In triangle ABC, ∠A=20∘ and ∠B=80∘. Let
D be a point on line segment AB such that
AD=BC. What is the measure (in degrees) of ∠ADC?
Given ΔABC,
A=20°
B=80° =>
C=80°
=> isosceles triangle with base BC=x.
Let E=mid-point of BC, then
ΔAEC is a right triangle right-angled at E
EC=x/2
By definition of cosine,
AC=(x/2)/cos(80°)
=x/(2cos(80°))
Consider ΔADC,
AD=x (given)
AC=x/(2cos(80°)) (from above)
∠DAC=20° (given),
we find DC by cosine rule
DC=sqrt(AD²+AC²-2*AD*AC*cos(20°) )
=sqrt(x²+x/(2cos(80°))-x²cos(80°))
=1.879x (approx.)
∠ ADC can be found by the sine rule:
sin(ADC)=(x/(2cos(80°))*sin(20°)/DC
=sin(20°)/(1-2cos(80°))
∠ADC=asin(sin(20°)/(1-2cos(80°)))
=31.6° approx.
Please check my arithmetic.
To find the measure of ∠ADC, we can use the fact that the sum of the angles in a triangle is 180 degrees.
Let's go step by step:
1. Start by drawing triangle ABC. Label the vertices as A, B, and C.
2. We are given that ∠A = 20∘ and ∠B = 80∘. So, draw these angles with their respective measurements.
3. Let D be a point on line segment AB such that AD = BC. Draw segment AD and segment DC.
4. Now, observe that triangle ACD and triangle BCD are congruent because AD = BC (given) and CD is common to both triangles.
5. Since the sum of the angles in a triangle is 180 degrees, the sum of ∠ACD and ∠DCB (which are corresponding angles in congruent triangles) is equal to ∠ACB.
6. Therefore, ∠ACD + ∠DCB = ∠ACB.
7. We know that ∠ACB is the exterior angle of triangle ADC, formed by extending side CD. By the Exterior Angle Theorem, the measure of the exterior angle is equal to the sum of the interior angles opposite to it.
8. So, ∠ACB = ∠ACD + ∠DCB.
9. Substituting the given values, we have 80∘ = ∠ACD + ∠DCB.
10. Since we want to find the measure of ∠ADC, we need to solve for ∠ACD.
11. Rearranging the equation, we get ∠ACD = 80∘ - ∠DCB.
12. Now, let's focus on ∠DCB. We have already observed that triangle DBC is congruent to triangle DAC (by SSS congruence, as AD = BC and CD is common). So, ∠DCB = ∠DAC.
13. Therefore, ∠ACD = 80∘ - ∠DAC.
14. We also know that ∠DAC is an interior angle of triangle ABC, which is 180 degrees - (∠A + ∠B) = 180∘ - (20∘ + 80∘) = 80∘.
15. Substituting this value in ∠ACD = 80∘ - ∠DAC, we get ∠ACD = 80∘ - 80∘ = 0∘.
Therefore, the measure of ∠ADC is 0 degrees.