Chapter 8 Using Congruence Theorems

Chapter 8 Using Congruence Theorems
8.1 Right Triangle Congruence Theorems
Bell Work:
Given: GE||KV, GE KV
Prove: ΔGVK ΔKEG
Statements
1. GE||KV
2. <EGK <VKG
3. GE KV
4. GK GK
5. ΔGVK ΔKEG
G
V
E
K
Reasons
1. Given
2. Alternate Interior Angles Theorem
3. Given
4. Reflexive Property
5. SAS
Right triangles are special. We already know a lot of information about them.
The legs of a right triangle are the sides that form the right angle.
The hypotenuse is the side opposite the right angle.
The Pythagorean Theorem states that a2 + b2 = c2, so it you know the
measures of any two sides of the triangle then you can always
find the measure of the third side.
Read the paragraph below Problem 1 Question 1 on page 600.
Since all right angles are congruent, proving two right triangles congruent should only involve having
two additional pieces of information.
We already have a set of angles congruent therefore proving two triangles congruent would be easiest
done using either SAS, ASA, or AAS.
Instruction:
If you are given that the two legs of one right triangle are congruent to two legs of another right
triangle is this enough information to be able to state that the triangles are congruent?
If so, which of our congruence theorems from the last chapter applies? SAS
We call this alternate version of SAS especially for right triangles the LL Right Triangle Congruence
Theorem. This theorem is stated at the top of page 605.
Leg-Leg (LL) Congruence Theorem
If two legs of one right triangle are congruent to two legs of another right triangle, then the
triangles are congruent .
Example: Complete question 6 on page 614 as a class.
Statements
Reasons
1. GU | bisector DB
1. Given
2. <GUD and <GUB are right angles
2. Definition of perpendicular
3. ΔGUD and ΔGUB are right triangles
3. Definition of right triangle
4. BU DU
4. Definition of segment bisector
5. GU GU
5. Reflexive Property
6. ΔGUD ΔGUB
6. LL
If you are given that one leg and one acute angle of one right triangle are congruent to the
corresponding leg and acute angle of another triangle, is this enough information to be able to
state that the triangles are congruent?
If so, which of our congruence theorems from the last chapter applies? ASA, AAS
We call this alternate version of ASA or AAS especially for right triangles the LA Right Triangle
Congruence Theorem. This theorem is stated at the top of page 609.
Leg-Angle (LA) Congruence Theorem
If a leg and an acute angle of one right triangle are congruent to a leg and an acute angle of
another right triangle, then the triangles are congruent.
Example: Complete alternate question 1 on page 611 as a class.
(Change the one given of P is the midpoint of CW to P is the midpoint of SD)
Statements
Reasons
1. CS | SD. WD | SD
1. Given
2. <CSP and <WDP are right angles
2. Definition of perpendicular
3. ΔCSP and ΔWDP are right triangles
3. Definition of right triangle
4. P is the midpoint of SD
4. Given
5. SP DP
5. Definition of midpoint
6. <CPS <WPD
6. Vertical Angle Theorem
7. ΔCSP ΔWDP
7. LA
If you are given that the hypotenuse and one acute angle of one right triangle are congruent to the
hypotenuse and corresponding acute angle of another triangle is this enough information to be able to
state that the triangles are congruent?
If so, which of our congruence theorems from the last chapter applies?
AAS
We call this alternate version of AAS especially for right triangle the HA Right Triangle Congruence
Theorem. This theorem is found at the top of page 607.
Hypotenuse-Angle (HA) Congruence Theorem
If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and
acute angle of another right triangle, then the triangles are congruent.
Example: Complete question 4 on page 612 as a class.
Statements
Reasons
1. ST | SR, AT | AR
1. Given
2. <TSR and <TAR are right angles
2. Definition of right angle
3. ΔTSR and ΔTAR are right triangles
3. Definition of right triangle.
4. <STR <ATR
4. Given
5. TR TR
5. Reflexive Property
6. ΔTSR ΔTAR
6. HA
Let’s go back to non-right triangles for a moment and investigate the SSA pattern.
Watch the duplication of triangle ABC using sides BC, AC, and <B and
triangle XYZ using sides YZ, XZ and <Y.
X’
X
A
Y
B
C
Z
X’
Y’
Y’
Z’
Z’
Notice that there is only 1 possible triangle resulting from ABC but two possible triangles resulting from
XYZ. This is why we say that SSA does not create congruent triangles.
But triangle ABC and triangle A’B’C’ are congruent so there must be something special about it.
The given angle is across from the largest side. When this happens SSA is a legitimate method of
proving triangles congruent.
This is useful as far as right triangles are concerned because in a right triangle the largest side is always
across from the given angle, the right angle. Therefore the SSA pattern will work as a method
of proving right triangles congruent. We call this the HL Right Triangle Theorem
Hypotenuse-Leg (HL) Congruence Theorem
If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and leg of
another right triangle, then the triangles are congruent.
Example: Complete question 3 on page 612 as a class.
Statements
Reasons
1. JA | MY
1. Given
2. <YMJ and <YMA are right angles
2. Definition of right angle
3. ΔYMJ and ΔYMA are right triangles
3. Definition of right triangle
4. JY AY
4. Given
5. YM YM
5. Reflexive Property
6. ΔYMJ ΔYMA
6. HL

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