What Is An Auxiliary Line

The following proof introduces you to a new thought: adding a line or segment (called an auxiliary line) to a proof diagram to help you do the proof. Some proofs are impossible to solve until you add a line to the diagram.

Auxiliary lines oftentimes create congruent triangles, or they intersect existing lines at correct angles. So if you're stumped by a proof, meet whether drawing an auxiliary line (or lines) could get you one of those things.

When y'all draw in an auxiliary line, but write something like

for the statement; then use the following postulate for the reason: "Two points determine a line (or ray or segment)."

Hither's an example proof:

Yous might come with a game program like the following:

• Take a look at the givens. The only affair yous can conclude from the single given is that the sides of GRAM are parallel (using the definition of a parallelogram). But it doesn't seem like you lot tin get anywhere from there. (For this proof, we're assuming that you oasis't yet learned the properties of the parallelogram, one of which is that opposite sides are congruent.)

• Jump to the end of the proof. What could be the justification for the final statement,

  

  At this point, no justification seems possible, so put on your thinking cap.

  

• Consider cartoon an auxiliary line.

  

• Show the triangles coinciding.

  

  That does information technology.

Hither'south the formal proof:

Statement one :

GRAM is a parallelogram.

Reason for statement 1 : Given.

Statement 2 :

Reason for statement ii : Two points decide a segment.

Statement 3 :

Reason for statement iii : Definition of parallelogram.

Statement 4 :

Reason for statement four : If two parallel lines,

are cut by a transversal (segment RM), then alternate interior angles are coinciding.

Statement v :

Reason for statement 5 : Definition of parallelogram.

Argument half dozen :

Reason for statement half dozen : Aforementioned every bit Reason 4, only this time

are the parallel lines.

Statement seven :

Reason for argument vii : Reflexive Property.

Statement eight :

Reason for statement 8 : ASA (using lines 4, 7, and 6).

Argument nine :

Reason for statement ix : CPCTC.

A skillful style to spot congruent alternating interior angles in a diagram is to look for pairs of so-chosen Z-angles. Look for a Z or backward Z — or a stretched-out Z or backward Z — as shown in the figures above and beneath. The angles in the crooks of the Z are congruent.

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What Is An Auxiliary Line,

Source: https://www.dummies.com/article/academics-the-arts/math/geometry/using-auxiliary-lines-in-proofs-187558/

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