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Unit 1 Geometry Help

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kiritosan:
In the diagram below, ABC is isosceles and AB is parallel to DE and GD/DH=BE/EC. Show that AF and HC are parallel.
Can I get some help with this question? Reasonings too :)

https://prnt.sc/rtqzxe

bingoman2000:
Sure, the easiest way to proceed is to first observe that triangle CDE is similar to triangle CAB (AA). Then, CE/CB = CD/CA. But, CA = CB since the triangle is isosceles, so CE = CD and BE = AD (since BE = CB-CE and AD = CA-CD). This then yields that BE/CE = AD/DC.

But we are told that GD/DH = BE/EC, so we now find that GD/DH = AD/DC. Since angle ADG = angle HDC (vertically opposite angles), this means that triangles CDH and ADG are similar. Thus, angle DAG = angle DCH, implying that AF and HC are indeed parallel.

Hope this helps :)

kiritosan:

--- Quote from: bingoman2000 on April 06, 2020, 05:16:28 pm ---Sure, the easiest way to proceed is to first observe that triangle CDE is similar to triangle CAB (AA). Then, CE/CB = CD/CA. But, CA = CB since the triangle is isosceles, so CE = CD and BE = AD (since BE = CB-CE and AD = CA-CD). This then yields that BE/CE = AD/DC.

But we are told that GD/DH = BE/EC, so we now find that GD/DH = AD/DC. Since angle ADG = angle HDC (vertically opposite angles), this means that triangles CDH and ADG are similar. Thus, angle DAG = angle DCH, implying that AF and HC are indeed parallel.

Hope this helps :)

--- End quote ---

Thanks makes a lot more sense now! Could you also help me with this one?
https://prnt.sc/rtxdk7

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