Proportionality theorem and its converse
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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.
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Triangles
Introduction
Sum of the angles of a triangle
Types of triangles
Altitude, Median and Angle Bisector
Congruence of triangles
Sides opposite congruent angles
Quadrilaterals
The document contains information about various properties of quadrilaterals, triangles, and parallelograms. It includes proofs of:
1) The sum of the interior angles of any quadrilateral is 360 degrees.
2) A diagonal of a parallelogram divides it into two congruent triangles.
3) If two triangles are on the same base and between the same parallels, they have equal areas.
Triangles.pptx
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.
7. gr. 12 euclidean geometry s
1. The document discusses concepts related to similar triangles including corresponding angles being proportional, the midpoint theorem, and proving triangles are similar.
2. It provides examples of using similarity to solve for missing side lengths or angle measures. Similar triangles are used to prove properties like the Pythagorean theorem.
3. Exercises are given to apply concepts like using proportional sides or corresponding angles to determine if triangles are similar or find missing values, and proving statements about triangles.
9 6 proportional parts and parallel lines
The document discusses proportional parts and parallel lines. It defines that if three or more parallel lines intersect two transversals, the lines divide the transversals proportionally. This means that the ratios of the parts of the first transversal will equal the ratios of the parts of the second transversal. It also states that if three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Examples are provided to demonstrate finding missing values using proportional reasoning with parallel lines.
Cogruence
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.
Congruent Triangles
The document discusses different theorems for proving triangles are congruent:
- Side-Side-Side (SSS) - If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
- Side-Angle-Side (SAS) - If two sides and the included angle of one triangle are congruent to those of another, the triangles are congruent.
- Angle-Side-Angle (ASA) - If two angles and the included side of one triangle are congruent to those of another, the triangles are congruent.
- Angle-Angle-Side (AAS) - If two angles and the non-included side of one triangle are congr
Congruence between triangles
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.
Geometry proofs
This document discusses various metric relationships in mathematics including proofs, congruency, similarity, angles, triangles, and the Pythagorean theorem. It defines key concepts such as complementary angles, supplementary angles, opposite angles, interior angles, corresponding angles, and more. Examples of exam questions are provided testing these concepts along with activities for further practice.
Triangles For Class 10 CBSE NCERT
The document describes how to determine if three lines will form a triangle based on their lengths. It states that if the sum of the lengths of the two shortest lines is less than the longest line, the lines will not form a triangle. If the sum is equal to the longest line, the two shortest lines will overlap the longest line. Only if the sum is greater than the longest line will the three lines form a triangle.
Chapter 1.3
Two figures are congruent if they have the same shape and size. To be congruent, triangles must have three corresponding sides of equal length or two corresponding sides of equal length and the angle between them equal, or one side of equal length and two corresponding angles equal. If two triangles are congruent, their corresponding sides and angles are equal. Similar triangles have equal corresponding angles and proportional corresponding sides, but may differ in size. Congruent triangles are always similar, but similar triangles are not always congruent.
a very nice ppt for learning theorems
The document contains proofs of several geometric theorems:
1) Intersecting lines L and K have vertically opposite angles 1 and 2 that are equal.
2) The interior angles of any triangle sum to 180 degrees.
3) The exterior angle of a triangle is equal to the sum of the two interior opposite angles.
4) In an isosceles triangle, the base angles are equal.
5) Corresponding angles are equal when two parallel lines are intersected by a transversal.
6) Angles opposite equal sides of a triangle are equal.
Similar to Proportionality theorem and its converse (20)
Congruence of Triangle 1.pptx Maths, Triangles for GRADE 8
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Proportionality theorem and its converse
- 1. PROPORTIONALITY THEOREM AND ITS CONVERSE
- 2. A B C D E Theorem : If a line is drawn parallel to one side of a triangle, then it divides other two sides proportionately
- 4. A B C D E To prove: EC AE BD AD
- 5. A B C D E N M Construction:
- 6. A B C D E N NEADADEArea 2 1 NEBDBDEArea 2 1 1...... BD AD BDE ADE
- 7. A B C D E M DMAEADEArea 2 1 DMECCDEArea 2 1 2...... EC AE CDE ADE
- 8. A B C D E 2...... EC AE CDE ADE 1...... BD AD BDE ADE CDEBDE EC AE BD AD
- 9. Converse: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
- 11. A B C D E To prove:
- 13. A B C D E F 1......given EC AE BD AD 2...... EF AE BD AD EF AE EC AE From ,2&1 EFEC
- 14. A B C D E AE AC AD AB TPCorollary ..:2 AE CE AD BD 11 AE CE AD BD AE AECE AD ADBD AE AC AD AB
- 15. A B C D E EC AC DB AB Corollary :1 EC AE BD AD EC AC BD AB 11 EC AE BD AD EC ECAE BD BDAD
- 16. A B C D E Corollary 3 BC DE AC AE AB AD TP ..
- 17. A B C AD/AB=AE/AC …….1 D E BF/BC=AE/AC DE/BC=AE/AC..2 F