Basic Geometry : Plane Geometry

Example Questions

Example Question #1591 : Basic Geometry

Refer to the above diagram.

True or false:  and  comprise a linear pair.

True

False

False

Explanation:

By definition, two angles form a linear pair if and only if

(1) they have the same vertex;

(2) they share a side; and,

(3) their interiors have no points in common.

is the angle with vertex ; its two sides are the rays  and , which have endpoint  and pass through  and , respectively.  has the same vertex; its two sides are the rays  and , which have endpoint  and pass through  and , respectively.  and  are indicated below in red and green, respectively:

The angles have the same vertex and they share a side. However, the interior of  is entirely contained in the interior of . The angles do not comprise a linear pair.

Example Question #1592 : Basic Geometry

Refer to the above diagram.

True

False

True

Explanation:

A quadrilateral is named after its four vertices in consecutive order, going clockwise or counterclockwise. Quadrilateral  is the figure in red, below:

, and  also name the four vertices in clockwise order. It follows that Quadrilateral  is another valid name for the figure.

Example Question #1593 : Basic Geometry

Refer to the above diagram.

True

False

True

Explanation:

A quadrilateral is named after its four vertices in consecutive order, going clockwise or counterclockwise. Quadrilateral  is the figure in red, below:

, , and  name the four vertices in counterclockwise order. It follows that Quadrilateral  is another valid name for the figure.

Example Question #1594 : Basic Geometry

Refer to the above diagram.

True, false, or inconclusive:

False

True

Inconclusive

Inconclusive

Explanation:

and  form a pair of vertical angles, and are consequently congruent whether or not it holds that . Therefore, whether the lines are parallel cannot be determined.

Example Question #1591 : Basic Geometry

Figure NOT drawn to scale.

Refer to the above diagram.

True, false, or inconclusive: .

Inconclusive

True

False

True

Explanation:

and  are both inside the two lines  and , and they appear on opposite sides of transversal . They are thus alternate interior angles, by definition, and since they are congruent, then by the Converse of the Alternate Interior Angles Theorem, it follows that ,

Example Question #1596 : Basic Geometry

Refer to the above diagram.

True or false:  and  refer to the same triangle.

True

False

True

Explanation:

The letters in the name of a triangle name its vertices, so  refers to the triangle with vertices , and . A triangle is named after its vertices in any order, so this triangle can also be called .

Example Question #1591 : Basic Geometry

Figure NOT drawn to scale.

Refer to the above diagram.

and  are supplementary.

True, false, or inconclusive: It follows that .

True

Inconclusive

False

Inconclusive

Explanation:

and  form a linear pair of angles and are supplementary regardless of whether or not

Example Question #1598 : Basic Geometry

Figure NOT drawn to scale.

Refer to the above diagram.

True, false, or inconclusive: it follows that

True

False

Inconclusive

True

Explanation:

and  form a linear pair of angles, and are therefore supplementary; the same holds for  and . Angles that are supplementary to congruent angles are themselves congruent, so, since , it follows that .

Example Question #1599 : Basic Geometry

If lines A and B are parallel, what is the measurement of ?

Explanation:

Notice that  and  are corresponding angles. This means that they possess the same angular measurements; thus, we can write the following:

Now, notice that  and the provided angle of  angle are vertical angles. Vertical angles share the same angle measurements; therefore, we may write the following:

If  and , then

Example Question #12 : Geometry

Angle  measures

is the bisector of

is the bisector of

What is the measure of ?