The document discusses the Hinge Theorem and its converse for comparing sides and angles of triangles. It provides examples of applying the Hinge Theorem and its converse to determine if one side or angle is greater than the other. It also gives an example problem of proving that one side is less than the other using the Hinge Theorem and properties of alternate interior angles for parallel lines cut by a transversal. The document concludes with assigning practice problems related to applying the Hinge Theorem and its converse.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Totally R*-Continuous and Totally R*-Irresolute Functionsinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Totally R*-Continuous and Totally R*-Irresolute Functionsinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
How to Split Bills in the Odoo 17 POS ModuleCeline George
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This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
2. Essential Questions
How do you apply the Hinge Theorem or its converse to
make comparisons in two triangles?
How do you prove triangle relationships using the Hinge
Theorem or its converse?
Tuesday, April 15, 14
4. Hinge Theorem
If two sides of a triangle are congruent to two sides of
another triangle, and the included angle of the first is larger
than the included angle of the second triangle, then the third
side of the first triangle is longer than the third side of the
second triangle.
Tuesday, April 15, 14
6. Converse of the Hinge Theorem
If two sides of a triangle are congruent to two sides of
another triangle, and the third side of the first triangle is
longer than the third side of the second triangle, then the
included angle measure of the first is larger than the
included angle measure of the second triangle.
Tuesday, April 15, 14
7. Example 1
Compare the given measures.
a. AD and BD
m∠ACD = 70°, m∠BCD = 68°
b. m∠ABD, m∠CDB
Tuesday, April 15, 14
8. Example 1
Compare the given measures.
a. AD and BD
m∠ACD = 70°, m∠BCD = 68°
AD > BD by the
Hinge Theorem
b. m∠ABD, m∠CDB
Tuesday, April 15, 14
9. Example 1
Compare the given measures.
a. AD and BD
m∠ACD = 70°, m∠BCD = 68°
AD > BD by the
Hinge Theorem
b. m∠ABD, m∠CDB
By the Converse of the Hinge
Theorem,
m∠ABD > m∠CDB
Tuesday, April 15, 14
10. Example 2
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. Matt Mitarnowski can tolerate the doctor
raising his right leg 35° and his left leg 65° from the table. Which
leg can Matt raise higher above the table? How do you know?
Tuesday, April 15, 14
11. Example 2
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. Matt Mitarnowski can tolerate the doctor
raising his right leg 35° and his left leg 65° from the table. Which
leg can Matt raise higher above the table? How do you know?
As the angle between the table and leg is greater, Matt
can lift his left leg higher.
Tuesday, April 15, 14
12. Example 3
Find the range of possible values for a.
Tuesday, April 15, 14
13. Example 3
Find the range of possible values for a.
9a +15<141
Tuesday, April 15, 14
14. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
Tuesday, April 15, 14
15. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
Tuesday, April 15, 14
16. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
Tuesday, April 15, 14
17. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
a <14
Tuesday, April 15, 14
18. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
a <14
9a +15>0
Tuesday, April 15, 14
19. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
a <14
9a +15>0
−15 −15
Tuesday, April 15, 14
20. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
a <14
9a +15>0
−15 −15
9a > −15
Tuesday, April 15, 14
21. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
a <14
9a +15>0
−15 −15
9a > −15
9 9
Tuesday, April 15, 14
22. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
a <14
9a +15>0
−15 −15
9a > −15
9 9
a > −
5
3
Tuesday, April 15, 14
23. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
a <14
9a +15>0
−15 −15
9a > −15
9 9
a > −
5
3
−
5
3
<a <14
Tuesday, April 15, 14
24. Example 3
Find the range of possible values for a.
9a +15<141
−15 −15
9a <126
9 9
a <14
9a +15>0
−15 −15
9a > −15
9 9
a > −
5
3
−
5
3
<a <14
If we are looking at the smaller angle, it
must be larger than 0°. If we are looking at
the larger angle, it must be less than 180°.
Tuesday, April 15, 14
25. Example 4
Prove the following.
Given: JK = HL; JH || KL;
m∠JKH + m∠HKL < m∠JHK + m∠KHL
Prove: JH < KL
Tuesday, April 15, 14
26. Example 4
Prove the following.
1. JK = HL; JH || KL;
m∠JKH + m∠HKL < m∠JHK + m∠KHL
Given: JK = HL; JH || KL;
m∠JKH + m∠HKL < m∠JHK + m∠KHL
Prove: JH < KL
Tuesday, April 15, 14
27. Example 4
Prove the following.
1. Given
1. JK = HL; JH || KL;
m∠JKH + m∠HKL < m∠JHK + m∠KHL
Given: JK = HL; JH || KL;
m∠JKH + m∠HKL < m∠JHK + m∠KHL
Prove: JH < KL
Tuesday, April 15, 14
28. Example 4
Prove the following.
1. Given
2. HK ≅ HK
1. JK = HL; JH || KL;
m∠JKH + m∠HKL < m∠JHK + m∠KHL
Given: JK = HL; JH || KL;
m∠JKH + m∠HKL < m∠JHK + m∠KHL
Prove: JH < KL
Tuesday, April 15, 14
29. Example 4
Prove the following.
1. Given
2. Reflexive2. HK ≅ HK
1. JK = HL; JH || KL;
m∠JKH + m∠HKL < m∠JHK + m∠KHL
Given: JK = HL; JH || KL;
m∠JKH + m∠HKL < m∠JHK + m∠KHL
Prove: JH < KL
Tuesday, April 15, 14
42. Problem Set
p. 371 #1-29 odd, 39, 47, 51
"Make visible what, without you, might perhaps never have been
seen." - Robert Bresson
Tuesday, April 15, 14
