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Ch 6 Ex 6.2

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Rashmi Taneja
Rashmi Taneja

This document discusses triangles and similarity. It provides examples of using the AAA, SAS, and SSS similarity criteria to determine if two triangles are similar. It also contains exercises involving applying similarity rules to pairs of triangles and finding missing angle measures. One exercise involves showing two triangles are similar using the fact that corresponding angles are equal for diagonals intersecting in a trapezium where the bases are parallel. The summary is provided in 3 sentences or less as requested.

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CH 6 TRIANGLES EX 6.2 CONTINUE + CONCEPT OF CONGRUENCY AND SIMILARITY Ex 6.1 and Ex 6.3
5. In the figure, DE || OQ and DF || OR. Show that EP || QR In? PQR ¡à E and F are two distinct points on PQ and PR respectively and ?? ?? = ?? ?? ;E and F are dividing the two sides PQ and PR in the same ratio in ¦¤PQR. ¡à EF || QR Sol. In ¦¤PQO DE || OQ [Given] ¡à Using the Basic Proportionality Theorem, we have: FROM (1) AND (2)
6. In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR. Sol. In ¦¤PQR, O is a point OP, OQ and OR and joined. We have A, B and C on OP, OQ. and OR respectively such that AB || PQ and AC || OR. Now, in OPQ, AB || PQ [Given]
Ch 6 Ex 6.2
Ch 6 Ex 6.2
Ch 6 Ex 6.2
Ch 6 Ex 6.2
Ch 6 Ex 6.2
Ch 6 Ex 6.2
For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other.
But we don't need to know all three sides and all three angles ...two or three out of the six is usually enough. Two triangles are similar if they have: ?all their angles equal ?corresponding sides are in the same ratio There are three ways to find if two triangles are similar: AA, SAS and SSS AA AA stands for "angle, angle" and means that the triangles have two of their angles equal. If two triangles have two of their angles equal, the triangles are similar.
SAS SAS stands for "side, angle, side" and means that we have two triangles where: ?the ratio between two sides is the same as the ratio between another two sides ?and we also know the included angles are equal. If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar. In this example we can see that: ?one pair of sides is in the ratio of 21 : 14 = 3 : 2 ?another pair of sides is in the ratio of 15 : 10 = 3 : 2 ?there is a matching angle of 75¡ã in between them So there is enough information to tell us that the two triangles are similar.
SSS SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio. If two triangles have three pairs of sides in the same ratio, then the triangles are similar. In this example, the ratios of sides are: ?a : x = 6 : 7.5 = 12 : 15 = 4 : 5 ?b : y = 8 : 10 = 4 : 5 ?c : z = 4 : 5 These ratios are all equal, so the two triangles are similar.
EXERCISE 6.1 1. Fill in the blanks using the correct b given in brackets: (i) All circles are ¡­¡­¡­ (congruent, similar) (ii) All squares are ¡­¡­¡­ (similar, congruent) (iii) All ¡­¡­¡­ triangles are similar. (isosceles, equilateral) (iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are ¡­¡­¡­ and (b) their corresponding sides are ¡­¡­¡­ (equal, proportional). Sol. (i) All circles are similar. (ii) All squares are similar. (iii) All equilateral triangles are similar. (iv) Two polygons of the same number of sides are similar if: (a) Their corresponding angles are equal and (b) Their corresponding sides are proportional. 2. Give two different examples of pair of. (i) similar figures (ii) non-similar figures. Sol. (i) (a) Any two circles are similar figures. (b) Any two squares are similar figures. (ii) (a) A circle and a triangle are non-similar figures. (b) An isosceles triangle and a scalene triangle are non-similar figures.
3. State whether the following quadrilaterals are similar or not: Sol. On observing the given figures, we find that: Their corresponding sides are proportional but their corresponding angles are Dot equal. ¡à The given figures are not similar. ? Similar Triangles Triangles are a special type of polygons. The study of their similarity is important. Two triangles are said to be similar if: (i) Their corresponding sides are proportional, and, (ii) Their corresponding angles are equal.
EXERCISE 6.3 1. State which pairs of triangles in the figures, are similar. Write the similarity criteria used by you for answering the question and also write the pairs of similar triangles in the symbolic form: Sol. (i) In ¦¤ABC and ¦¤PQR We have: ¡ÏA = ¡ÏP = 60¡ã ¡ÏB = ¡ÏQ = 80¡ã ¡ÏC = ¡ÏR = 40¡ã ¡à The corresponding angles are equal, ¡à Using the AAA similarity rule, ¦¤ABC ~ ¦¤PQR
Ch 6 Ex 6.2
2. In the figure, ¦¤ODC ~ ¦¤OBA, ¡ÏBOC = 125¡ã and ¡ÏCDO = 70¡ã. Find ¡ÏDOC, ¡ÏDCO and ¡ÏOAB.
Sol. We have: ¡ÏBOC = 125¡ã and ¡ÏCDO = 70¡ã Thus, from (1), (2) and (3) ¡ÏDOC = 55¡ã, ¡ÏDCO = 55¡ã and ¡ÏOAB = 55¡ã. since, ¡ÏDOC + ¡ÏBOC = 180¡ã [Linear Pair] ? ¡ÏDOC 180¡ã ¨C 125¡ã = 55¡ã ...(1) In ¦¤DOC Using the angle sum property, we get ¡ÏDOC + ¡ÏODC + ¡ÏDCO = 180¡ã ? 55¡ã + 70¡ã + ¡ÏDCO = 180¡ã ?¡ÏDCO =180¡ã ¨C 55¡ã ¨C 70¡ã = 55¡ã ...(2) Again, ¦¤ODC ~ ¦¤OBA [Given] ¡à Their corresponding angles are equal And ¡ÏOCD = ¡ÏOAB = 55¡ã ...(3)
3. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that Sol. We have a trapezium ABCD in which AB || DC. The diagonals AC and BD intersect at O. In ¦¤OAB and ¦¤OCD AB || DC [Given] and BD intersects them ¡à¡ÏOBA = ¡ÏODC ...(1) [Alternate angles] similarly, ¡ÏOAB = ¡ÏOCD ...(2) ¡àUsing AA similarity rule, ¦¤OAB ~ ¦¤OCD

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Ch 6 Ex 6.2

  • 1. CH 6 TRIANGLES EX 6.2 CONTINUE + CONCEPT OF CONGRUENCY AND SIMILARITY Ex 6.1 and Ex 6.3
  • 2. 5. In the figure, DE || OQ and DF || OR. Show that EP || QR In? PQR ¡à E and F are two distinct points on PQ and PR respectively and ?? ?? = ?? ?? ;E and F are dividing the two sides PQ and PR in the same ratio in ¦¤PQR. ¡à EF || QR Sol. In ¦¤PQO DE || OQ [Given] ¡à Using the Basic Proportionality Theorem, we have: FROM (1) AND (2)
  • 3. 6. In the figure, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. Show that BC || QR. Sol. In ¦¤PQR, O is a point OP, OQ and OR and joined. We have A, B and C on OP, OQ. and OR respectively such that AB || PQ and AC || OR. Now, in OPQ, AB || PQ [Given]
  • 10. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other.
  • 11. But we don't need to know all three sides and all three angles ...two or three out of the six is usually enough. Two triangles are similar if they have: ?all their angles equal ?corresponding sides are in the same ratio There are three ways to find if two triangles are similar: AA, SAS and SSS AA AA stands for "angle, angle" and means that the triangles have two of their angles equal. If two triangles have two of their angles equal, the triangles are similar.
  • 12. SAS SAS stands for "side, angle, side" and means that we have two triangles where: ?the ratio between two sides is the same as the ratio between another two sides ?and we also know the included angles are equal. If two triangles have two pairs of sides in the same ratio and the included angles are also equal, then the triangles are similar. In this example we can see that: ?one pair of sides is in the ratio of 21 : 14 = 3 : 2 ?another pair of sides is in the ratio of 15 : 10 = 3 : 2 ?there is a matching angle of 75¡ã in between them So there is enough information to tell us that the two triangles are similar.
  • 13. SSS SSS stands for "side, side, side" and means that we have two triangles with all three pairs of corresponding sides in the same ratio. If two triangles have three pairs of sides in the same ratio, then the triangles are similar. In this example, the ratios of sides are: ?a : x = 6 : 7.5 = 12 : 15 = 4 : 5 ?b : y = 8 : 10 = 4 : 5 ?c : z = 4 : 5 These ratios are all equal, so the two triangles are similar.
  • 14. EXERCISE 6.1 1. Fill in the blanks using the correct b given in brackets: (i) All circles are ¡­¡­¡­ (congruent, similar) (ii) All squares are ¡­¡­¡­ (similar, congruent) (iii) All ¡­¡­¡­ triangles are similar. (isosceles, equilateral) (iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are ¡­¡­¡­ and (b) their corresponding sides are ¡­¡­¡­ (equal, proportional). Sol. (i) All circles are similar. (ii) All squares are similar. (iii) All equilateral triangles are similar. (iv) Two polygons of the same number of sides are similar if: (a) Their corresponding angles are equal and (b) Their corresponding sides are proportional. 2. Give two different examples of pair of. (i) similar figures (ii) non-similar figures. Sol. (i) (a) Any two circles are similar figures. (b) Any two squares are similar figures. (ii) (a) A circle and a triangle are non-similar figures. (b) An isosceles triangle and a scalene triangle are non-similar figures.
  • 15. 3. State whether the following quadrilaterals are similar or not: Sol. On observing the given figures, we find that: Their corresponding sides are proportional but their corresponding angles are Dot equal. ¡à The given figures are not similar. ? Similar Triangles Triangles are a special type of polygons. The study of their similarity is important. Two triangles are said to be similar if: (i) Their corresponding sides are proportional, and, (ii) Their corresponding angles are equal.
  • 16. EXERCISE 6.3 1. State which pairs of triangles in the figures, are similar. Write the similarity criteria used by you for answering the question and also write the pairs of similar triangles in the symbolic form: Sol. (i) In ¦¤ABC and ¦¤PQR We have: ¡ÏA = ¡ÏP = 60¡ã ¡ÏB = ¡ÏQ = 80¡ã ¡ÏC = ¡ÏR = 40¡ã ¡à The corresponding angles are equal, ¡à Using the AAA similarity rule, ¦¤ABC ~ ¦¤PQR
  • 18. 2. In the figure, ¦¤ODC ~ ¦¤OBA, ¡ÏBOC = 125¡ã and ¡ÏCDO = 70¡ã. Find ¡ÏDOC, ¡ÏDCO and ¡ÏOAB.
  • 19. Sol. We have: ¡ÏBOC = 125¡ã and ¡ÏCDO = 70¡ã Thus, from (1), (2) and (3) ¡ÏDOC = 55¡ã, ¡ÏDCO = 55¡ã and ¡ÏOAB = 55¡ã. since, ¡ÏDOC + ¡ÏBOC = 180¡ã [Linear Pair] ? ¡ÏDOC 180¡ã ¨C 125¡ã = 55¡ã ...(1) In ¦¤DOC Using the angle sum property, we get ¡ÏDOC + ¡ÏODC + ¡ÏDCO = 180¡ã ? 55¡ã + 70¡ã + ¡ÏDCO = 180¡ã ?¡ÏDCO =180¡ã ¨C 55¡ã ¨C 70¡ã = 55¡ã ...(2) Again, ¦¤ODC ~ ¦¤OBA [Given] ¡à Their corresponding angles are equal And ¡ÏOCD = ¡ÏOAB = 55¡ã ...(3)
  • 20. 3. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at the point O. Using a similarity criterion for two triangles, show that Sol. We have a trapezium ABCD in which AB || DC. The diagonals AC and BD intersect at O. In ¦¤OAB and ¦¤OCD AB || DC [Given] and BD intersects them ¡à¡ÏOBA = ¡ÏODC ...(1) [Alternate angles] similarly, ¡ÏOAB = ¡ÏOCD ...(2) ¡àUsing AA similarity rule, ¦¤OAB ~ ¦¤OCD