that $∠ D X E$ has no angle marker and its measure seems to be greater than the measure of $∠ B X C .$ Therefore, the angles are not congruent.

$∠ B X C$ and $∠ D X E$
Adjacent	Congruent	Vertical
$\times$	$\times$	$?$

Lastly, note that $E,$ $X,$ and $C$ lie on the same line but $D,$ $X,$ and $B$ do not. Therefore, $∠ B X C$ and $∠ D X E$ are not vertical angles. As such, there is no relation between these angles.

$∠ B X C$ and $∠ D X E$
Adjacent	Congruent	Vertical
$\times$	$\times$	$\times$

c From Part A,

∠ E X A

and

∠ C X D

are vertical angles, which also means they are congruent. This means that they have the same measure. Since the measure of

∠ C X D

10 0^{\circ},

the measure of

∠ E X A

is also

10 0^{\circ} .

Vertical Angles	Supplementary Angles
$∠ 1$ and $∠ 3$ $∠ 2$ and $∠ 4$	$∠ 1$ and $∠ 2,$ $∠ 2$ and $∠ 3$ $∠ 3$ and $∠ 4,$ $∠ 4$ and $∠ 1$
$∠ 5$ and $∠ 7$ $∠ 8$ and $∠ 9$	$∠ 5$ and $∠ 6,$ $∠ 6$ and $∠ 7$ $∠ 7$ and $∠ 8,$ $∠ 8$ and $∠ 5$

Vertical Angles

Supplementary Angles

∠ 1

and

∠ 3

∠ 2

and

∠ 4

∠ 1

and

∠ 2,

∠ 2

and

∠ 3

∠ 3

and

∠ 4,

∠ 4

and

∠ 1

∠ 5

and

∠ 7

∠ 8

and

∠ 9

∠ 5

and

∠ 6,

∠ 6

and

∠ 7

∠ 7

and

∠ 8,

∠ 8

and

∠ 5

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Catch-Up and Review

Interior and Exterior of an Angle

Answer

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution