The document discusses congruent triangles and the different tests that can be used to prove triangles are congruent. It states that in order to prove triangles are congruent, three pieces of information are required. It then lists and describes the four main tests: (1) Side-Side-Side, (2) Side-Angle-Side, (3) Angle-Angle-Side, and (4) Right Angle-Hypotenuse-Side. It provides an example proof using these tests and also defines different types of triangles like isosceles and equilateral triangles. Finally, it discusses some triangle terminology like altitude and median.
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.
This document discusses different methods for proving that two triangles are congruent, including:
- Side-Side-Side (SSS), where all three sides of one triangle are congruent to the corresponding sides of the other triangle.
- Side-Angle-Side (SAS), where two sides and the angle between them in one triangle are congruent to the corresponding parts in the other triangle.
- Angle-Side-Angle (ASA), where two angles and the side between them in one triangle are congruent to the corresponding parts in the other.
- Angle-Angle-Side (AAS), where two angles and a non-included side are congruent between the triangles.
It notes that you need
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 provides information about congruence of triangles from a geometry textbook. It includes definitions of congruent figures and associating real numbers with lengths and angles. It describes the one-to-one correspondence test for congruence of triangles. It discusses sufficient conditions for congruence including SAS, SSS, ASA, and SAA. It presents activities and examples verifying these tests and exploring properties of isosceles and equilateral triangles. The document encourages critical thinking through "Think it Over" prompts and upgrading the chapter with additional content.
This document defines and describes properties of various quadrilaterals:
- Rectangles have four right angles and opposite sides of equal length. The area formula is length x width.
- Parallelograms have two pairs of parallel sides. The opposite angles are equal and adjacent angles sum to 180 degrees. Diagonals bisect each other.
- Trapezoids have one pair of parallel sides. Isosceles trapezoids have two pairs of equal angles and equal or equal length diagonals. Right trapezoids contain one right angle. The area of any trapezoid is half the product of the height and sum of the parallel sides.
(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.
Theorem: Angles opposite to equal sides of an isosceles triangle are equal.RameshSiyol
The document proves that the angles opposite to equal sides of an isosceles triangle are equal using congruence rules of triangles. It takes an isosceles triangle ABC with equal sides AB and AC. It draws a line AD bisecting angle A. This shows that angles BAD and CAD are equal. It then proves that triangles ABD and ACD are congruent by the Side-Angle-Side rule. Therefore, corresponding angles ABC and ACB are equal, proving the theorem.
1) A quadrilateral is a geometric figure with four sides, four angles, and two diagonals. The sum of the angles is always 360 degrees.
2) There are six types of quadrilaterals: trapezoid, parallelogram, rectangle, rhombus, square, and kite. A parallelogram has both pairs of opposite sides parallel. A rectangle has one right angle. A square is both a rectangle and rhombus with all sides equal.
3) Theorems include: the diagonals of a parallelogram bisect each other; if the diagonals of a quadrilateral bisect each other it is a parallelogram; a quadrilateral is
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.
This document discusses different methods for proving that two triangles are congruent, including:
- Side-Side-Side (SSS), where all three sides of one triangle are congruent to the corresponding sides of the other triangle.
- Side-Angle-Side (SAS), where two sides and the angle between them in one triangle are congruent to the corresponding parts in the other triangle.
- Angle-Side-Angle (ASA), where two angles and the side between them in one triangle are congruent to the corresponding parts in the other.
- Angle-Angle-Side (AAS), where two angles and a non-included side are congruent between the triangles.
It notes that you need
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 provides information about congruence of triangles from a geometry textbook. It includes definitions of congruent figures and associating real numbers with lengths and angles. It describes the one-to-one correspondence test for congruence of triangles. It discusses sufficient conditions for congruence including SAS, SSS, ASA, and SAA. It presents activities and examples verifying these tests and exploring properties of isosceles and equilateral triangles. The document encourages critical thinking through "Think it Over" prompts and upgrading the chapter with additional content.
This document defines and describes properties of various quadrilaterals:
- Rectangles have four right angles and opposite sides of equal length. The area formula is length x width.
- Parallelograms have two pairs of parallel sides. The opposite angles are equal and adjacent angles sum to 180 degrees. Diagonals bisect each other.
- Trapezoids have one pair of parallel sides. Isosceles trapezoids have two pairs of equal angles and equal or equal length diagonals. Right trapezoids contain one right angle. The area of any trapezoid is half the product of the height and sum of the parallel sides.
(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.
Theorem: Angles opposite to equal sides of an isosceles triangle are equal.RameshSiyol
The document proves that the angles opposite to equal sides of an isosceles triangle are equal using congruence rules of triangles. It takes an isosceles triangle ABC with equal sides AB and AC. It draws a line AD bisecting angle A. This shows that angles BAD and CAD are equal. It then proves that triangles ABD and ACD are congruent by the Side-Angle-Side rule. Therefore, corresponding angles ABC and ACB are equal, proving the theorem.
1) A quadrilateral is a geometric figure with four sides, four angles, and two diagonals. The sum of the angles is always 360 degrees.
2) There are six types of quadrilaterals: trapezoid, parallelogram, rectangle, rhombus, square, and kite. A parallelogram has both pairs of opposite sides parallel. A rectangle has one right angle. A square is both a rectangle and rhombus with all sides equal.
3) Theorems include: the diagonals of a parallelogram bisect each other; if the diagonals of a quadrilateral bisect each other it is a parallelogram; a quadrilateral is
This document provides information about different types of quadrilaterals and prisms. It defines a quadrilateral as a four-sided polygon and lists the main types: square, rectangle, rhombus, parallelogram, kite, trapezoid, and cyclic and irregular quadrilaterals. It gives key properties and formulas for calculating areas and perimeters of each type. The document also defines types of prisms as polyhedrons with flat faces, listing regular, irregular, right, and oblique prisms. It provides the formulas for calculating the surface area and volume of right prisms and discusses frustums of prisms, which have non-parallel cutting planes.
The document presents two triangle congruence theorems:
1) The Hypotenuse-Leg (HyL) Congruence Theorem states that if the hypotenuse and a leg of one right triangle are congruent to the corresponding hypotenuse and leg of another triangle, then the triangles are congruent.
2) The Hypotenuse-Acute Angle (HyA) Congruence Theorem states that if the hypotenuse and an acute angle of one right triangle are congruent to the corresponding hypotenuse and acute angle of another right triangle, then the triangles are congruent.
Proofs of sample triangle congruences are provided for each theorem.
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.
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.
The document discusses geometry proofs, including givens and conclusions, triangle congruencies, triangle congruency shortcuts using SSS, SAS, ASA, AAS, and HL, writing two-column proofs, and applying the CPCTC principle that corresponding parts of congruent triangles are congruent. It provides examples of two-column proofs using different congruency rules and reasoning to prove that two triangles are congruent.
The document discusses different rules for determining if two triangles are congruent, including:
- The ASA (Angle-Side-Angle) rule, which states two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle. An example proof of this rule is provided.
- The SSS (Side-Side-Side) rule, which states two triangles are congruent if three sides of one triangle are equal to the corresponding three sides of the other triangle. An example proof is also provided.
- The Hypotenuse-Leg rule, which states two right triangles are congruent if the hypotenuse and one side of one
Isosceles triangle and its two theoremsRameshSiyol
The document discusses isosceles triangles, which are triangles that have two equal sides. It provides two theorems about isosceles triangles: 1) the angles opposite the equal sides are also equal, and 2) the sides opposite equal angles are equal. Examples of isosceles triangles in real life are also mentioned.
A quadrilateral is a shape with four sides, four angles, and four vertices. There are six types of quadrilaterals: trapezium, parallelogram, rectangle, rhombus, square, and kite. A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. The diagonals of a parallelogram bisect each other and divide the parallelogram into two congruent triangles. Opposite sides and opposite angles of a parallelogram are equal.
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 provides information about quadrilaterals including definitions, types of quadrilaterals, and properties of quadrilaterals. It begins by defining a quadrilateral and listing the eight main types: parallelogram, rectangle, rhombus, square, trapezium, isosceles trapezium, kite I, and kite II. It then presents eight theorems proving various properties of quadrilaterals, such as the sum of interior angles equaling 360°, opposite sides being congruent, and diagonals bisecting each other. It concludes by providing three example problems solving for angles and side lengths of various quadrilaterals. The overall document serves as a comprehensive overview of quadril
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.
(1) If the diameter of a circle is the hypotenuse of a right triangle, the third vertex lies on the circle.
(2) If two points on the same side of a line segment subtend the same angle, the four points are concyclic.
(3) If a pair of opposite angles in a quadrilateral are supplementary, the quadrilateral is cyclic.
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.
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.
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.
The document discusses tests for determining if triangles are similar. There are three tests: (1) corresponding sides are in proportion (SSS), (2) two pairs of corresponding sides are in proportion and included angles are equal (SAS), (3) all three angles are equal (AA). An example problem finds the length of side AD using the properties of similar triangles. The side lengths are in the same ratio as the corresponding sides between the two triangles.
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.
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.
1. There are four conditions that can be used to prove that two triangles are congruent: SAS, ASA, SSS, and RHS.
2. SAS means that two sides and the angle between them are equal in both triangles. ASA means that two angles and the side between them are equal. SSS means that all three sides are equal. RHS means that there is a right angle, equal hypotenuses, and one other equal side.
3. Some properties of triangles include: the angles opposite equal sides of an isosceles triangle are equal; the sides opposite equal angles are equal; and altitudes drawn to equal sides of a triangle are equal.
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.
This document defines and describes different types of triangles based on side lengths and angle measures. It also discusses properties of similar triangles, including the AAA, SSS, SAS, and AA similarity criteria. Properties of special right triangles and the Pythagorean theorem are also covered. Types of triangles discussed include scalene, isosceles, equilateral, acute, obtuse, and right triangles.
This document provides information about different types of quadrilaterals and prisms. It defines a quadrilateral as a four-sided polygon and lists the main types: square, rectangle, rhombus, parallelogram, kite, trapezoid, and cyclic and irregular quadrilaterals. It gives key properties and formulas for calculating areas and perimeters of each type. The document also defines types of prisms as polyhedrons with flat faces, listing regular, irregular, right, and oblique prisms. It provides the formulas for calculating the surface area and volume of right prisms and discusses frustums of prisms, which have non-parallel cutting planes.
The document presents two triangle congruence theorems:
1) The Hypotenuse-Leg (HyL) Congruence Theorem states that if the hypotenuse and a leg of one right triangle are congruent to the corresponding hypotenuse and leg of another triangle, then the triangles are congruent.
2) The Hypotenuse-Acute Angle (HyA) Congruence Theorem states that if the hypotenuse and an acute angle of one right triangle are congruent to the corresponding hypotenuse and acute angle of another right triangle, then the triangles are congruent.
Proofs of sample triangle congruences are provided for each theorem.
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.
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.
The document discusses geometry proofs, including givens and conclusions, triangle congruencies, triangle congruency shortcuts using SSS, SAS, ASA, AAS, and HL, writing two-column proofs, and applying the CPCTC principle that corresponding parts of congruent triangles are congruent. It provides examples of two-column proofs using different congruency rules and reasoning to prove that two triangles are congruent.
The document discusses different rules for determining if two triangles are congruent, including:
- The ASA (Angle-Side-Angle) rule, which states two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle. An example proof of this rule is provided.
- The SSS (Side-Side-Side) rule, which states two triangles are congruent if three sides of one triangle are equal to the corresponding three sides of the other triangle. An example proof is also provided.
- The Hypotenuse-Leg rule, which states two right triangles are congruent if the hypotenuse and one side of one
Isosceles triangle and its two theoremsRameshSiyol
The document discusses isosceles triangles, which are triangles that have two equal sides. It provides two theorems about isosceles triangles: 1) the angles opposite the equal sides are also equal, and 2) the sides opposite equal angles are equal. Examples of isosceles triangles in real life are also mentioned.
A quadrilateral is a shape with four sides, four angles, and four vertices. There are six types of quadrilaterals: trapezium, parallelogram, rectangle, rhombus, square, and kite. A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. The diagonals of a parallelogram bisect each other and divide the parallelogram into two congruent triangles. Opposite sides and opposite angles of a parallelogram are equal.
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 provides information about quadrilaterals including definitions, types of quadrilaterals, and properties of quadrilaterals. It begins by defining a quadrilateral and listing the eight main types: parallelogram, rectangle, rhombus, square, trapezium, isosceles trapezium, kite I, and kite II. It then presents eight theorems proving various properties of quadrilaterals, such as the sum of interior angles equaling 360°, opposite sides being congruent, and diagonals bisecting each other. It concludes by providing three example problems solving for angles and side lengths of various quadrilaterals. The overall document serves as a comprehensive overview of quadril
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.
(1) If the diameter of a circle is the hypotenuse of a right triangle, the third vertex lies on the circle.
(2) If two points on the same side of a line segment subtend the same angle, the four points are concyclic.
(3) If a pair of opposite angles in a quadrilateral are supplementary, the quadrilateral is cyclic.
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.
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.
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.
The document discusses tests for determining if triangles are similar. There are three tests: (1) corresponding sides are in proportion (SSS), (2) two pairs of corresponding sides are in proportion and included angles are equal (SAS), (3) all three angles are equal (AA). An example problem finds the length of side AD using the properties of similar triangles. The side lengths are in the same ratio as the corresponding sides between the two triangles.
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.
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.
1. There are four conditions that can be used to prove that two triangles are congruent: SAS, ASA, SSS, and RHS.
2. SAS means that two sides and the angle between them are equal in both triangles. ASA means that two angles and the side between them are equal. SSS means that all three sides are equal. RHS means that there is a right angle, equal hypotenuses, and one other equal side.
3. Some properties of triangles include: the angles opposite equal sides of an isosceles triangle are equal; the sides opposite equal angles are equal; and altitudes drawn to equal sides of a triangle are equal.
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.
This document defines and describes different types of triangles based on side lengths and angle measures. It also discusses properties of similar triangles, including the AAA, SSS, SAS, and AA similarity criteria. Properties of special right triangles and the Pythagorean theorem are also covered. Types of triangles discussed include scalene, isosceles, equilateral, acute, obtuse, and right triangles.
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.
This document discusses properties of quadrilaterals. It begins with definitions of different types of quadrilaterals based on the number of pairs of parallel sides they have: trapezoids have 1 pair, parallelograms have 2 pairs. It then discusses properties of parallelograms and special types of parallelograms like rectangles, rhombi, and squares. Several theorems about the properties of parallelograms are then proved, such as opposite sides being equal, opposite angles being equal, and diagonals bisecting each other.
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.
This document discusses different ways to prove that two triangles are congruent using congruence postulates. It explains that triangles are congruent if corresponding parts are congruent, but you do not need all six. There are four main postulates covered: side-side-side (SSS), side-angle-side (SAS), angle-side-angle (ASA), and angle-angle-side (AAS). It warns that there is no side-side-angle (SSA) or angle-angle-angle (AAA) postulate to prove congruence.
The document discusses several angle theorems related to cyclic quadrilaterals:
1) Opposite angles of a cyclic quadrilateral are supplementary. This is proven using properties of angles on a circumference.
2) The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
3) Angles subtended at the circumference by the same or equal arcs are equal. This is proven using properties of angles on a circumference.
k-12 curriculum grade 9 geometry. Geometry is the study of different types of shapes, figures and sizes in Maths or in real life. In geometry, we learn about different angles, transformations and similarities in the figures. The basics of geometry depend on majorly point, line, angles and plane.
The document discusses different methods for proving triangles congruent:
1. SSS (side-side-side)
2. SAS (side-angle-side)
3. ASA (angle-side-angle)
4. AAS (angle-angle-side)
5. HL (hypotenuse-leg) for right triangles only.
It also covers built-in information in triangles like shared sides, parallel lines, and vertical angles that can be used for proofs. The document provides examples of congruence proofs using these methods and concepts like corresponding parts of congruent triangles.
The document discusses different methods for proving triangles congruent:
1. SSS (side-side-side)
2. SAS (side-angle-side)
3. ASA (angle-side-angle)
4. AAS (angle-angle-side)
5. HL (hypotenuse-leg) for right triangles only.
It provides examples of using these methods to determine if pairs of triangles are congruent or not. It also discusses using corresponding parts of congruent triangles (CPCTC) to prove additional parts are congruent if two triangles are already shown to be congruent.
10.17 Triangle Congruence Proofs Day 2.pptmikeebio1
The document discusses different methods for proving triangles congruent:
1. SSS (side-side-side)
2. SAS (side-angle-side)
3. ASA (angle-side-angle)
4. AAS (angle-angle-side)
5. HL (hypotenuse-leg) for right triangles only.
It provides examples of using these methods to determine if pairs of triangles are congruent or not. It also discusses using corresponding parts of congruent triangles (CPCTC) to prove additional parts congruent, such as adding auxiliary lines.
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.
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.
Similar to 11X1 T07 03 congruent triangles (2010) (20)
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.
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.
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)Nigel Simmons
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
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.
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.
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/
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.
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.
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.
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.
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
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
3. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
4. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
5. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
6. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
7. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
8. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
9. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
10. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
(2) Side-Angle-Side (SAS)
11. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
(2) Side-Angle-Side (SAS)
12. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
(2) Side-Angle-Side (SAS)
13. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
(2) Side-Angle-Side (SAS)
14. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
(2) Side-Angle-Side (SAS)
15. Congruent Triangles
In order to prove congruent triangles you require three pieces of
information.
Hint: Look for a side that is the same in both triangles first.
TESTS
(1) Side-Side-Side (SSS)
(2) Side-Angle-Side (SAS)
NOTE:
must be
included
angle
22. (3) Angle-Angle-Side (AAS)
NOTE:
sides must be in
the same position
(4) Right Angle-Hypotenuse-Side (RHS)
23. (3) Angle-Angle-Side (AAS)
NOTE:
sides must be in
the same position
(4) Right Angle-Hypotenuse-Side (RHS)
24. (3) Angle-Angle-Side (AAS)
NOTE:
sides must be in
the same position
(4) Right Angle-Hypotenuse-Side (RHS)
25. (3) Angle-Angle-Side (AAS)
NOTE:
sides must be in
the same position
(4) Right Angle-Hypotenuse-Side (RHS)
26. (3) Angle-Angle-Side (AAS)
NOTE:
sides must be in
the same position
(4) Right Angle-Hypotenuse-Side (RHS)
27. e.g. (1985)
C
D
In the diagram ABCD is a quadrilateral.
The diagonals AC and BD intersect at
S right angles, and DAS BAS
A B
28. e.g. (1985)
C
D
In the diagram ABCD is a quadrilateral.
The diagonals AC and BD intersect at
S right angles, and DAS BAS
A B
(i) Prove DA = AB
29. e.g. (1985)
C
D
In the diagram ABCD is a quadrilateral.
The diagonals AC and BD intersect at
S right angles, and DAS BAS
A B
(i) Prove DA = AB
DAS BAS given A
30. e.g. (1985)
C
D
In the diagram ABCD is a quadrilateral.
The diagonals AC and BD intersect at
S right angles, and DAS BAS
A B
(i) Prove DA = AB
DAS BAS given A
AS is common S
31. e.g. (1985)
D C
In the diagram ABCD is a quadrilateral.
The diagonals AC and BD intersect at
S right angles, and DAS BAS
A B
(i) Prove DA = AB
DAS BAS given A
AS is common S
DSA BSA 90 given A
32. e.g. (1985)
D C
In the diagram ABCD is a quadrilateral.
The diagonals AC and BD intersect at
S right angles, and DAS BAS
A B
(i) Prove DA = AB
DAS BAS given A
AS is common S
DSA BSA 90 given A
DAS BAS AAS
33. e.g. (1985)
D C
In the diagram ABCD is a quadrilateral.
The diagonals AC and BD intersect at
S right angles, and DAS BAS
A B
(i) Prove DA = AB
DAS BAS given A
AS is common S
DSA BSA 90 given A
DAS BAS AAS
DA AB matching sides in 's
36. (ii) Prove DC = CB
DA AB proven S
DAS BAS given A
37. (ii) Prove DC = CB
DA AB proven S
DAS BAS given A
AC is common S
38. (ii) Prove DC = CB
DA AB proven S
DAS BAS given A
AC is common S
DAC BAC SAS
39. (ii) Prove DC = CB
DA AB proven S
DAS BAS given A
AC is common S
DAC BAC SAS
DC CB matching sides in 's
40. (ii) Prove DC = CB
DA AB proven S
DAS BAS given A
AC is common S
DAC BAC SAS
DC CB matching sides in 's
Types Of Triangles
Isosceles Triangle
A
B C
41. (ii) Prove DC = CB
DA AB proven S
DAS BAS given A
AC is common S
DAC BAC SAS
DC CB matching sides in 's
Types Of Triangles
Isosceles Triangle
A
AB AC sides in isoscelesABC
B C
42. (ii) Prove DC = CB
DA AB proven S
DAS BAS given A
AC is common S
DAC BAC SAS
DC CB matching sides in 's
Types Of Triangles
Isosceles Triangle
A
AB AC sides in isoscelesABC
B C ' s in isoscelesABC
B C
43. (ii) Prove DC = CB
DA AB proven S
DAS BAS given A
AC is common S
DAC BAC SAS
DC CB matching sides in 's
Types Of Triangles
Isosceles Triangle
A
AB AC sides in isoscelesABC
B C ' s in isoscelesABC
B C
Equilateral Triangle
A
B C
44. (ii) Prove DC = CB
DA AB proven S
DAS BAS given A
AC is common S
DAC BAC SAS
DC CB matching sides in 's
Types Of Triangles
Isosceles Triangle
A
AB AC sides in isoscelesABC
B C ' s in isoscelesABC
B C
Equilateral Triangle
A AB AC BC sides in equilateralABC
B C
45. (ii) Prove DC = CB
DA AB proven S
DAS BAS given A
AC is common S
DAC BAC SAS
DC CB matching sides in 's
Types Of Triangles
Isosceles Triangle
A
AB AC sides in isoscelesABC
B C ' s in isoscelesABC
B C
Equilateral Triangle
A AB AC BC sides in equilateralABC
A B C 60 ' s in equilateralABC
B C
47. Triangle Terminology
Altitude: (perpendicular height)
Perpendicular from one side passing
through the vertex
48. Triangle Terminology
Altitude: (perpendicular height)
Perpendicular from one side passing
through the vertex
49. Triangle Terminology
Altitude: (perpendicular height)
Perpendicular from one side passing
through the vertex
Median: Line joining vertex to the
midpoint of the opposite side
50. Triangle Terminology
Altitude: (perpendicular height)
Perpendicular from one side passing
through the vertex
Median: Line joining vertex to the
midpoint of the opposite side
51. Triangle Terminology
Altitude: (perpendicular height)
Perpendicular from one side passing
through the vertex
Median: Line joining vertex to the
midpoint of the opposite side
Right Bisector: Perpendicular drawn
from the midpoint of a side
52. Triangle Terminology
Altitude: (perpendicular height)
Perpendicular from one side passing
through the vertex
Median: Line joining vertex to the
midpoint of the opposite side
Right Bisector: Perpendicular drawn
from the midpoint of a side