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Question based on the theorem:
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Q. In an isosceles triangle ABC with AB = AC, D and E are points on BC such that BE = CD. Show that AD = AE.
Congruent triangles. Proving congruent triangles.
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
Q. In an isosceles triangle ABC with AB = AC, D and E are points on BC such t...RameshSiyol
Question based on the theorem:
Angles opposite to equal sides of an isosceles triangle are equal.
Q. In an isosceles triangle ABC with AB = AC, D and E are points on BC such that BE = CD. Show that AD = AE.
Congruent triangles. Proving congruent triangles.
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
NCERT Solutions of Class 9 chapter 7-Triangles are created here for helping the students of class 9 in helping their preparations for CBSE board exams. All NCERT Solutions of Class 9 of chapter 7-Triangles are solved by an expert of maths in such a way that every student can understand easily without the help of anybody.
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This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
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The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
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4. POSTULATES & THEOREMS
Ruler Postulate: The points on any line or segment can be put
into one-to-one correspondence with real numbers
Segment Addition Postulate:
5. POSTULATES & THEOREMS
Ruler Postulate: The points on any line or segment can be put
into one-to-one correspondence with real numbers
You can measure the distance between two points
Segment Addition Postulate:
6. POSTULATES & THEOREMS
Ruler Postulate: The points on any line or segment can be put
into one-to-one correspondence with real numbers
You can measure the distance between two points
Segment Addition Postulate: If A, B, and C are collinear, then B is
between A and C if and only if (IFF) AB + BC = AC
7. THEOREM 2.2 - PROPERTIES OF
SEGMENT CONGRUENCE
Reflexive Property of Congruence:
Symmetric Property of Congruence:
Transitive Property of Congruence:
8. THEOREM 2.2 - PROPERTIES OF
SEGMENT CONGRUENCE
Reflexive Property of Congruence:
Symmetric Property of Congruence:
Transitive Property of Congruence:
9. THEOREM 2.2 - PROPERTIES OF
SEGMENT CONGRUENCE
Reflexive Property of Congruence:
Symmetric Property of Congruence: If AB ≅ CD, then CD ≅ AB
Transitive Property of Congruence:
10. THEOREM 2.2 - PROPERTIES OF
SEGMENT CONGRUENCE
Reflexive Property of Congruence:
Symmetric Property of Congruence: If AB ≅ CD, then CD ≅ AB
Transitive Property of Congruence: If AB ≅ CD and CD ≅ EF, then
AB ≅ EF
15. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
16. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC
17. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC Reflexive property of equality
18. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC Reflexive property of equality
4. AB + BC = AC
19. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC Reflexive property of equality
4. AB + BC = AC Segment Addition
20. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC Reflexive property of equality
4. AB + BC = AC Segment Addition
5. CD + BC = AC
21. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC Reflexive property of equality
4. AB + BC = AC Segment Addition
5. CD + BC = AC Substitution prop. of equality
25. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
7. AC = BD
6. CD + BC = BD Segment Addition
26. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
7. AC = BD Substitution property of equality
6. CD + BC = BD Segment Addition
27. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
7. AC = BD Substitution property of equality
6. CD + BC = BD Segment Addition
8. AC ≅ BD
28. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
7. AC = BD Substitution property of equality
6. CD + BC = BD Segment Addition
8. AC ≅ BD Def. of ≅ segments
32. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF
33. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF Def. of ≅ segments
34. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF Def. of ≅ segments
3. AB = EF
35. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF Def. of ≅ segments
3. AB = EF
Transitive property
of Equality
36. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF Def. of ≅ segments
3. AB = EF
Transitive property
of Equality
4. AB ≅ EF
37. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF Def. of ≅ segments
3. AB = EF
Transitive property
of Equality
4. AB ≅ EF Def. of ≅ segments
38. EXAMPLE 2
Matt Mitarnowski is designing a badge for his club. The length of
the top edge of the badge is equal to the length of the left edge of the
badge. The top edge of the badge is congruent to the right edge of the
badge, and the right edge of the badge is congruent to the bottom edge
of the badge. Prove that the bottom edge of the badge is congruent to
the left edge of the badge.
39. EXAMPLE 2
Matt Mitarnowski is designing a badge for his club. The length of
the top edge of the badge is equal to the length of the left edge of the
badge. The top edge of the badge is congruent to the right edge of the
badge, and the right edge of the badge is congruent to the bottom edge
of the badge. Prove that the bottom edge of the badge is congruent to
the left edge of the badge.
A B
CD
40. EXAMPLE 2
Matt Mitarnowski is designing a badge for his club. The length of
the top edge of the badge is equal to the length of the left edge of the
badge. The top edge of the badge is congruent to the right edge of the
badge, and the right edge of the badge is congruent to the bottom edge
of the badge. Prove that the bottom edge of the badge is congruent to
the left edge of the badge.
A B
CD
Given: AB = AD, AB ≅ BC, and BC ≅ CD
41. EXAMPLE 2
Matt Mitarnowski is designing a badge for his club. The length of
the top edge of the badge is equal to the length of the left edge of the
badge. The top edge of the badge is congruent to the right edge of the
badge, and the right edge of the badge is congruent to the bottom edge
of the badge. Prove that the bottom edge of the badge is congruent to
the left edge of the badge.
A B
CD
Prove: AD ≅ CD
Given: AB = AD, AB ≅ BC, and BC ≅ CD
45. EXAMPLE 2
1. AB = AD, AB ≅ BC, and
BC ≅ CD
Given
2. AB ≅ AD
A B
CD
46. EXAMPLE 2
1. AB = AD, AB ≅ BC, and
BC ≅ CD
Given
2. AB ≅ AD Def. of ≅ segments
A B
CD
47. EXAMPLE 2
1. AB = AD, AB ≅ BC, and
BC ≅ CD
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD
A B
CD
48. EXAMPLE 2
1. AB = AD, AB ≅ BC, and
BC ≅ CD
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD Transitive property
A B
CD
49. EXAMPLE 2
1. AB = AD, AB ≅ BC, and
BC ≅ CD
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD Transitive property
4. AD ≅ AB
A B
CD
50. EXAMPLE 2
1. AB = AD, AB ≅ BC, and
BC ≅ CD
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD Transitive property
4. AD ≅ AB Symmetric property
A B
CD
51. EXAMPLE 2
1. AB = AD, AB ≅ BC, and
BC ≅ CD
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD Transitive property
4. AD ≅ AB Symmetric property
A B
CD
5. AD ≅ CD
52. EXAMPLE 2
1. AB = AD, AB ≅ BC, and
BC ≅ CD
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD Transitive property
4. AD ≅ AB Symmetric property
A B
CD
5. AD ≅ CD Transitive property
