11X1 T06 05 Similar Triangles
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The document discusses the properties of similar triangles. It outlines three tests for determining if triangles are similar: (1) the sides are proportional (SSS), (2) two pairs of sides are proportional and angles are equal (SAS), (3) the angles are equal (AA). It provides an example of using similar triangles to find an unknown side length.
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The document discusses congruent triangles and the different tests used to prove triangles are congruent. There are four tests: side-side-side (SSS), side-angle-side (SAS), angle-angle-side (AAS), and right angle-hypotenuse-side (RHS). It also provides an example proof using AAS and SAS to show two sides of a quadrilateral are congruent. Finally, it defines isosceles and equilateral triangles.
11 x1 t07 03 congruent triangles (2013)
The document discusses congruent triangles and the different tests that can be used to prove triangles are congruent. There are four main tests: side-side-side (SSS), side-angle-side (SAS), angle-angle-side (AAS), and right angle-hypotenuse-side (RHS). It also provides examples of applying the SAS test to prove corresponding sides and angles of two triangles are congruent. Additionally, it defines different types of triangles such as isosceles and equilateral triangles based on their side lengths and angle measures.
11 x1 t08 05 similar triangles
(1) There are three tests to determine if triangles are similar: corresponding sides are proportional (SSS), two pairs of corresponding sides are proportional and included angles are equal (SAS), or all three angles are equal (AA).
(2) To find the missing side AD of a similar triangle, set up a proportion using the ratio of corresponding sides from the given triangles.
(3) For similar shapes, if sides are in ratio a:b, then area is in ratio a^2:b^2 and volume is in ratio a^3:b^3.
11 x1 t07 05 similar triangles (2012)
(1) There are three tests to determine if triangles are similar: corresponding sides are proportional (SSS), two pairs of corresponding sides are proportional and included angles are equal (SAS), or all three angles are equal (AA).
(2) To find the missing side AD of a similar triangle, set up a proportion using the ratio of corresponding sides from the given triangles.
(3) For similar shapes, if sides are in ratio a:b, then area is in ratio a^2:b^2 and volume is in ratio a^3:b^3.
Triangles and its all types
This document defines and describes different types of triangles based on side lengths and angle measures. It also discusses properties of triangles related to similarity, including the AAA, SSS, SAS, and RHS similarity criteria. Properties of special right triangles and the Pythagorean theorem are also covered. Types of triangles described include scalene, isosceles, equilateral, acute, obtuse, and right triangles.
Digit l textbook 131
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.
R.TANUJ Maths Triangles for Class IX
1. The document discusses different types of triangles based on their sides and angles. It defines triangle congruence and presents several triangle congruence theorems including SAS, ASA, AAS, SSS, and RHS.
2. Properties of triangles such as corresponding angles and sides of congruent triangles being equal are explained. Inequalities in triangles and relationships between sides and angles are also covered.
3. Objectives of the lesson include defining triangle congruence, stating criteria for congruence, and properties of triangles like sum of angles and relationships between sides and angles.
class-9-math-triangles_1595671835220.pdf
1. A triangle has three sides, three angles, and three vertices.
2. There are five criteria to determine if two triangles are congruent: side-angle-side, angle-side-angle, angle-angle-side, side-side-side, and right-angle-hypotenuse-side.
3. Properties of triangles include: angles opposite equal sides are equal; sides opposite equal angles are equal; the longer the side, the larger the opposite angle; and the sum of any two sides is greater than the third side.
Triangles
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).
Similar Triangles PPT and examples.ppt
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.
8 3similar Triangles
The document discusses similar triangles and how to determine if triangles are similar. It explains that similar triangles have congruent angles, while congruent triangles have both congruent angles and sides. It provides examples of using the Angle-Angle similarity criterion to determine if triangles are similar and setting up ratio proportions to calculate missing sides of similar triangles. The lesson objectives are to apply properties of similar triangles and prove that triangles are similar.
Shivam goyal ix e
The document discusses various properties of quadrilaterals:
- It defines the six types of quadrilaterals - trapezium, parallelogram, rectangle, rhombus, square, and kite.
- It provides examples and definitions for each type.
- Several theorems regarding the properties of parallelograms are presented, including that the diagonals of a parallelogram bisect each other and that opposite sides of a parallelogram are equal.
- Additional theorems state that a quadrilateral is a parallelogram if opposite sides are equal or if opposite angles are equal.
7th pre alg similarity & triangles
This document discusses similarity between shapes and triangles. It provides the following key points:
1. Two shapes are similar if corresponding angles are equal and corresponding sides are proportional. All regular polygons and rectangles can be similar or not similar depending on if they meet these criteria.
2. If two shapes are similar, one is an enlargement of the other. The scale factor represents the ratio of corresponding sides.
3. Similar triangles only require equal corresponding angles, not proportional sides. Unknown sides of similar triangles can be found using scale factors determined from corresponding sides.
4. Parallel lines drawn to the sides of a triangle form similar triangles. This property can be used to solve problems involving similar triangles.
congruenttriangles-130611002549-.docx
The document discusses various properties and theorems related to triangles. It defines different types of triangles based on sides and angles. It introduces concepts like congruence of triangles and corresponding parts. It describes the four main congruence rules: SAS, ASA, AAS, and SSS. It also discusses properties like angles opposite to equal sides are equal, sides opposite to equal angles are equal, sum of angles of a triangle is 180 degrees, and theorems related to inequality of sides and angles.
Proving triangle congruency
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.
8 3 Similar Triangles
The document discusses similar triangles and how to determine if two triangles are similar. It explains that two triangles are similar if corresponding angles are congruent. It provides examples of using the Angle-Angle similarity criterion to show triangles are similar and using proportions to find missing sides of similar triangles. The lesson covered congruent triangles, similar triangles, determining similarity using corresponding angles, and applying similarity to find unknown lengths.
Traingle class 10 aditi sharma presentation.pptx
1. Two triangles are congruent if their corresponding sides and angles are equal. This means the triangles have the same shape and size.
2. There are four criteria for congruence: SAS, ASA, AAS, and SSS. SAS means two sides and the included angle are equal. ASA means two angles and the included side are equal. AAS means two angles and one side are equal. SSS means all three sides are equal.
3. Properties of triangles include: angles opposite equal sides are equal, sides opposite equal angles are equal, and the sum of any two sides is greater than the third side.
Similar to 11X1 T06 05 Similar Triangles (20)
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Goodbye slideshare UPDATE
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
Goodbye slideshare
NOTE: there is no sound, i.e. it is not a slidecast, as slideshare are going to get rid of this feature.
11 x1 t01 03 factorising (2014)
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
11 x1 t01 01 algebra & indices (2014)
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
12 x1 t01 01 log laws (2013)
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
X2 t02 03 roots & coefficients (2013)
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
X2 t02 02 multiple roots (2013)
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
X2 t02 01 factorising complex expressions (2013)
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
11 x1 t16 07 approximations (2013)
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
11 x1 t16 05 volumes (2013)
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
11 x1 t16 04 areas (2013)
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
11 x1 t16 02 definite integral (2013)
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
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11X1 T06 05 Similar Triangles
- 3. Similar Triangles TESTS (1) Corresponding sides are in proportion (SSS – with ratio a:b)
- 4. Similar Triangles TESTS (1) Corresponding sides are in proportion (SSS – with ratio a:b) (2) Two pairs of corresponding sides are in proportion AND the included angles are equal (SAS – with ratio a:b)
- 5. Similar Triangles TESTS (1) Corresponding sides are in proportion (SSS – with ratio a:b) (2) Two pairs of corresponding sides are in proportion AND the included angles are equal (SAS – with ratio a:b) (3) All three angles are the same as the three angles in the other (AA)
- 6. Similar Triangles TESTS (1) Corresponding sides are in proportion (SSS – with ratio a:b) (2) Two pairs of corresponding sides are in proportion AND the included angles are equal (SAS – with ratio a:b) (3) All three angles are the same as the three angles in the other (AA) e.g. Find AD A B C D E 15 cm 21 cm 24 cm
- 7. Similar Triangles TESTS (1) Corresponding sides are in proportion (SSS – with ratio a:b) (2) Two pairs of corresponding sides are in proportion AND the included angles are equal (SAS – with ratio a:b) (3) All three angles are the same as the three angles in the other (AA) e.g. Find AD A B C D E 15 cm 21 cm 24 cm
- 8. Similar Triangles TESTS (1) Corresponding sides are in proportion (SSS – with ratio a:b) (2) Two pairs of corresponding sides are in proportion AND the included angles are equal (SAS – with ratio a:b) (3) All three angles are the same as the three angles in the other (AA) e.g. Find AD A B C D E 15 cm 21 cm 24 cm
- 9. Similar Triangles TESTS (1) Corresponding sides are in proportion (SSS – with ratio a:b) (2) Two pairs of corresponding sides are in proportion AND the included angles are equal (SAS – with ratio a:b) (3) All three angles are the same as the three angles in the other (AA) e.g. Find AD A B C D E 15 cm 21 cm 24 cm
- 10. A A B C D E 24 cm 36 cm 15 cm
- 11. A A B C D E 24 cm 36 cm 15 cm
- 12. A A B C D E 24 cm 36 cm 15 cm
- 13. A A B C D E 24 cm 36 cm 15 cm In similar shapes;
- 14. A A B C D E 24 cm 36 cm 15 cm In similar shapes;
- 15. A A B C D E 24 cm 36 cm 15 cm In similar shapes;
- 16. A A B C D E 24 cm 36 cm 15 cm In similar shapes;
- 17. A A B C D E 24 cm 36 cm 15 cm In similar shapes; Exercise 8H; 2bd, 4ab, 6bc, 8, 12, 16, 18, 20, 21, 24*