CBSE Class 12-science Answered
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![question image](http://images.topperlearning.com/topper/new-ate/top_mob1695475304256889890c25d696e-b461-437c-a3d5-a785890c74c3.jpg)
Asked by sdmbotch1123 | 23 Sep, 2023, 18:51: PM
![](https://images.topperlearning.com/topper/tinymce/imagemanager/files/e090149238eae0973beee19a5e85f509651430473a6715.89686398f1.png)
Figure shows two angles
ABC and
DEF so that their arms ar parallel. AB || DE and BC || EF .
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
Let BG and EH are bisectors of respective angles. When arms are parallel, then
ABC =
DEF .
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
Also their bisected angles also equal.
Let us extend bisector HE to meet BC at I as shown in figure.
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
Since
HEF =
GBC , we also get
GBC =
EIC .
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
Hence GB || EI or bisectors are parallel .
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![](https://images.topperlearning.com/topper/tinymce/imagemanager/files/28c2bb6f7c8770b9a3c867a0869cef9f6514384a9afee0.18637406f2.png)
Other possibility of two angles with their arms are parallel is shown in above figure.
AB || DE and BC || EF
Bisector of
ABC is drawn and is extended so that it meets DE at H and futher extension meets the bisector of
DEFat J .
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
If
HBG = θ , then
BGH = (180-2θ) because sum of interior angles between parallel line is 180.
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
Hence we get ,
BHG =
EHJ = θ (
sum of angles of triangle is 180 )
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style because end style](https://images.topperlearning.com/topper/tinymce/cache/e9d7aeadd4b70e0d3942ad3f92a71cc3.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
Hence bisected angles
DEI =
FEI = 90-θ
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
Hence
EJH = 180 - (90-θ) - θ = 90o
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
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Answered by Thiyagarajan K | 27 Sep, 2023, 19:59: PM
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