concept. example 1 use the hinge theorem and its converse a. compare the measures ad and bd. answer:...
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![Page 2: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/2.jpg)
Use the Hinge Theorem and Its Converse
A. Compare the measures AD and BD.
Answer: By the Hinge Theorem, mACD > mBCD, so AD > DB.
In ΔACD and ΔBCD, AC BC, CD CD, and ACD > BCD.
![Page 3: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/3.jpg)
Use the Hinge Theorem and Its Converse
B. Compare the measures ABD and BDC.
Answer: By the Converse of the Hinge Theorem, ABD > BDC.
In ΔABD and ΔBCD, AB CD, BD BD, and AD > BC.
![Page 4: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/4.jpg)
A. A
B. B
C. C
D. D
A. FG > GH
B. FG < GH
C. FG GH
D. not enough information
A. Compare the lengths of FG and GH.
![Page 5: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/5.jpg)
A. A
B. B
C. C
D. D
A. mJKM > mKML
B. mJKM < mKML
C. mJKM = mKML
D. not enough information
B. Compare JKM and KML.
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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.
![Page 9: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/9.jpg)
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°.
![Page 10: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/10.jpg)
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.
![Page 11: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/11.jpg)
A. A
B. BA. Meena’s kite
B. Rita’s kite
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?
![Page 12: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/12.jpg)
Apply Algebra to the Relationships in Triangles
ALGEBRA Find the range of possible values for a.
From the diagram we know that
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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.
![Page 14: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/14.jpg)
Apply Algebra to the Relationships in Triangles
The two inequalities can be written as the compound
inequality
![Page 15: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/15.jpg)
A. A
B. B
C. C
D. D
Find the range of possible values of n.
A. 6 < n < 12
B.
C. n > 6
D. 6 < n < 18.3
__17
![Page 16: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/16.jpg)
Which reason correctly completes the following proof?Given:Prove: AC > DC
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Statements Reasons
2. 2. Reflexive Property
3. mABC > mABD + mDBC
3. Given
1. 1. Given
5. AC > DC 5. ?
4. mABC > mDBC 4. Definition of Inequality
![Page 18: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/18.jpg)
A. A
B. B
C. C
D. D
A. Substitution
B. Isosceles Triangle Theorem
C. Hinge Theorem
D. none of the above
![Page 19: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/19.jpg)
Prove Relationships Using Converse of Hinge Theorem
Given:
Prove:
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Prove Relationships Using Converse of Hinge TheoremProof:
Statements Reasons
1. 1. Given2. 2. Reflexive Property3. 3. Given
4. 4. Given
5. 5. Substitution
6. 6. Converse of Hinge Theorem
![Page 21: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/21.jpg)
Which reason correctly completes the following proof?Given: X is the midpoint of
ΔMCX is isosceles.CB > CM
Prove:
![Page 22: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/22.jpg)
Statements Reasons
4. CB > CM 4. Given5. mCXB > mCXM 5. ?
1. X is the midpoint of MB; ΔMCX is isosceles
1. Given
2. 2. Definition of midpoint
3. 3. Reflexive Property
6. 6. Definition of isosceles triangle
7. 7. Isosceles Triangle Theorem
8. mCXB > mCMX 8. Substitution
![Page 23: Concept. Example 1 Use the Hinge Theorem and Its Converse A. Compare the measures AD and BD. Answer: By the Hinge Theorem, m ACD > m BCD, so AD >](https://reader036.vdocuments.site/reader036/viewer/2022081513/56649cf65503460f949c503a/html5/thumbnails/23.jpg)
A. A
B. B
C. C
D. D
A. Converse of Hinge Theorem
B. Definition of Inequality
C. Substitution
D. none of the above