Angle Theorems

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Angle Theorems

Alternate Exterior Angles Theorem

If a transversal intersects two parallel lines, then the alternate exterior angles are congruent.

Converse also true: If a transversal intersects two lines and the alternate exterior angles are congruent, then the lines are

parallel.

The alternate exterior angles have the same degree measures because the lines are parallel to each other.

Alternate Interior Angles Theorem

If a transversal intersects two parallel lines, then the alternate interior angles are congruent.

Converse also true: If a transversal intersects two lines and the alternate interior angles are congruent, then the lines areparallel.

The alternate interior angles have the same degree measures because the lines are parallel to each other.

Congruent Complements Theorem

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If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent.

Congruent Supplements Theorem

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent.

Right Angles Theorem

All right angles are congruent.

Same-Side Interior Angles Theorem

If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.

Converse also true: If a transversal intersects two lines and the interior angles on the same side of the transversal are

supplementary, then the lines are parallel.

The sum of the degree measures of the same-side interior

angles is 180.

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Vertical Angles Theorem

If two angles are vertical angles, then they have equal measures.

The vertical angles have equal degree measures. There are two pairs of vertical angles.

Exercises

(1) Given: mDGH = 131

Find: mGHK

First, we must rely on the information we are given to begin our proof. In this exercise, we note that the measure ofDGHis131.

From the illustration provided, we also see that linesDJandEKare parallel to each other. Therefore, we can utilize some of the

angle theorems above in order to find the measure ofGHK.

We realize that there exists a relationship betweenDGHandEHI: they are corresponding angles. Thus, we can utilize the

Corresponding Angles Postulate to determine thatDGHEHI.

Directly opposite fromEHIisGHK. Since they are vertical angles, we can use the Vertical Angles Theorem, to see that

EHIGHK.

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Now, by transitivity, we have thatDGHGHK.

Congruent angles have equal degree measures, so the measure ofDGHis equal to the measure ofGHK.

Finally, we use substitution to conclude that the measure ofGHKis 131. This argument is organized in two-column proof

form below.

(2) Given: m1 = m3

Prove: mPTR = mSTQ

We begin our proof with the fact that the measures of1 and3 are equal.

In our second step, we use the Reflexive Property to show that2 is equal to itself.

Though trivial, the previous step was necessary because it set us up to use the Addition Property of Equality by showing that

adding the measure of2 to two equal angles preserves equality.

Then, by the Angle Addition Postulate we see thatPTR is the sum of1 and2, whereasSTQ is the sum of3 and2.

Ultimately, through substitution, it is clear that the measures ofPTR andSTQ are equal. The two-column proof for this

exercise is shown below.

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(3) Given: mDCJ = 71, mGFJ = 46

Prove: mAJH = 117

We are given the measure ofDCJandGFJto begin the exercise. Also, notice that the three lines that run horizontally in the

illustration are parallel to each other. The diagram also shows us that the final steps of our proof may require us to add up the two

angles that composeAJH.

We find that there exists a relationship betweenDCJandAJI: they are alternate interior angles. Thus, we can use the

Alternate Interior Angles Theorem to claim that they are congruent to each other.

By the definition ofcongruence, their angles have the same measures, so they are equal.

Now, we substitute the measure ofDCJwith 71 since we were given that quantity. This tells us thatAJIis also 71.

SinceGFJandHJIare also alternate interior angles, we claim congruence between them by the Alternate Interior Angles

Theorem.

The definition of congruent angles once again proves that the angles have equal measures. Since we knew the measure ofGFJ

we just substitute to show that 46is the degree measure ofHJI.

As predicted above, we can use the Angle Addition Postulate to get the sum ofAJIandHJIsince they composeAJH.

Ultimately, we see that the sum of these two angles gives us 117. The two-column proof for this exercise is shown below.

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(4) Given: m1 = 4x + 9, m2 = 7(x + 4)

Find: m3

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In this exercise, we are not given specific degree measures for the angles shown. Rather, we must use some algebrato help us

determine the measure of3. As always, we begin with the information given in the problem. In this case, we are given

equations for the measures of1 and2. Also, we note that there exists two pairs of parallel lines in the diagram.

By the Same-Side Interior Angles Theorem, we know that that sum of1 and2 is 180 because they are supplementary.

Aftersubstituting these angles by the measures given to us and simplifying, we have 11x + 37 = 180. In order to solve forx, we

first subtract both sides of the equation by 37, and then divide both sides by 11.

Once we have determined that the value ofxis 13, we plug it back in to the equation for the measure of2 with the intention ofeventually using the Corresponding Angles Postulate. Plugging 13 in forxgives us a measure of119 for2.

Finally, we conclude that3 must have this degree measure as well since2 and3 are congruent. The two-column proof

that shows this argument is shown below.

Congruent triangles

Congruent triangles have the same size and the same shape. When triangles are congruent, all

corresponding sides and corresponding angles are also congruent or equal

For examples, the two triangles below are congruent

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Corresponding angles are angles in the same position

Corresponding sides are sides that are in the same position

The two triangles above have a side with 3 markings. These sides are at the same position and thus

are corresponding

Congruent sides are sides that have equal measures

Congruent angles are angles that have equal sides and equal measures

In the triangle above, if we pull out the side with one and three markings and the included angle, we

get the following:

The above 45 degrees angle is a good example of congruent angles because the sides are equal and

the angles are equal

Included side: A side between two angles

Included angle: An angle between two sides

There are three postulates and two theorems that are used to identify if two triangles are congruent

With these postulates and theorems, you don't have to check if all corresponding angles and all sides

are congruent

If the triangles meet the condition of the postulate or theorem, then, you have congruent triangles.

They are the SSS postulate, SAS postulate, ASA postulate, AAS theorem, and Hypotenuse-Leg theorem

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SSS postulate:

If three sides of a triangle are congruent to three sides of a second triangle, then the two triangles are

congruent

Example:

ASA postulate:

If two angles and the side between these two angles (included side) of one triangle are congruent to

the corresponding angles and the included side of a second triangle, then the two triangles are

congruent

Example:

SAS postulate:

If two sides and the angle between these two sides (included angle) of one triangle are congruent to

the corresponding two sides and the included angle of a second triangle, then the two triangles are

congruent

Example:

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AAS theorem:

If two angles and a side not between these two angles of one triangle are congruent to two angles and

the corresponding side not between these two angles of a second triangle, then the two triangles are

congruent

Example:

Hypotenuse-Leg theorem:

If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and corresponding leg of

a second right triangle, then the two triangles are congruent

Example:

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Congruence Of Triangles

Copyright 1997, Jim Loy

In Plane Geometry, two objects are congruent if all of their corresponding parts are congruent. In the first

diagram, the two triangles have two sides which are congruent, and the angle between these sides is also

congruent. Euclid proved that they are congruent triangles (Theorem I.4, called "Side-Angle-Side" ofSAS). But, he was not happy with the proof, as he avoided similar proofs in other situations. The way he proved it, is to move

one triangle until it is superimposed on the other triangle. Such a trick (superposition: placing one triangle on top of another, to

see if they are congruent), is not considered legal, now. It involves some complicated assumptions. So, now this (SAS) is given

as an assumption (postulate). It is the Side-Angle-Side Postulate.

Euclid proved his Side-Side-Side (SSS) Theorem (I.8) and his Angle-Side-Angle (ASA, diagram at the

right) Theorem (I.26) in a similar way. In SSS, if a triangle has all three sides conguent to the

corresponding sides of a second triangle, then they are congruent. In ASA, if a triangle has two angles

and the side in between the angles congruent to the corresponding parts of a second triangle, then they are congruent. These too,

are now assumed as postulates.

It is easy to construct a triangle given the lengths of the three sides (diagram on the left), or two sides and

the included angle, or two angles and the included side. Given the ease of these constructions (especiallySSS), it seems strange that the corresponding congruence "theorems" cannot be proven, except by slightly

shady means (superposition).

All three of these congruence postulates are equivalent. You can assume any one of them, and prove the other two from there.

SeeCongruence Of Triangles, Part IIfor a proof of this.

Euclid also proved an Angle-Angle-Side (AAS) as a corollary (a minor theorem derived directly from the parent theorem) to

Theorem I.26. The situation SSA (or ASS) is not a theorem or postulate, as it is often ambiguous, there are often two differentnon-congruent triangles with two congruent sides and a congruent angle at the end of one of the sides.

Note: In the above, I used the term "congruent" instead of "equal" when comparing sides and angles. Numbers are equal. Linesegments (sides) and angles are congruent. Calling them "equal" is a sloppy way of saying that their measures (lengths or sizes)

are equal. Nevertheless, I do use this more informal terminology in some of my articles.

Addendum:

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In answer to email about proving these congruence theorems/assumptions, I wrote this:

The issue is this (using SAS as an example). If you take your angle and two segments, and move it to somewhere else, does the

triangle that you get, by drawing the third side of your moved object, change shape (is it congruent to the original triangle)? We

can say that it doesn't change shape (that would seem to be right), but we can't prove it. Moving things is a complicated process;

do we need postulates that say what changes and what does not change when we move a geometric figure? It is simpler to makecongruence assumptions. Then we know that a triangle over here is congruent to a triangle over there. And motion never enters

into it.

Congruence Of Triangles, Part II

Copyright 1999, Jim Loy

This is a continuation of the articleCongruence Of Triangles. Read that article first. In that part, I stated, "All three of these

congruence postulates are equivalent. You can assume any one of them, and prove the other two from there." Someone namedJason phoned me to ask how to do that. So, here is how I do it. It can be done with three proofs. But, the way I do it is with four

proofs. See that article (Congruence Of Triangles), for the definitions of SAS, ASA, and SSS, as well as "congruence."

1. Assume SAS and prove ASA.

We have two triangles, as shown in the diagram. Angle A=angle EDF, angle B=angleE, and AB=DE (Note that I am using=to mean "congruent"). Prove that the two

triangles are congruent.

1. If BC=EF, then the two triangles are congruent, by SAS. So, assume that BCnot=EF.

2. There is a point G, on EF (perhaps extended), on the same side of the line DE, so that EG=BC.

3. Triangle ABC=triangle DEG [by SAS].

4. Angle EDF=angle EDG [definition of congruent triangles].

5. So DF and DG are the same line.

6. DF and EF intersect at both F and G. So, F & G are the same point [two distinct lines intersect in only one point].

7. So EF=EG=BC, which contradicts our assumption that BC not=EF.

8. So BC=EF, and the triangles are congruent by SAS, as mentioned in step #1.

2. Assume ASA and prove SAS.

We have two triangles, as shown in the diagram. Angle A=angle D, AB=DE, and

AC=DF. Prove that the two triangles are congruent.

1. If angle B=angle DEF, then the two triangles are congruent, by ASA. So,

assume that angle B not=angle DEF.

2. Draw a line EG, so that angle DEG=angle B (with G on the same side of DE

as F).

3. EG intersects line DEF (perhaps extended) at some point H.

4. Triangle ABC=triangle DEG [ASA].5. AC=DH [definition of congruent triangles].

6. DH=DF [both=AC]

7. H and F are the saem point. And EH and EF are the same line. And angle DEG and angle DEF are the same angle.

8. That contradicts our assumption that angle B not=angle DEF.

9. So angle B=angle DEF, and the 2 triangles are congruent by ASA, as mentioned in step #1.

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3. Assume SSS and prove SAS.

We have two triangles, as shown in the diagram. Angle A=angle EDF, AB=DE, and

AC=DF. Prove that the two triangles are congruent.

1. If BC=EF, then the two triangles are congruent, by SSS. So, assume that BC

not=EF.2. Find the point G (on the same side of line DE as F is) so that DF=DG and

BC=EG. We do this by drawing arcs of those radii and finding where they intersect, if they do intersect.

3. Triangle ABC=triangle DEG [SSS].4. Angle A=angle EDG [definition of congruence of triangles].

5. Angle EDG=angle EDF [both=angle A].

6. So, they are the same angle. And F and G are the same point. And EF=BC.

7. This contradicts our assumption that BC not=EF.

8. So BC=EF, and the triangles are congruent by SSS, as mentioned in step #1.

4. Assume SAS and prove SSS.

We have two triangles, as shown in the diagram. AB=DE, AC=DF, and BC=EF. Prove

that the two triangles are congruent.

1. If angle A=angle EDF, then the two triangles are congruent, by SAS. So,

assume that angle A not=angle EDF.

2. Draw a line DG, so that angle BAC=angle EDG.

3. Find the point H (on the same side of DE that F is) so that DH=AC.4. F & H are not the same point [DF and DG can only intersect at D].

5. Draw EH and FH.

6. Triangle ABC=triangle DEH [SAS].

7. EH=BC [definition of congruent triangles].

8. EH=EF [both=BC].

9. DH=DF [both=AC].

10. Both triangle DFH and triangle EFH are isosceles [definition].

11. The perpendicular bisector of FH must pass through both D and E [a Euclid theorem, which I will track down later].

12. So DE is the perpendicular bisector of FH [definition].

13. But that is impossible, since H and F are on the same side of line DE [step #3].14. This contradicts our assumption that angle A not=angle EDF.

15. So angle A=angle EDF, and the triangles are congruent by SAS, as mentioned in step #1.

Note: Above I showed that ASA implies SAS and vice versa. I also showed that SAS

implies SSS and vice versa. So, you can assume any of the three, and then go by a

chain of above proofs to prove any of the other two. See the left half of this diagram.

This kind of chain can be done in three proofs, rather than four. See the right half of

the diagram, to see how that could be done. I had difficulty coming up with any proof

involving ASA and SSS. So, I did it with four proofs.

Notice that all four of the above proofs are fairly similar. That is further evidence that these three theorems/postulates are

equivalent. And none of them is much more obvious than any of the others.