UNIT 4.6 CONGRUENCE IN RIGHT UNIT 4.6 CONGRUENCE IN RIGHT TRIANGLESTRIANGLES

The Triangle Congruence The Triangle Congruence Postulates &TheoremsPostulates &Theorems

LAHALLHL

FOR RIGHT TRIANGLES ONLY

AASASASASSSS

FOR ALL TRIANGLES

HL (Hypotenuse, Leg)HL (Hypotenuse, Leg)

If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . .

A

C

B

F

E

D

then the 2 triangles are

CONGRUENT!

HA (Hypotenuse, Angle)HA (Hypotenuse, Angle)

If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . .

then the 2 triangles are

CONGRUENT!

F

E

D

A

C

B

LA (Leg, Angle)LA (Leg, Angle)

If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . .

then the 2 triangles are

CONGRUENT!

A

C

B

F

E

D

LL (Leg, Leg)LL (Leg, Leg)

If both pair of legs of two RIGHT triangles are congruent, . . .

then the 2 triangles are

CONGRUENT!

A

C

B

F

E

D

Answers will be after Example 7Answers will be after Example 7

Example 1Example 1

Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

F

E

D

A

C

B

Example 2Example 2

Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

A

C

B

F

E

D

Example 3Example 3

Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

D

A

C

B

Why are the two triangles congruent?

What are the corresponding vertices?

A

B

C

D

EF

Why are the two triangles congruent?

What are the corresponding angles?

A

B

C

D

Given:

B C

DA

CDABADBC

Are the triangles congruent?

Why?

Given: QR PS

RSSR

Are the Triangles Congruent?

QSR PRS = 90°

Q

RS

P

T

mQSR = mPRS = 90°

PS QR

Why?

HL – Pair of sides including the Hypotenuse and one Leg

HA – Pair of hypotenuses and one acute angle

LL – Both pair of legs

LA – One pair of legs and one pair of acute angles

Example 1 – SAS

Example 2 – SSA not congruent

Example 3 – SSS

Example 4 – ASA, A = F, D = B, E = C

Example 5 – SSS, A C, ADB CDB, ABD CBD

Example 6 – SSS

Example 7 - HL

(a)

(b)

(c)

(d)

Let's take a closer look at all of the diagrams to determine which of them show a pair of congruent triangles by the HL Theorem.

In (a), it appears as though we might be able to use the HL Theorem. However, upon careful examination, we note that the angles at vertices A and D are not right angles. Because a square is not used to indicate that the angles are right angles, we cannot use the HL Theorem. Recall that the only type of triangle for which this theorem holds is a right triangle, so we cannot APPLY it in this situation.

Figure (b) does show two triangles that are congruent, but not by the HL Theorem. We are given that segment FG is congruent to segment HG and that segment EG is congruent to segment IG. We also have right angles that form at G. Because we have two sides and the included angle of one triangle congruent to the corresponding parts of the other triangle, we know that the triangles are congruent by the SAS Postulate. However, we are not given any information regarding the hypotenuses of ?EGF and ?IHG, so we cannot APPLY the HL Theorem to prove that the triangles are congruent.

Now, let's look at (c). Notice that we have two right angles in the figure: ?JLK and ?JLM. Also, we have been given the fact that segment JK is congruent to segment JM. These segments are actually the hypotenuses of the triangles because they lie on the side opposite of the right angle. Moreover, the two triangles in the figure share segment JL. By transitivity, we know that the segment is congruent to itself. Thus, we can apply the HL Theorem to prove that ?JKL??JML, since we know that the triangles are right triangles, their hypotenuses are congruent, and they have a pair of legs that are congruent.

Finally, we have the figure for (d). We have been given that there are right angles at vertices O and Q. We can also imply that ?NPO and ?RPQ are congruent because they are vertical angles. This will not help us try to prove that the triangles are congruent by the HL Theorem, however. What we are looking for is information about the legs or hypotenuses of the triangles. Since we cannot deduce any more facts from the diagram that will help us, we cannot apply the HL Theorem in this situation.

Therefore, we can only apply the HL Theorem in (c) to show that the triangles are congruent.

We want to examine the information that has been given to us in the problem. We know that segment RV is perpendicular to segment SK, and that segments SR and KR are congruent. Let's try to deduce more information from the given statements that may help us prove that

Since we were given that RV and SK are perpendicular, we know that there exist right angles at RVS and RVK. This fact is a key component of our proof because we know that triangle RSV and triangle RKV are right triangles. Thus, we can try to use the HL Theorem to prove that they are congruent to each other.

We have already been given that the hypotenuses are congruent, so all that is left to show is that a pair of legs of the triangles is congruent. Since they both share segment RV, we can use the Transitive Property to say that the segment is congruent to itself.

In all, we have found right angles, congruent hypotenuses, and congruent legs between the triangles, so we APPLY the HL Theorem to say that . Our new diagram and the two-column geometric proof are shown below.

Leg-Leg (LL) Theorem

If the legs of one right triangle are congruent to the legs of another right triangle, then the two right triangles are congruent.

This statement is the same as the SAS Postulate we've learned about because it involves two sides of triangles, as well as the included angle (which is the right angle).

Leg-Acute (LA) Angle TheoremIf a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the two right triangles are congruent.

This statement is equivalent to the ASA Postulate we've learned about because it involves right angles (which are congruent), a pair of sides with the same measure, and congruent acute angles.

Hypotenuse-Acute (HA) Angle TheoremIf the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.

This statement is the same as the AAS Postulate because it includes right angles (which are congruent), two congruent acute angles, and a pair of congruent hypotenuses.

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