Mixing
Why homologous sides are sides opposite to homologous angles?
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
In Plane and solid geometry by Fletcher Durell, he mentions in page 34 that
91. Homologous angles of two mutually equiangular triangles are corresponding angles in those triangles.
Homologous sides of two mutually equiangular triangles are sides
opposite homologous angles in those triangles.
I hope to see that, length of an opposite side of the given angle doesn't have to be influenced by it's opposite angle. So why does the author say that "homologous sides are sides opposite to homologous angles"?
I'm asking this question because in some proofs like in Proposition XVI : it says triangle $F'BH$ = triangle $BHC$, so $F'H$ = $CH$ (homologous sides of equal triangles). What is the author trying to say by this statement?
triangle
asked Jul 16 at 22:03
justin
217113
add a comment |Â
up vote
0
down vote
favorite
In Plane and solid geometry by Fletcher Durell, he mentions in page 34 that
91. Homologous angles of two mutually equiangular triangles are corresponding angles in those triangles.
Homologous sides of two mutually equiangular triangles are sides
opposite homologous angles in those triangles.
I hope to see that, length of an opposite side of the given angle doesn't have to be influenced by it's opposite angle. So why does the author say that "homologous sides are sides opposite to homologous angles"?
I'm asking this question because in some proofs like in Proposition XVI : it says triangle $F'BH$ = triangle $BHC$, so $F'H$ = $CH$ (homologous sides of equal triangles). What is the author trying to say by this statement?
triangle
asked Jul 16 at 22:03
justin
217113
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
In Plane and solid geometry by Fletcher Durell, he mentions in page 34 that
91. Homologous angles of two mutually equiangular triangles are corresponding angles in those triangles.
Homologous sides of two mutually equiangular triangles are sides
opposite homologous angles in those triangles.
I hope to see that, length of an opposite side of the given angle doesn't have to be influenced by it's opposite angle. So why does the author say that "homologous sides are sides opposite to homologous angles"?
I'm asking this question because in some proofs like in Proposition XVI : it says triangle $F'BH$ = triangle $BHC$, so $F'H$ = $CH$ (homologous sides of equal triangles). What is the author trying to say by this statement?
triangle
asked Jul 16 at 22:03
justin
217113
In Plane and solid geometry by Fletcher Durell, he mentions in page 34 that
91. Homologous angles of two mutually equiangular triangles are corresponding angles in those triangles.
Homologous sides of two mutually equiangular triangles are sides
opposite homologous angles in those triangles.
I hope to see that, length of an opposite side of the given angle doesn't have to be influenced by it's opposite angle. So why does the author say that "homologous sides are sides opposite to homologous angles"?
I'm asking this question because in some proofs like in Proposition XVI : it says triangle $F'BH$ = triangle $BHC$, so $F'H$ = $CH$ (homologous sides of equal triangles). What is the author trying to say by this statement?
triangle
asked Jul 16 at 22:03
justin
217113
asked Jul 16 at 22:03
justin
217113
asked Jul 16 at 22:03
justin
217113
asked Jul 16 at 22:03
justin
217113
217113
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
It is a definition, a name to refer to these sides. It does not imply that their length are pairwise equal in general (although the ratio of the lengths is the same for all three pairs).
In the example, it is important to note that it is not "homologous sides of mutually equiangular triangles" but "homologous sides of equal triangles". Equal triangles, in this context, seems to mean isometric.
answered Jul 16 at 22:26
Arnaud Mortier
19.2k22159
Can the term 'corresponding' be used in lieu of 'homologous' or whether 'homologous' has some special meaning other than 'corresponding'?
– justin
Jul 17 at 21:19
@justin Of course you could use that instead, it would also make sense. But the authors of the book want to make everything clear and state which term they are going to use, once and for all.
– Arnaud Mortier
Jul 17 at 21:26
Is there any specific reason, in choosing the side opposite of the homologous angle to be the homologous side of an equianglular triangle?
– justin
Jul 17 at 21:32
@justin If you look at two such triangles, it' clear: homologous sides are those that you would naturally put into pairs if you had to form a correspondence between the sides of the two triangles. The ratio of the lengths is the same for all pairs.
– Arnaud Mortier
Jul 17 at 21:38
Does this mean that for equiangluar triangles, the ratio of the corresponding sides would always be in a proportion?
– justin
Jul 17 at 21:41
 |Â
show 6 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
It is a definition, a name to refer to these sides. It does not imply that their length are pairwise equal in general (although the ratio of the lengths is the same for all three pairs).
In the example, it is important to note that it is not "homologous sides of mutually equiangular triangles" but "homologous sides of equal triangles". Equal triangles, in this context, seems to mean isometric.
answered Jul 16 at 22:26
Arnaud Mortier
19.2k22159
Can the term 'corresponding' be used in lieu of 'homologous' or whether 'homologous' has some special meaning other than 'corresponding'?
– justin
Jul 17 at 21:19
@justin Of course you could use that instead, it would also make sense. But the authors of the book want to make everything clear and state which term they are going to use, once and for all.
– Arnaud Mortier
Jul 17 at 21:26
Is there any specific reason, in choosing the side opposite of the homologous angle to be the homologous side of an equianglular triangle?
– justin
Jul 17 at 21:32
@justin If you look at two such triangles, it' clear: homologous sides are those that you would naturally put into pairs if you had to form a correspondence between the sides of the two triangles. The ratio of the lengths is the same for all pairs.
– Arnaud Mortier
Jul 17 at 21:38
Does this mean that for equiangluar triangles, the ratio of the corresponding sides would always be in a proportion?
– justin
Jul 17 at 21:41
 |Â
show 6 more comments
up vote
0
down vote
It is a definition, a name to refer to these sides. It does not imply that their length are pairwise equal in general (although the ratio of the lengths is the same for all three pairs).
In the example, it is important to note that it is not "homologous sides of mutually equiangular triangles" but "homologous sides of equal triangles". Equal triangles, in this context, seems to mean isometric.
answered Jul 16 at 22:26
Arnaud Mortier
19.2k22159
Can the term 'corresponding' be used in lieu of 'homologous' or whether 'homologous' has some special meaning other than 'corresponding'?
– justin
Jul 17 at 21:19
@justin Of course you could use that instead, it would also make sense. But the authors of the book want to make everything clear and state which term they are going to use, once and for all.
– Arnaud Mortier
Jul 17 at 21:26
Is there any specific reason, in choosing the side opposite of the homologous angle to be the homologous side of an equianglular triangle?
– justin
Jul 17 at 21:32
@justin If you look at two such triangles, it' clear: homologous sides are those that you would naturally put into pairs if you had to form a correspondence between the sides of the two triangles. The ratio of the lengths is the same for all pairs.
– Arnaud Mortier
Jul 17 at 21:38
Does this mean that for equiangluar triangles, the ratio of the corresponding sides would always be in a proportion?
– justin
Jul 17 at 21:41
 |Â
show 6 more comments
up vote
0
down vote
up vote
0
down vote
It is a definition, a name to refer to these sides. It does not imply that their length are pairwise equal in general (although the ratio of the lengths is the same for all three pairs).
In the example, it is important to note that it is not "homologous sides of mutually equiangular triangles" but "homologous sides of equal triangles". Equal triangles, in this context, seems to mean isometric.
answered Jul 16 at 22:26
Arnaud Mortier
19.2k22159
It is a definition, a name to refer to these sides. It does not imply that their length are pairwise equal in general (although the ratio of the lengths is the same for all three pairs).
In the example, it is important to note that it is not "homologous sides of mutually equiangular triangles" but "homologous sides of equal triangles". Equal triangles, in this context, seems to mean isometric.
answered Jul 16 at 22:26
Arnaud Mortier
19.2k22159
answered Jul 16 at 22:26
Arnaud Mortier
19.2k22159
answered Jul 16 at 22:26
Arnaud Mortier
19.2k22159
answered Jul 16 at 22:26
Arnaud Mortier
19.2k22159
19.2k22159
Can the term 'corresponding' be used in lieu of 'homologous' or whether 'homologous' has some special meaning other than 'corresponding'?
– justin
Jul 17 at 21:19
@justin Of course you could use that instead, it would also make sense. But the authors of the book want to make everything clear and state which term they are going to use, once and for all.
– Arnaud Mortier
Jul 17 at 21:26
Is there any specific reason, in choosing the side opposite of the homologous angle to be the homologous side of an equianglular triangle?
– justin
Jul 17 at 21:32
@justin If you look at two such triangles, it' clear: homologous sides are those that you would naturally put into pairs if you had to form a correspondence between the sides of the two triangles. The ratio of the lengths is the same for all pairs.
– Arnaud Mortier
Jul 17 at 21:38
Does this mean that for equiangluar triangles, the ratio of the corresponding sides would always be in a proportion?
– justin
Jul 17 at 21:41
 |Â
show 6 more comments
Can the term 'corresponding' be used in lieu of 'homologous' or whether 'homologous' has some special meaning other than 'corresponding'?
– justin
Jul 17 at 21:19
@justin Of course you could use that instead, it would also make sense. But the authors of the book want to make everything clear and state which term they are going to use, once and for all.
– Arnaud Mortier
Jul 17 at 21:26
Is there any specific reason, in choosing the side opposite of the homologous angle to be the homologous side of an equianglular triangle?
– justin
Jul 17 at 21:32
@justin If you look at two such triangles, it' clear: homologous sides are those that you would naturally put into pairs if you had to form a correspondence between the sides of the two triangles. The ratio of the lengths is the same for all pairs.
– Arnaud Mortier
Jul 17 at 21:38
Does this mean that for equiangluar triangles, the ratio of the corresponding sides would always be in a proportion?
– justin
Jul 17 at 21:41
Can the term 'corresponding' be used in lieu of 'homologous' or whether 'homologous' has some special meaning other than 'corresponding'?
– justin
Jul 17 at 21:19
Can the term 'corresponding' be used in lieu of 'homologous' or whether 'homologous' has some special meaning other than 'corresponding'?
– justin
Jul 17 at 21:19
@justin Of course you could use that instead, it would also make sense. But the authors of the book want to make everything clear and state which term they are going to use, once and for all.
– Arnaud Mortier
Jul 17 at 21:26
@justin Of course you could use that instead, it would also make sense. But the authors of the book want to make everything clear and state which term they are going to use, once and for all.
– Arnaud Mortier
Jul 17 at 21:26
Is there any specific reason, in choosing the side opposite of the homologous angle to be the homologous side of an equianglular triangle?
– justin
Jul 17 at 21:32
Is there any specific reason, in choosing the side opposite of the homologous angle to be the homologous side of an equianglular triangle?
– justin
Jul 17 at 21:32
@justin If you look at two such triangles, it' clear: homologous sides are those that you would naturally put into pairs if you had to form a correspondence between the sides of the two triangles. The ratio of the lengths is the same for all pairs.
– Arnaud Mortier
Jul 17 at 21:38
@justin If you look at two such triangles, it' clear: homologous sides are those that you would naturally put into pairs if you had to form a correspondence between the sides of the two triangles. The ratio of the lengths is the same for all pairs.
– Arnaud Mortier
Jul 17 at 21:38
Does this mean that for equiangluar triangles, the ratio of the corresponding sides would always be in a proportion?
– justin
Jul 17 at 21:41
Does this mean that for equiangluar triangles, the ratio of the corresponding sides would always be in a proportion?
– justin
Jul 17 at 21:41
 |Â
show 6 more comments
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2853905%2fwhy-homologous-sides-are-sides-opposite-to-homologous-angles%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password