The document discusses two postulates for proving triangle congruence: the Angle-Side-Angle (ASA) postulate and the Angle-Angle-Side (AAS) postulate. It includes examples of using these postulates to prove triangles are congruent by showing that corresponding angles and sides are congruent. The examples demonstrate how to set up and solve proofs of triangle congruence using angle and side relationships.
LL- If the two legs of a right triangle are congruent to the two legs of another triangle, then the triangles are congruent.
LA-If the leg and an acute angle of one right triangle are congruent to the corresponding leg and an acute angle of another right triangle, then the right triangles are congruent.
LL- If the two legs of a right triangle are congruent to the two legs of another triangle, then the triangles are congruent.
LA-If the leg and an acute angle of one right triangle are congruent to the corresponding leg and an acute angle of another right triangle, then the right triangles are congruent.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
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.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
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.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
2. ESSENTIAL QUESTIONS
How do you use the ASA Postulate to test for triangle
congruence?
How do you use the AAS Postulate to test for triangle
congruence?
4. VOCABULARY
1. Included Side: The side between two consecutive
angles in a triangle
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence:
Theorem 4.5 - Angle-Angle-Side (AAS) Congruence:
5. VOCABULARY
1. Included Side: The side between two consecutive
angles in a triangle
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: If
two angles and the included side of one triangle are
congruent to two angles and included side of a
second triangle, then the triangles are congruent
Theorem 4.5 - Angle-Angle-Side (AAS) Congruence:
6. VOCABULARY
1. Included Side: The side between two consecutive
angles in a triangle
Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: If
two angles and the included side of one triangle are
congruent to two angles and included side of a
second triangle, then the triangles are congruent
Theorem 4.5 - Angle-Angle-Side (AAS) Congruence: If
two angles and the nonincluded side of one triangle
are congruent to the corresponding angles and
nonincluded side of a second triangle, then the
triangles are congruent
7. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
Given: L is the midpoint of WE, WR ! ED
8. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
Given: L is the midpoint of WE, WR ! ED
1. L is the midpoint of WE, WR ! ED
9. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
1. Given
Given: L is the midpoint of WE, WR ! ED
1. L is the midpoint of WE, WR ! ED
10. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
1. Given
Given: L is the midpoint of WE, WR ! ED
2. WL ≅ EL
1. L is the midpoint of WE, WR ! ED
11. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
1. Given
Given: L is the midpoint of WE, WR ! ED
2. WL ≅ EL 2. Midpoint Thm.
1. L is the midpoint of WE, WR ! ED
12. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
1. Given
Given: L is the midpoint of WE, WR ! ED
2. WL ≅ EL 2. Midpoint Thm.
3. ∠WLR ≅ ∠ELD
1. L is the midpoint of WE, WR ! ED
13. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
1. Given
Given: L is the midpoint of WE, WR ! ED
2. WL ≅ EL 2. Midpoint Thm.
3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm.
1. L is the midpoint of WE, WR ! ED
14. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
1. Given
Given: L is the midpoint of WE, WR ! ED
2. WL ≅ EL 2. Midpoint Thm.
3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm.
4. ∠LWR ≅ ∠LED
1. L is the midpoint of WE, WR ! ED
15. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
1. Given
Given: L is the midpoint of WE, WR ! ED
2. WL ≅ EL 2. Midpoint Thm.
3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm.
4. ∠LWR ≅ ∠LED 4. Alternate Interior Angles Thm
1. L is the midpoint of WE, WR ! ED
16. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
1. Given
Given: L is the midpoint of WE, WR ! ED
2. WL ≅ EL 2. Midpoint Thm.
3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm.
4. ∠LWR ≅ ∠LED 4. Alternate Interior Angles Thm
5. △WRL ≅△EDL
1. L is the midpoint of WE, WR ! ED
17. EXAMPLE 1
Prove the following.
Prove: △WRL ≅△EDL
1. Given
Given: L is the midpoint of WE, WR ! ED
2. WL ≅ EL 2. Midpoint Thm.
3. ∠WLR ≅ ∠ELD 3. Vertical Angles Thm.
4. ∠LWR ≅ ∠LED 4. Alternate Interior Angles Thm
5. △WRL ≅△EDL 5. ASA
1. L is the midpoint of WE, WR ! ED
19. EXAMPLE 2
Prove the following.
Prove: LN ≅ MN
Given: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM
20. EXAMPLE 2
Prove the following.
Prove: LN ≅ MN
Given: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
21. EXAMPLE 2
Prove the following.
Prove: LN ≅ MN
Given: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠N ≅ ∠N
22. EXAMPLE 2
Prove the following.
Prove: LN ≅ MN
Given: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠N ≅ ∠N 2. Reflexive
23. EXAMPLE 2
Prove the following.
Prove: LN ≅ MN
Given: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠N ≅ ∠N 2. Reflexive
3. △JNM ≅△KNL
24. EXAMPLE 2
Prove the following.
Prove: LN ≅ MN
Given: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠N ≅ ∠N 2. Reflexive
3. △JNM ≅△KNL 3. AAS
25. EXAMPLE 2
Prove the following.
Prove: LN ≅ MN
Given: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠N ≅ ∠N 2. Reflexive
3. △JNM ≅△KNL 3. AAS
4. LN ≅ MN
26. EXAMPLE 2
Prove the following.
Prove: LN ≅ MN
Given: ∠NKL ≅ ∠NJM, KL ≅ JM
1. ∠NKL ≅ ∠NJM, KL ≅ JM 1. Given
2. ∠N ≅ ∠N 2. Reflexive
3. △JNM ≅△KNL 3. AAS
4. LN ≅ MN 4. Corresponding Parts
of Congruent Triangles
are Congruent (CPCTC)
27. EXAMPLE 3
On a template design for a certain envelope, the top
and bottom flaps are isosceles triangles with congruent
bases and base angles. If EV = 8 cm and the height of
the isosceles triangle is 3 cm, find PO.
28. EXAMPLE 3
On a template design for a certain envelope, the top
and bottom flaps are isosceles triangles with congruent
bases and base angles. If EV = 8 cm and the height of
the isosceles triangle is 3 cm, find PO.
EV ≅ PL, so each segment has a measure
of 8 cm. If an auxiliary line is drawn
from point O perpendicular to PL, you
will have a right triangle formed.
29. EXAMPLE 3
On a template design for a certain envelope, the top
and bottom flaps are isosceles triangles with congruent
bases and base angles. If EV = 8 cm and the height of
the isosceles triangle is 3 cm, find PO.
EV ≅ PL, so each segment has a measure
of 8 cm. If an auxiliary line is drawn
from point O perpendicular to PL, you
will have a right triangle formed.
In the right triangle, we have one leg (the height) of 3 cm.
The auxiliary line will bisect PL, as point O is equidistant
from P and L.
