Download A. m - TeacherWeb

Five-Minute Check (over Lesson 5–5)
Then/Now
Theorems: Inequalities in Two Triangles
Example 1: Use the Hinge Theorem and its Converse
Proof: Hinge Theorem
Example 2: Real-World Example: Use the Hinge Theorem
Example 3: Apply Algebra to the Relationships in Triangles
Example 4: Prove Triangle Relationships Using Hinge Theorem
Example 5: Prove Relationships Using Converse of Hinge Theorem
You used inequalities to make comparisons in
one triangle. (Lesson 5–3)
• Apply the Hinge Theorem or its converse to
make comparisons in two triangles.
• Prove triangle relationships using the Hinge
Theorem or its converse.
Use the Hinge Theorem and Its Converse
A. Compare the measures AD and BD.
In ΔACD and ΔBCD, AC  BC, CD  CD, and
mACD > mBCD.
Answer: By the Hinge Theorem, mACD > mBCD,
so AD > DB.
Use the Hinge Theorem and Its Converse
B. Compare the measures mABD and mBDC.
In ΔABD and ΔBCD, AB  CD,
BD  BD, and AD > BC.
Answer: By the Converse of
the Hinge Theorem,
mABD > mBDC.
A. Compare the lengths of
FG and GH.
A. FG > GH
B. FG < GH
C. FG = GH
D. not enough information
B. Compare mJKM and mKML.
A. mJKM > mKML
B. mJKM < mKML
C. mJKM = mKML
D. not enough information
Use the Hinge Theorem
HEALTH Doctors use a straight-leg-raising test to
determine the amount of pain felt in a person’s
back. The patient lies flat on the examining table,
and the doctor raises each leg until the patient
experiences pain in the back area. Nitan can
tolerate the doctor raising his right leg 35° and his
left leg 65° from the table. Which leg can Nitan raise
higher above the table?
Understand Using the angles given in the problem,
you need to determine which leg can be
risen higher above the table.
Use the Hinge Theorem
Plan
Draw a diagram of the situation.
Solve
Since Nitan’s legs are the same length
and his left leg and the table is the same
length in both situations, the Hinge
Theorem says his left leg can be risen
higher, since 65° > 35°.
Use the Hinge Theorem
Answer: Nitan can raise his left leg higher above the
table.
Check
Nitan’s left leg is pointed 30° more towards
the ceiling, so it should be higher that his
right leg.
Meena and Rita are both flying kites in a field near
their houses. Both are using strings that are
10 meters long. Meena’s kite string is at an angle of
75° with the ground. Rita’s kite string is at an angle
of 65° with the ground. If they are both standing at
the same elevation, which kite is higher in the air?
A. Meena’s kite
B. Rita’s kite
Apply Algebra to the Relationships in Triangles
ALGEBRA Find the range of possible values for a.
From the diagram we know that
Apply Algebra to the Relationships in Triangles
Converse of the Hinge Theorem
Substitution
Subtract 15 from each side.
Divide each side by 9.
Recall that the measure of any angle is always
greater than 0.
Subtract 15 from each side.
Divide each side by 9.
Apply Algebra to the Relationships in Triangles
The two inequalities can be written as the compound
inequality
Find the range of possible
values of n.
A. 6 < n < 25
B.
C. n > 6
D. 6 < n < 18.3
Prove Triangle Relationships Using Hinge
Theorem
Write a two-column proof.
Given: JK = HL; JH || KL
mJKH + mHKL < mJHK + mKHL
Prove: JH < KL
Statements
Reasons
1. JK = HL
2. HK = HK
1. Given
2. Reflexive Property
3. mJKH + mHKL <
mJHK + mKHL,
JH || KL
3. Given
Prove Triangle Relationships Using Hinge
Theorem
Statements
Reasons
4. mHKL = mJHK
4. Alternate Interior
angles are
5. mJKH + mJHK <
mJHK + mKHL
5. Substitution
6. mJKH < mKHL
6. Subtraction
Property of
Inequality
7. JH < KL
7. Hinge Theorem
Which reason correctly completes the following
proof?
Given:
Prove: AC > DC
Statements
Reasons
1.
2.
1. Given
2. Reflexive Property
3. mABC =
mABD + mDBC
3. Angle Addition
Postulate
4. mABC > mDBC
4. Definition of
Inequality
5. AC > DC
5.
?
A. Substitution
B. Isosceles Triangle Theorem
C. Hinge Theorem
D. none of the above
Prove Relationships Using Converse of Hinge
Theorem
Given:
Prove:
Answer:
Proof:
Statements
Prove Relationships Using Converse of Hinge
Theorem
Reasons
1.
2.
3.
1. Given
2. Reflexive Property
3. Given
4.
4. Given
5.
6.
5. Substitution
6. SSS Inequality
Which reason correctly completes the following
proof?
Given: X is the midpoint of
ΔMCX is isosceles.
CB > CM
Prove:
Statements
1. X is the midpoint of
MB; ΔMCX is isosceles
2.
3.
4. CB > CM
5. mCXB > mCXM
6.
7.
8. mCXB > mCMX
Reasons
1. Given
2. Definition of midpoint
3. Reflexive Property
4. Given
5. ?
6. Definition of isosceles
triangle
7. Isosceles Triangle
Theorem
8. Substitution
A. Converse of Hinge Theorem
B. Definition of Inequality
C. Substitution
D. none of the above