The document discusses congruence and similarity of triangles and figures. It defines congruence as two figures having equal corresponding sides and angles. Similarity requires proportional corresponding sides and equal corresponding angles. The conditions for congruence of triangles are: side-side-side, side-angle-side, angle-side-angle, and angle-angle-side. Congruence is reflexive, symmetric, and transitive. The conditions for similarity of triangles are: corresponding sides proportional and two or more equal corresponding angles.
This document contains information about proving triangles congruent using various postulates and theorems of geometry including:
- SSS (side-side-side) postulate
- SAS (side-angle-side) postulate
- ASA (angle-side-angle) postulate
- AAS (angle-angle-side) theorem
- Hypotenuse-Leg theorem
It also defines key terms like hypotenuse and legs of a right triangle and presents the Pythagorean theorem.
The document discusses various properties and theorems related to triangles. It begins by defining different types of triangles based on side lengths and angle measures. It then covers the four congruence rules for triangles: SAS, ASA, AAS, and SSS. The document proceeds to prove several theorems about relationships between sides and angles of triangles, such as opposite sides/angles of isosceles triangles being equal, larger sides having greater opposite angles, and the sum of any two angles being greater than the third side. It concludes by proving that the perpendicular from a point to a line is the shortest segment.
This document covers the topics of congruence, similarity, and ratios between similar shapes. It includes 4 tests for determining if triangles are congruent based on side lengths and angles. It also discusses identifying similar shapes and using corresponding parts of similar triangles to determine unknown lengths and angles. Finally, it examines how linear dimensions, areas, and volumes are scaled between similar cuboids based on common scale factors.
1. Triangles are congruent if all corresponding sides and angles are congruent. They will have the same shape and size but may be mirror images.
2. There are four main postulates and theorems used to prove triangles congruent: SSS (three sides), SAS (two sides and included angle), ASA (two angles and included side), and AAS (two angles and non-included side).
3. Corresponding parts of congruent triangles are also congruent based on the CPCTC theorem. This allows using previously proven congruent parts in future proofs.
This document discusses different ways to prove that two triangles are congruent, including:
1. Side-Side-Side (SSS) - If all three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
2. Side-Angle-Side (SAS) - If one angle and the sides that form it in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent.
3. Angle-Side-Angle (ASA) - If two angles and the included side in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent.
This document presents the proportionality theorem and its converse. The proportionality theorem states that if a line is drawn parallel to one side of a triangle, then it divides the other two sides proportionately. The converse states that if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. The document provides constructions and proofs of both the theorem and its converse using ratios of corresponding parts of triangles.
The document provides information about triangle congruence, including:
1. There are three postulates for proving triangles are congruent: side-side-side (SSS), side-angle-side (SAS), and angle-side-angle (ASA).
2. The SSS postulate states that if three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.
3. The SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.
4. The ASA postulate states that if two angles
1) The document discusses properties and theorems related to triangles, including similarity, congruence, medians, altitudes, angle bisectors, and the Pythagorean theorem.
2) It provides criteria for establishing similarity between triangles, including having equal corresponding angles or proportional corresponding sides.
3) Theorems are presented on parallel lines cutting across two sides of a triangle in the same ratio, and on perpendiculars drawn from right triangle vertices cutting the hypotenuse.
This document discusses the concept of CPCTC (Corresponding Parts of Congruent Triangles are Congruent). It states that once two triangles are proven to be congruent using SSS, SAS, ASA, AAS, or HL, then all corresponding parts of those triangles are also congruent due to CPCTC. It provides two examples of using CPCTC to prove corresponding angles are congruent after first showing the triangles are congruent using given information.
This document provides an introduction to congruent triangles and the different methods to prove triangles are congruent: SSS, SAS, and ASA. It includes examples of using side and angle correspondences to show triangles are congruent according to the three congruence rules. Students are asked to complete a KWL chart on triangles as an exit ticket.
The document discusses different types of triangles and the properties used to determine if two triangles are congruent. It defines triangles and their components like sides and angles. It then explains the different types of triangles based on side lengths and angle measures. The properties used to prove triangle congruence are side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), side-side-side (SSS), and hypotenuse-side (RHS).
The document discusses congruent triangles and the different tests that can be used to prove triangles are congruent, including side-side-side (SSS), side-angle-side (SAS), angle-angle-side (AAS), and right angle-hypotenuse-side (RHS). It also provides examples of isosceles, equilateral, and right triangles and defines triangle terminology such as altitude, median, and right bisector.
Triangles are congruent if they have the same three sides and three angles. Congruence can be proven using various criteria: SAS (side-angle-side), ASA (angle-side-angle), SSS (side-side-side), RHS (right angle-hypotenuse-side). Properties of triangles include: angles opposite equal sides of an isosceles triangle are equal; sides opposite equal angles are equal; the angle opposite the longer side is larger; the side opposite the longer angle is longer; the sum of any two sides is greater than the third side.
This document provides guidance on writing proofs to show that two triangles are congruent. It explains that a two-column proof lists given information, deduced information, and the statement to be proved, with reasons for each step. A basic three-step method is outlined: 1) Mark given information on the diagram, 2) Identify the congruence theorem and additional needed information, 3) Write the statements and reasons, with the last statement being what is to be proved. An example proof is provided using the Side-Side-Side congruence theorem to prove two triangles are congruent. Common theorems that can be used in proofs are also listed.
Two triangles are congruent if they have the same size and shape. To prove triangles are congruent, it is sufficient to show that some corresponding parts, like sides or angles, are congruent. There are four main ways to prove triangles are congruent: side-angle-side (SAS), side-side-side (SSS), angle-side-angle (ASA), and angle-angle-side (AAS). Each involves showing that specific corresponding parts - two sides and an angle, three sides, two angles and a side, or two angles and a side - are congruent.
This document provides information on proving triangle congruence using various postulates and properties. It discusses the six corresponding parts used to determine if two triangles are congruent, as well as five postulates for proving congruence: SSS, SAS, ASA, SAA/AAS, and the third angle theorem. Examples are given of applying each postulate, along with exercises to identify the postulate used and complete triangle congruence proofs. Key details include identifying the six corresponding parts of triangles as sides and angles, discussing the five postulates for proving congruence based on sides and angles, and providing examples of setting up triangle congruence proofs.
This document defines and provides examples of congruent angles, congruent segments, and congruent triangles in geometry. It explains that angles are congruent if they have the same measure, segments are congruent if they have the same length, and triangles are congruent if corresponding angles and sides are congruent. The document provides exercises asking the reader to identify corresponding parts of congruent triangles without diagrams as well as included angles and sides within triangles.
The document discusses different methods for proving triangles are congruent: SSS, SAS, ASA, AAS, and HL. It states that if the specified corresponding parts of two triangles are congruent using one of these methods, then the triangles are congruent. The document provides examples of congruence proofs using these methods and their reasoning.
This document discusses different methods for proving that triangles are congruent: SSS, SAS, ASA, AAS, and HL. It defines congruent polygons as having the same size and shape, meaning all corresponding sides are the same length and all corresponding angles have the same measure. The order of letters in a congruence statement indicates which angles or sides are being shown as congruent to prove the triangles are congruent.
SIMILAR TRIANLES MATHEMATICS IN EMGLISH.pptssuser2b2e9e
Similar triangles are triangles that have the same shape but not necessarily the same size. To prove that two triangles are similar, it is not necessary to show that all six corresponding angle and side ratios are equal, as some of these are dependent on each other due to the properties of triangles. There are three special combinations that can be used to prove similarity: 1) if two angles of one triangle are congruent to two angles of another (AA similarity postulate), 2) if three pairs of corresponding sides are proportional (SSS similarity theorem), or 3) if one angle of one triangle is congruent to one angle of another, and the lengths of the sides including these angles are proportional (SAS similarity theorem).
The document discusses similar polygons and triangles. It provides examples of similar figures and the criteria for determining if two triangles are similar: corresponding angles are equal and the ratios of corresponding sides are equal. Three similarity theorems are described: 1) AAA - two pairs of congruent angles, 2) SAS - two pairs of proportional sides and a congruent included angle, 3) SSS - three pairs of proportional sides. The document also shows examples of using similarity to solve for missing values.
There are four types of lines that can be drawn in a triangle:
1. A perpendicular bisector of a side bisects the side at a 90 degree angle.
2. An angle bisector bisects the interior angle of the triangle into two equal angles.
3. A height or altitude drops perpendicular from a vertex to the opposite side.
4. A median connects a vertex to the midpoint of the opposite side.
Triangles and Types of triangles&Congruent Triangles (Congruency Rule)pkprashant099
This document defines and describes different types of triangles:
- Equilateral triangles have three equal sides and three equal angles.
- Isosceles triangles have at least two equal sides.
- Scalene triangles have no equal sides.
- Right triangles have one 90 degree angle.
- Acute triangles have all angles less than 90 degrees.
- Obtuse triangles have one angle greater than 90 degrees.
It also describes three theorems used to prove triangle congruence: SSS (three equal sides), SAS (two equal sides and the included angle), and ASA (two equal angles and one included side).
The document is a lecture on similar triangles. It defines similar triangles as having the same shape but different sizes, and discusses how similar triangles have corresponding angles that are congruent and corresponding sides that are proportional. It provides examples of similar triangles and statements showing their similarity. It also covers using proportions of corresponding sides to solve for missing sides in similar triangles and several proportionality principles related to similar triangles, including the basic proportionality theorem involving parallel lines cutting across a triangle.
This document provides information about Module 17 on similar triangles. The key points covered are:
1. The module discusses the definition of similar triangles, similarity theorems, and how to determine if two triangles are similar or find missing lengths using properties of similar triangles.
2. Students are expected to learn how to apply the definition of similar triangles, verify the AAA, SAS, and SSS similarity theorems, and use proportionality theorems to calculate lengths of line segments.
3. Several examples and exercises are provided to help students practice determining if triangles are similar, citing the appropriate similarity theorem, finding missing lengths, and applying properties of similar triangles.
Similar triangles are triangles that have the same shape but different sizes. Triangles are similar if the ratios of corresponding sides are equal, or if their angles are congruent based on the AAA, AA, SAS, or SSS similarity postulates/theorems. Right triangles are similar if the ratios of corresponding legs or a leg and hypotenuse are equal, according to the LL or HL similarity theorems.
This document provides information about congruent triangles. It defines congruent triangles as two triangles that have the same shape and size, with corresponding sides and angles being equal. It describes several triangle congruence theorems including SSS, SAS, ASA, AAS, and RHS, which establish that triangles are congruent if certain combinations of sides and/or angles are equal. It also discusses isosceles triangles, angle bisectors, and provides examples applying the congruence theorems to prove triangles are congruent or not.
Kesebangunan dua segitiga dan contoh soalnyaMakna Pujarka
The document discusses properties of congruent triangles in three sentences or less:
Two triangles are congruent if (1) their corresponding sides are proportional or (2) their corresponding angles are equal in measure. Several examples demonstrate how to prove triangles are congruent by showing their corresponding sides are proportional or corresponding angles are equal. Proportionality of corresponding sides and equality of corresponding angles are used to determine missing side lengths in various triangle scenarios.
Here are the key steps:
1. ∆MAN ~ ∆MON (by AAA similarity theorem)
2. There is 1 triangle similar to ∆MAN
For the polyominoes activity:
- A polyomino made of 1 square would require 4 sticks
- A polyomino made of 2 squares would require 6 sticks
- A polyomino made of 3 squares would require 8 sticks
- A polyomino made of 4 squares would require 10 sticks (as in the example given)
- Continuing the pattern, a polyomino made of n squares would require 2n + 2 sticks
For the rectangle counting activity:
- There are 4 rectangles in the given diagram (
5.1 Introduction 5.2 Ratio And Proportionality 5.3 Similar Polygons 5.4 Basic Proportionality Theorem 5.5 Angle Bisector Theorem 5.6 Similar Triangles 5.7 Properties Of Similar Triangles
This document discusses congruence of triangles and the different postulates used to prove triangles are congruent. It introduces the Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), and Angle-Angle-Side (AAS) postulates. It explains that to prove triangles are congruent using SSS, all three sides must be equal; with SAS, two sides and the included angle must be equal; with ASA, two angles and the included side must be equal; and with AAS, two angles and a non-included side must be equal. The document emphasizes there is no Side-Side-Angle (SSA) or Triple-Angle
1. The angles labeled (2a)° and (5a + 5)° are supplementary and add up to 180°. Solving for a gives a = 25. Similarly, the angles labeled (4b +10)° and (2b – 10)° are supplementary and add up to 180°. Solving for b gives b = 30. Therefore, a + b = 25 + 30 = 55.
2. Two parallel lines intersected by a transversal form eight angles. The acute angles are equal and the obtuse angles are equal. The acute angles are supplementary to the obtuse angles. Solving the equation relating the angles gives the answer.
3. The relationship between angles formed when
The document is an acknowledgement from a group of 5 students - Abhishek Mahto, Lakshya Kumar, Mohan Kumar, Ritik Kumar, and Vivek Singh of class X E. They are thanking their principal Dr. S.V. Sharma and math teacher Mrs. Shweta Bhati for their guidance and support in completing their project on triangles and similarity. They also thank their parents and group members for their advice and assistance during the project.
This document discusses different ways to prove that two triangles are congruent. It outlines the Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Leg-Leg (L-L), Leg-Acute Angle (L-AA), Hypotenuse-Acute Angle (H-AA), Hypotenuse-Leg (H-L) postulates. Each postulate states that if certain corresponding parts of two triangles (sides, angles, legs, hypotenuses) are congruent, then the triangles are congruent. Examples are provided to illustrate each postulate. Key terms like included angle and included side are also defined.
6.5 prove triangles similar by sss and sasdetwilerr
The document discusses proving triangles similar using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) similarity theorems. It provides examples of determining if triangles are similar and explaining the reasoning. It also covers finding missing side lengths of similar triangles. The guided practice questions have students identify similar triangles and explain the similarity statements using the given information about side lengths or angles.
This document discusses two postulates for proving triangles congruent:
1. Angle-Side-Angle (ASA) postulate - If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
2. Angle-Angle-Side (AAS) postulate - If two angles and one non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the triangles are congruent.
The document provides examples of applying these postulates to determine if pairs of triangles are congruent. It also notes there are no AAA or SSA postulates to prove triangles congruent.
The document contains information about triangles, including:
1) If two triangles have proportional sides and equal angles, they are similar triangles.
2) In a right triangle, a perpendicular line from the right angle to the hypotenuse divides it into two right triangles that are similar to each other and to the original triangle.
3) A line dividing two sides of a triangle proportionally is parallel to the third side.
The document discusses proportion and similar triangles in geometry. It defines proportion as an equation stating that two ratios are equal, and provides examples of using cross products to check for proportion. It then defines similar polygons and triangles as those with congruent corresponding angles and proportional corresponding sides. The document provides different methods to prove triangles are similar, including SAS, SSS, and AA similarity. It also discusses how corresponding parts of similar triangles, such as perimeters, altitudes, angle bisectors, and medians are proportional. Several theorems and examples involving parallel lines cutting across triangles proportionally are presented.
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The @api.onchange decorator in Odoo is indeed used to trigger a method when a field's value changes. It's commonly used for validating data or triggering actions based on the change of a specific field. When the field value changes, the function decorated with @api.onchange will be called automatically.
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Its resistance in one direction is low (ideally zero) and high (ideally infinite) resistance in the other direction.
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Life of Ah Gong and Ah Kim ~ A Story with Life Lessons (Hokkien, English & Ch...OH TEIK BIN
A PowerPoint Presentation of a fictitious story that imparts Life Lessons on loving-kindness, virtue, compassion and wisdom.
The texts are in Romanized Hokkien, English and Chinese.
For the Video Presentation with audio narration in Hokkien, please check out the Link:
https://vimeo.com/manage/videos/987932748
Odoo 17 Project Module : New Features - Odoo 17 SlidesCeline George
The Project Management module undergoes significant enhancements, aimed at providing users with more robust tools for planning, organizing, and executing projects effectively.
2. Similar triangles are triangles that have the same
shape but not necessarily the same size.
A
C
B
D
F
E
ABC DEF
When we say that triangles are similar there are several
repercussions that come from it.
A D
B E
C F
AB
DE
BC
EF
AC
DF= =
3. 1. SSS Similarity Theorem
3 pairs of proportional sides
Six of those statements are true as a result of the
similarity of the two triangles. However, if we need to
prove that a pair of triangles are similar how many of
those statements do we need? Because we are working
with triangles and the measure of the angles and sides
are dependent on each other. We do not need all six.
There are three special combinations that we can use
to prove similarity of triangles.
2. SAS Similarity Theorem
2 pairs of proportional sides and congruent
angles between them
3. AA Similarity Theorem
2 pairs of congruent angles
4. 1. SSS Similarity Theorem
3 pairs of proportional sidesA
B C
E
F D
251
4
5
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12
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5. 2. SAS Similarity Theorem
2 pairs of proportional sides and congruent
angles between them
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H I
L
J K
660
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.
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GHm
660
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7
.
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10.5
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70
mH = mK
GHI LKJ
6. The SAS Similarity Theorem does not work unless
the congruent angles fall between the proportional
sides. For example, if we have the situation that is
shown in the diagram below, we cannot state that the
triangles are similar. We do not have the information
that we need.
G
H I
L
J K
7
10.5
50
50
Angles I and J do not fall in between sides GH and HI and
sides LK and KJ respectively.
7. 3. AA Similarity Theorem
2 pairs of congruent angles
M
N O
Q
P R
70
70
50
50
mN = mR
mO = mP MNO QRP
8. It is possible for two triangles to be similar when
they have 2 pairs of angles given but only one of
those given pairs are congruent.
87
34
34
S
T
U
XY
Z
mT = mX
mS = 180- (34 + 87)
mS = 180- 121
mS = 59
mS = mZ
TSU XZY
59
5959
34
34