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Applications of Line and Angle Theorems

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Applications of Line and Angle Theorems
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What can you say about the relationship between  \overline{AB} and \overline{CD} ? What does this have to do with kites?

Watch This

http://www.youtube.com/watch?v=rQcgfcqwv0A James Sousa: Parallel Line Properties

Guidance

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,  \angle 1 and  \angle 3 are vertical angles.  \angle 2 and  \angle 4 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.

  • \angle 1  and  \angle 5 are corresponding angles because their locations are corresponding. If lines are parallel, then corresponding angles are congruent . Other examples of corresponding angles are  \angle 2 and \angle 6\angle 3 and \angle 7 , and  \angle 4 and \angle 8 .
  • \angle 4  and  \angle 5 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  \angle 3 and \angle 6 .
  • \angle 3  and  \angle 5 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  \angle 4 and \angle 6 .
  • \angle 1  and  \angle 7 are alternate exterior angles . If lines are parallel, then alternate exterior angles are congruent . Another example of alternate exterior angles is  \angle 2 and \angle 8 .
  • \angle 2  and  \angle 7 are same side exterior angles . If lines are parallel, then same side exterior angles are supplementary . Another example of same side exterior angles is  \angle 1 and \angle 8 .

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  C moves along the perpendicular bisector, it will always be true that \overline{AC} \cong \overline{CB} .

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

Example A

If  m \angle KIB=105^\circ , what is:

  1. m \angle JIE ?
  2. m \angle DEI ?
  3. m \angle GEI ?

Solution:

  1. m \angle JIE=105^\circ because it is a vertical angle with  \angle KIB and vertical angles are congruent.
  2. m \angle DEI=105^\circ because it is a corresponding angle with  \angle KIB and corresponding angles are congruent when lines are parallel (note that the \gg markings indicate that the lines are parallel).
  3. m \angle GEI=75^\circ because it forms a straight line with  \angle DEI and so those angles are supplementary.

Example B

Find the length of \overline{CB} .

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

Example C

Parallelogram  ABCD is shown below. Prove that \Delta ABC \cong \Delta CDA .

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

Statements

Reasons

\overline{AD} \ \| \ \overline{BC}  and \overline{AB} \ \| \ \overline{DC}

Definition of a parallelogram

\angle DAC \cong \angle BCA , \angle ACD \cong \angle CAB

Alternate interior angles are congruent if lines are parallel

\overline{AC} \cong \overline{AC}

Reflexive Property

\Delta ABC \cong \Delta CDA

ASA \cong

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.

Concept Problem Revisited

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

Quadrilateral  ACBD 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.

Vocabulary

A kite is a convex quadrilateral with two pairs of adjacent congruent sides such that not all sides are congruent.

A parallelogram is a quadrilateral with two pairs of parallel sides.

A transversal is a line that intersects multiple other lines.

The transitive property states that if two objects are equal/congruent to the same object, they are equal/congruent to each other.

Two angles are supplementary if the sum of their measures is 180^\circ .

A postulate is a statement that is assumed to be true without proof.

A theorem is a true statement that must/can be proven.

A proof is a mathematical argument that shows step by step why a statement must be true. All proofs must contain statements and reasons.

A paragraph proof is a proof that is written out in words/sentences.

A two-column proof organizes statements and reasons into columns.

A flow diagram proof organizes statements in boxes with reasons underneath. Arrows show the flow of logic from the original assumptions and given statements to the conclusion.

The reflexive property states that anything is congruent to itself.

CPCTC is an abbreviation for “corresponding parts of congruent triangles are congruent”. It is used to show that two angles or line segments are congruent after it has been shown that two triangles are congruent.

The converse of a statement switches the “if” part of the statement (known as the hypothesis ) with the “then” part of the statement (known as the conclusion ).

Guided Practice

In the diagram below,  m \angle ABC=50^\circ and m \angle KIJ=80^\circ .

  1. Find m \angle EBI .
  2. Find m \angle BIE .
  3. Find m \angle BEI .
  4. Find m \angle GEI .

Answers:

  1. m \angle EBI=50^\circ because it is a vertical angle with  \angle ABC and vertical angles are congruent.
  2. m \angle BIE=80^\circ because it is a vertical angle with  \angle KIJ and vertical angles are congruent.
  3. m \angle BEI=50^\circ because  \angle EBI\angle BIE , and  \angle BEI form a triangle, and the sum of the measures of the interior angles of a triangle is 180^\circ .
  4. m \angle GEI=80^\circ because it is a corresponding angle with  \angle KIJ and corresponding angles are congruent when lines are parallel.

Practice

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,  C is the midpoint of \overline{BD} . Prove that \Delta ABC \cong \Delta EDC .

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

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

13. The measures of two angles are given below. Solve for x .

14. The measures of two angles are given below. Solve for x .

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

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