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Applications of Line and Angle Theorems

Find angles and line segments, and determine if shapes are congruent and lines are parallel.

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Applications of Line and Angle Theorems

What can you say about the relationship between \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{CD}\end{align*}? What does this have to do with kites?

Applying Line and Angle Theorems

There are four categories of theorems to remember that have to do with lines and angles.

1) When two lines intersect, two pairs of vertical angles are formed.

In the diagram above, \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 3\end{align*} are vertical angles. \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 4\end{align*} are also vertical angles. Vertical angles are always congruent.

2) When two lines are cut by a transversal, many different angle pairs are formed. If the two lines are parallel, these angle pairs have special properties.

  • \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 5\end{align*} are corresponding angles because their locations are corresponding. If lines are parallel, then corresponding angles are congruent. Other examples of corresponding angles are \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 6\end{align*}\begin{align*}\angle 3\end{align*} and \begin{align*}\angle 7\end{align*}, and \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 8\end{align*}.
  • \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 5\end{align*} are same side interior angles because they are inside the lines and on the same side of the transversal. If lines are parallel, then same side interior angles are supplementary. Another example of same side interior angles is \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 6\end{align*}.
  • \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 5\end{align*} are alternate interior angles because they are inside the lines and on opposite sides of the transversal. If lines are parallel, then alternate interior angles are congruent. Another example of alternate interior angles is \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 6\end{align*}.
  • \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 7\end{align*} are alternate exterior angles. If lines are parallel, then alternate exterior angles are congruent. Another example of alternate exterior angles is \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 8\end{align*}.
  • \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 7\end{align*} are same side exterior angles. If lines are parallel, then same side exterior angles are supplementary. Another example of same side exterior angles is \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 8\end{align*}.

3) The converses of all of the above theorems and postulates are also true and are ways to show that lines are parallel. For example, if corresponding angles are congruent then lines must be parallel. Similarly, if same side interior angles are supplementary then lines must be parallel.

4) When a line segment is bisected by a perpendicular line, the points on the perpendicular bisector are exactly those equidistant from the segment's endpoints.

For the figure above, as \begin{align*}C\end{align*} moves along the perpendicular bisector, it will always be true that \begin{align*}\overline{AC} \cong \overline{CB}\end{align*}.

If you remember all of the above postulates and theorems, you can use them to help solve problems.

Measuring Angles 

If \begin{align*}m \angle KIB=105^\circ\end{align*}, what is:

a. \begin{align*}m \angle JIE\end{align*}

\begin{align*}m \angle JIE=105^\circ\end{align*} because it is a vertical angle with \begin{align*}\angle KIB\end{align*} and vertical angles are congruent.

b. \begin{align*}m \angle DEI\end{align*}

\begin{align*}m \angle DEI=105^\circ\end{align*} because it is a corresponding angle with \begin{align*}\angle KIB\end{align*} and corresponding angles are congruent when lines are parallel (note that the \begin{align*}\gg\end{align*} markings indicate that the lines are parallel).

c. \begin{align*}m \angle GEI\end{align*}

\begin{align*}m \angle GEI=75^\circ\end{align*} because it forms a straight line with \begin{align*}\angle DEI\end{align*} and so those angles are supplementary.

Measuring Line Segments 

Find the length of \begin{align*}\overline{CB}\end{align*}.

The markings in the picture indicate that \begin{align*}D\end{align*} is the midpoint of \begin{align*}\overline{AB}\end{align*} and \begin{align*}\angle CDB\end{align*} is a right angle. This means that \begin{align*}\overleftrightarrow{C D}\end{align*} is the perpendicular bisector of \begin{align*}\overline{AB}\end{align*}. Therefore, \begin{align*}C\end{align*} must be equidistant from \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, and \begin{align*}CB=2 \ cm\end{align*}.

Writing a Proof

Parallelogram \begin{align*}ABCD\end{align*} is shown below. Prove that \begin{align*}\Delta ABC \cong \Delta CDA\end{align*}.

Recall that the definition of a parallelogram is a quadrilateral with two pairs of parallel sides. Since this is a parallelogram, you know that \begin{align*}\overline{AD} \ \| \ \overline{BC}\end{align*} and \begin{align*}\overline{AB} \ \| \ \overline{DC}\end{align*} (Remember that \begin{align*}\|\end{align*} means parallel). With parallel lines comes lots of congruent angles. These angles will help you to show that the triangles are congruent.

Statements

Reasons

\begin{align*}\overline{AD} \ \| \ \overline{BC}\end{align*} and \begin{align*}\overline{AB} \ \| \ \overline{DC}\end{align*}

Definition of a parallelogram

\begin{align*}\angle DAC \cong \angle BCA\end{align*}, \begin{align*}\angle ACD \cong \angle CAB\end{align*}

Alternate interior angles are congruent if lines are parallel

\begin{align*}\overline{AC} \cong \overline{AC}\end{align*}

Reflexive Property

\begin{align*}\Delta ABC \cong \Delta CDA\end{align*}

\begin{align*}ASA \cong \end{align*}

If you have trouble seeing the alternate interior angles, try extending the lines that form the parallelogram and focusing on one pair of parallel lines at a time.

Examples

Example 1

Earlier, you were asked what does the relationship between \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{CD}\end{align*} have to do with kites

From the markings in the picture, you can see that \begin{align*}\overline{AC} \cong \overline{CB}\end{align*} and \begin{align*}\overline{AD} \cong \overline{DB}\end{align*}. This means that both \begin{align*}C\end{align*} and \begin{align*}D\end{align*} are equidistant from \begin{align*}A\end{align*} and \begin{align*}B\end{align*}. Therefore, both \begin{align*}C\end{align*} and \begin{align*}D\end{align*} are on the perpendicular bisector of \begin{align*}\overline{AB}\end{align*}. Therefore, \begin{align*}\overline{CD}\end{align*} must BE the perpendicular bisector of \begin{align*}\overline{AB}\end{align*}.

Quadrilateral \begin{align*}ACBD\end{align*} is a kite because it has two pairs of adjacent congruent sides. This shows that one of the diagonals of a kite is the perpendicular bisector of the other diagonal.

In the diagram below, \begin{align*}m \angle ABC=50^\circ\end{align*} and \begin{align*}m \angle KIJ=80^\circ\end{align*}.

Example 2

Find \begin{align*}m \angle EBI\end{align*}.

\begin{align*}m \angle EBI=50^\circ\end{align*} because it is a vertical angle with \begin{align*}\angle ABC\end{align*} and vertical angles are congruent.

Example 3

Find \begin{align*}m \angle BIE\end{align*}.

\begin{align*}m \angle BIE=80^\circ\end{align*} because it is a vertical angle with \begin{align*}\angle KIJ\end{align*} and vertical angles are congruent.

Example 4

Find \begin{align*}m \angle BEI\end{align*}.

\begin{align*}m \angle BEI=50^\circ\end{align*} because \begin{align*}\angle EBI\end{align*}\begin{align*}\angle BIE\end{align*}, and \begin{align*}\angle BEI\end{align*} form a triangle, and the sum of the measures of the interior angles of a triangle is \begin{align*}180^\circ \end{align*}.

Example 5

Find \begin{align*}m \angle GEI\end{align*}.

\begin{align*}m \angle GEI=80^\circ\end{align*} because it is a corresponding angle with \begin{align*}\angle KIJ\end{align*} and corresponding angles are congruent when lines are parallel.

Review

1. Draw an example of vertical angles.

Use the diagram below for questions 2-4.

2. Give an example of same side interior angles. Name each angle with three letters.

3. Give an example of alternate interior angles. Name each angle with three letters.

4. Give an example of corresponding angles. Name each angle with three letters.

5. If lines are not parallel, are corresponding angles still congruent?

For 6-9, determine whether or not the lines are parallel based on the given angle measures. Explain your answer in each case.

6.

7.

8.

9.

10. In the diagram below, \begin{align*}C\end{align*} is the midpoint of \begin{align*}\overline{BD}\end{align*}. Prove that \begin{align*}\Delta ABC \cong \Delta EDC\end{align*}.

11. Extend your proof from #10 to prove that \begin{align*}\overline{AC} \cong \overline{CE}\end{align*}.

12. Which two line segments must be parallel in the picture below?

13. The measures of two angles are given below. Solve for \begin{align*}x\end{align*}.

14. The measures of two angles are given below. Solve for \begin{align*}x\end{align*}.

15. \begin{align*}D\end{align*} is the midpoint of \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{AC} \cong \overline{CB}\end{align*}. Find the length of \begin{align*}\overline{AB}\end{align*}.

 

Review (Answers)

To see the Review answers, open this  PDF file and look for section 4.3. 

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Vocabulary

TermDefinition
converse If a conditional statement is p \rightarrow q (if p, then q), then the converse is q \rightarrow p (if q, then p. Note that the converse of a statement is not true just because the original statement is true.
Kite A kite is a quadrilateral with distinct adjacent congruent sides.
Reflexive Property of Congruence \overline{AB} \cong \overline{AB} or \angle B \cong \angle B

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