如图,在△ABC中,∠ABC的平分线与∠ACE的平分线交于D点,若∠A=80°,求∠D的度数.

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如图,在△ABC中,∠ABC的平分线与∠ACE的平分线交于D点,若∠A=80°,求∠D的度数.

如图,在△ABC中,∠ABC的平分线与∠ACE的平分线交于D点,若∠A=80°,求∠D的度数.
如图,在△ABC中,∠ABC的平分线与∠ACE的平分线交于D点,若∠A=80°,求∠D的度数.

如图,在△ABC中,∠ABC的平分线与∠ACE的平分线交于D点,若∠A=80°,求∠D的度数.
∵∠ACE=∠A+∠ABC,CD平分∠ACE
∴∠DCE=∠ACE/2=(∠A+∠ABC)/2
∵BD平分∠ABC
∴∠DBC=∠ABC/2
∴∠DCE=∠D+∠DBC=∠D+∠ABC/2
∴∠D+∠ABC/2=(∠A+∠ABC)/2
∴∠D=∠A/2=80/2=40°

∵BD平分∠ABC,CD平分∠ACE
∴∠DBC=1/2∠ABC
∠ACD=1/2∠ACE=1/2(180°-∠ACB)
∴∠D=180°-∠DBC-∠ACB-∠ACD
=180°-1/2∠ABC-∠ACB-1/2(180°-∠ACB)
=180°-1/2∠ABC-∠ACB-90°+1/2∠ACB
=90°...

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∵BD平分∠ABC,CD平分∠ACE
∴∠DBC=1/2∠ABC
∠ACD=1/2∠ACE=1/2(180°-∠ACB)
∴∠D=180°-∠DBC-∠ACB-∠ACD
=180°-1/2∠ABC-∠ACB-1/2(180°-∠ACB)
=180°-1/2∠ABC-∠ACB-90°+1/2∠ACB
=90°-1/2∠ABC-1/2∠ACB
=90°-1/2(∠ABC+∠ACB)
=90°-1/2(180°-∠A)
=1/2∠A
=40°

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